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First-Order Differential Equations
CHAPTER 2
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Contents
2.1 Solution Curves Without a Solution2.2 Separable Variables2.3 Linear Equations2.4 Exact Equations2.5 Solutions by Substitutions2.6 A Numerical Methods2.7 Linear Models2.8 Nonlinear Model2.9 Modeling with Systems of First-Order DEs
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2.1 Solution Curve Without a Solution
Introduction: Begin our study of first-order DE with analyzing a DE qualitatively.
SlopeA derivative dy/dx of y = y(x) gives slopes of tangent lines at points.
Lineal ElementAssume dy/dx = f(x, y(x)). The value f(x, y) represents the slope of a line, or a line element is called a lineal element. See Fig2.1
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Fig2.1
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Direction Field
If we evaluate f over a rectangular grid of points, and draw a lineal element at each point (x, y) of the grid with slope f(x, y), then the collection is called a direction field or a slope field of the following DE dy/dx = f(x, y)
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Example 1
The direction field of dy/dx = 0.2xy is shown in Fig2.2(a) and for comparison with Fig2.2(a), some representative graphs of this family are shown in Fig2.2(b).
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Fig2.2
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Example 2
Use a direction field to draw an approximate solution curve for dy/dx = sin y, y(0) = −3/2.
Solution:Recall from the continuity of f(x, y) and f/y = cos y. Theorem 2.1 guarantees the existence of a unique solution curve passing any specified points in the plane. Now split the region containing (-3/2, 0) into grids. We calculate the lineal element of each grid to obtain Fig2.3.
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Fig2.3
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Increasing/DecreasingIf dy/dx > 0 for all x in I, then y(x) is increasing in I.If dy/dx < 0 for all x in I, then y(x) is decreasing in I.
DEs Free of the Independent variable dy/dx = f(y) (1)
is called autonomous. We shall assume f and f are continuous on some I.
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Critical Points
The zeros of f in (1) are important. If f(c) = 0, then c is a critical point, equilibrium point or stationary point.Substitute y(x) = c into (1), then we have 0 = f(c) = 0.
If c is a critical point, then y(x) = c, is a solution of (1).
A constant solution y(x) = c of (1) is called an equilibrium solution.
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Example 3
The following DEdP/dt = P(a – bP)
where a and b are positive constants, is autonomous.From f(P) = P(a – bP) = 0, the equilibrium solutions are P(t) = 0 and P(t) = a/b.
Put the critical points on a vertical line. The arrows in Fig 2.4 indicate the algebraic sign of f(P) = P(a – bP). If the sign is positive or negative, then P is increasing or decreasing on that interval.
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Fig2.4
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Solution Curves
If we guarantee the existence and uniqueness of (1), through any point (x0, y0) in R, there is only one solution curve. See Fig 2.5(a).
Suppose (1) possesses exactly two critical points, c1, and c2, where c1 < c2. The graph of the equilibrium solution y(x) = c1, y(x) = c2 are horizontal lines and split R into three regions, say R1, R2 and R3 as in Fig 2.5(b).
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Fig 2.5
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Some discussions without proof:
(1) If (x0, y0) in Ri, i = 1, 2, 3, when y(x) passes through (x0, y0), will remain in the same subregion. See Fig 2.5(b).
(2) By continuity of f , f(y) can not change signs in a subregion.
(3) Since dy/dx = f(y(x)) is either positive or negative in Ri, a solution y(x) is monotonic.
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(4)If y(x) is bounded above by c1, (y(x) < c1), the graph of y(x) will approach y(x) = c1;If c1 < y(x) < c2, it will approach y(x) = c1 and y(x) = c2;If c2 < y(x) , it will approach y(x) = c2;
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Example 4
Referring to example 3, P = 0 and P = a/b are two critical points, so we have three intervals for P:
R1 : (-, 0), R2 : (0, a/b), R3 : (a/b, )
Let P(0) = P0 and when a solution pass through P0, we
have three kind of graph according to the interval where P0 lies on. See Fig 2.6.
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Fig 2.6
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Example 5
The DE: dy/dx = (y – 1)2 possesses the single critical point 1. From Fig 2.7(a), we conclude a solution y(x) is increasing in - < y < 1 and 1 < y < , where - < x < . See Fig 2.7.
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Fig2.7
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Attractors and Repellers
See Fig 2.8(a). When y0 lies on either side of c, it will approach c. This kind of critical point is said to be asymptotically stable, also called an attractor.
See Fig 2.8(b). When y0 lies on either side of c, it will move away from c. This kind of critical point is said to be unstable, also called a repeller.
See Fig 2.8(c) and (d). When y0 lies on one side of c, it will be attracted to c and repelled from the other side. This kind of critical point is said to be semi-stable.
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Fig 2.8
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Autonomous DEs and Direction Field
Fig 2.9 shows the direction field of dy/dx = 2y – 2.It can be seen that lineal elements passing through points on any horizontal line must have the same slope. Since the DE has the form dy/dx = f(y), the slope depends only on y.
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Fig 2.9
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2.2 Separable Variables
Introduction: Consider dy/dx = f(x, y) = g(x). The DEdy/dx = g(x)
(1)can be solved by integration. Integrating both sides to get y = g(x) dx = G(x) + c.eg: dy/dx = 1 + e2x, then
y = (1 + e2x) dx = x + ½ e2x + c
A first-order DE of the formdy/dx = g(x)h(y)
is said to be separable or to have separable variables.
DEFINITION 2.1Separable Equations
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Rewrite the above equation as
(2)
where p(y) = 1/h(y). When h(y) = 1, (2) reduces to (1).
)()( xgdxdy
yp
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If y = (x) is a solution of (2), we must have
and
(3)
But dy = (x) dx, (3) is the same as
(4)
dxxxP )('))((
)()('))(( xgxxP
cxGyHdxxgdyyP )()(or )()(
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Example 1
Solve (1 + x) dy – y dx = 0.
Solution:Since dy/y = dx/(1 + x), we have
Replacing by c, gives y = c(1 + x).
)1(
1
1lnln
1
1
111 1ln1ln
1
xe
exeeey
cxy
xdx
ydy
c
ccxcx
1ce
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Example 2
Solve
Solution:
We also can rewrite the solution asx2 + y2 = c2, where c2 = 2c1
Apply the initial condition, 16 + 9 = 25 = c2
See Fig2.18.
3)4( , yyx
dxdy
1
22
22 and c
xyxdxydy
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Fig2.18
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Losing a Solution
When r is a zero of h(y), then y = r is also a solution of dy/dx = g(x)h(y). However, this solution will not show up after integration. That is a singular solution.
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Example 3
Solve dy/dx = y2 – 4.
Solution:Rewrite this DE as
(5)then
dxdyyy
dxy
dy
22or
44
1
4
1
2
22
,422
ln
,241
2ln41
24
2
1
cxeyy
cxyy
cxyy
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Example 3 (2)
Replacing exp(c2) by c and solving for y, we have
(6)
If we rewrite the DE as dy/dx = (y + 2)(y – 2), from the previous discussion, we have y = 2 is a singular solution.
x
x
ce
cey
4
4
1
12
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Example 4
Solve
Solution:Rewrite this DE as
using sin 2x = 2 sin x cos x, then (ey – ye-y) dy = 2 sin x dx
from integration by parts,ey + ye-y + e-y = -2 cos x + c (7)
From y(0) = 0, we have c = 4 to getey + ye-y + e-y = 4 −2 cos x (8)
0)0( ,2sin)(cos 2 yxedxdy
yex yy
dxxx
dye
yey
y
cos2sin2
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Use of Computers
Let G(x, y) = ey + ye-y + e-y + 2 cos x. Using some computer software, we plot the level curves of G(x, y) = c. The resulting graphs are shown in Fig2.19 and Fig2.20.
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Fig2.19 Fig2.20
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If we solve dy/dx = xy½ , y(0) = 0
(9)The resulting graphs are shown in Fig2.21.
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Fig2.21
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2.3 Linear Equations
Introduction: Linear DEs are friendly to be solved. We can find some smooth methods to deal with.
A first-order DE of the forma1(x)(dy/dx) + a0(x)y = g(x) (1)
is said to be a linear equation in y. When g(x) = 0, (1) is said to be homogeneous; otherwise it is nonhomogeneous.
DEFINITION 2.2Linear Equations
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Standard FormThe standard for of a first-order DE can be written as
dy/dx + P(x)y = f(x)(2)
The PropertyDE (2) has the property that its solution is sum of two solutions, y = yc + yp, where yc is a solution of the homogeneous equation
dy/dx + P(x)y = 0(3)and yp is a particular solution of (2).
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Verification
Now (3) is also separable. Rewrite (3) as
Solving for y gives
)()()(
])[(][
xfyxPdx
dyyxP
dxdy
yyxPyydxd
Pp
cc
pcoc
0)( dxxPy
dy
)(1)(
xcyceydxxP
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Let yp = u(x) y1(x), where y1(x) is defined as above.We want to find u(x) so that yp is also a solution. Substituting yp into (2) gives
)()(
or
)()(
111
111
xfdxdu
yyxPdxdy
u
xfuyxPdxdu
ydxdy
u
Variation of Parameters
CH2_44
Since dy1/dx + P(x)y1 = 0, so that y1(du/dx) = f(x) Rearrange the above equation,
From the definition of y1(x), we have
(4)
dxxyxf
udxxyxf
du)()(
and )()(
11
dxxfeeceyyydxxPdxxPdxxP
pc
)()()()(
CH2_45
Solving Procedures
If (4) is multiplied by(5)
then(6)
is differentiated(7)
we get(8)
Dividing (8) by gives (2).
dxxPe
)(
dxxfecye dxxPdxxP )()()(
)()()(
xfeyedxd dxxPdxxP
)()()()()(
xfeyexPdxdy
edxxPdxxPdxxP
dxxPe )(
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We call y1(x) = is an integrating factor and we should only memorize this to solve problems.
dxxPe )(
Integrating Factor
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Example 1
Solve dy/dx – 3y = 6.
Solution:Since P(x) = – 3, we have the integrating factor is
then
is the same as
So e-3xy = -2e-3x + c, a solution is y = -2 + ce-3x, - < x < .
xdxee 3)3(
xxx eyedxdy
e 333 63
xx eyedxd 33 6][
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The DE of example 1 can be written as
so that y = –2 is a critical point.
)2(3 ydxdy
Notes
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Equation (4) is called the general solution on some interval I. Suppose again P and f are continuous on I. Writing (2) as Suppose again P and f are continuous on I. Writing (2) as y = F(x, y) we identify
F(x, y) = – P(x)y + f(x), F/y = – P(x)which are continuous on I.Then we can conclude that there exists one and only one solution of
(9) 00)( ),()( yxyxfyxPdxdy
General Solutions
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Example 2
Solve
Solution:Dividing both sides by x, we have
(10)So, P(x) = –4/x, f(x) = x5ex, P and f are continuous on (0, ).Since x > 0, we write the integrating factor as
xexydxdy
x 64
xexyxdx
dy 54
4lnln4/4 4
xeee xxxdx
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Example 2
Multiply (10) by x-4,
Using integration by parts, it follows that the general solution on (0, ) is
x-4y = xex – ex + c or y = x5ex – x4ex + cx4
xxy xeyxdxd
xexdxdy
x ][ ,4 454
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Example 3
Find the general solution of
Solution:Rewrite as
(11)
So, P(x) = x/(x2 – 9). Though P(x) is continuous on (-, -3), (-3, 3) and (3, ), we shall solve this DE on the first and third intervals. The integrating factor is
092
y
x
xdxdy
929ln2
1)9/(2
2
1)9/(
222
xeee
xxxdxxxdx
0)9( 2 xydxdy
x
CH2_53
Example 3 (2)
then multiply (11) by this factor to get
and
Thus, either for x > 3 or x < -3, the general solution is
Notes: x = 3 and x = -3 are singular points of the DE and is discontinuous at these points.
cyx 92
092 yxdxd
9/ 2 xcy
9/ 2 xcy
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Example 4
Solve
Solution:We first have P(x) = 1 and f(x) = x, and are continuous on (-, ). The integrating factor is , so
gives exy = xex – ex + c and y = x – 1 + ce-x
Since y(0) = 4, then c = 5. The solution isy = x – 1 + 5e-x, – < x < (1
2)
4)0( , yxydxdy
xxdxee /
xx xeyedxd ][
CH2_55
Notes: From the above example, we find yc = ce-x and yp = x – 1
we call yc is a transient term, since yc 0 as x .Some solutions are shown in Fig2.24.
Fig2.24
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Example 5
Solve , where
Solution:First we see the graph of f(x) in Fig2.25.
Fig2.25
1 ,0
10 ,1)(
x
xxf
0)0( ),( yxfydxdy
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Example 5 (2)
We solve this problem on 0 x 1 and 1 < x < .For 0 x 1,
then y = 1 + c1e-x
Since y(0) = 0, c1 = -1, y = 1 - e-x
For x > 1, dy/dx + y = 0 then y = c2e-x
xx eyedxd
ydxdy ][ ,1
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Example 5
We have
Furthermore, we want y(x) is continuous at x = 1, that is, when x 1+, y(x) = y(1) implies c2 = e – 1.As in Fig2.26, the function
(13)
are continuous on [0, ).
1 ,)1(
10 ,1
xee
xey
x
x
1 ,
10 ,1
2 xec
xey
x
x
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Fig2.26
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We are interested in the error function and complementary error function
and (14)
Since , we find erf(x) + erfc(x) = 1
dtexx t
0
22)(erf
dtex
x
t
22)(erfc
1)/2(0
2
dte t
Functions Defined by Integrals
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Example 6
Solve dy/dx – 2xy = 2, y(0) = 1.
Solution:We find the integrating factor is exp{-x2},
we get (15)
Applying y(0) = 1, we have c = 1. See Fig2.27
22
2][ xx eedxd
222
02 xx tx cedteey
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Fig2.27
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2.4 Exact Equations
Introduction: Though ydx + xdy = 0 is separable, we can solve it in an alternative way to get the implicit solution xy = c.
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Differential of a Function of Two Variables
If z = f(x, y), its differential or total differential is
(1)Now if z = f(x, y) = c,
(2)
eg: if x2 – 5xy + y3 = c, then (2) gives (2x – 5y) dx + (-5x + 3y2) dy = 0
(3)
dyyf
dxxf
dz
0
dyyf
dxxf
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An expression M(x, y) dx + N(x, y) dy is an exact
differential in a region R corresponding to the
differential of some function f(x, y). A first-order DE
of the form M(x, y) dx + N(x, y) dy = 0
is said to be an exact equation, if the left side is an
exact differential.
DEFINITION 2.3Exact Equation
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Let M(x, y) and N(x, y) be continuous and have
continuous first partial derivatives in a region R defined
by a < x < b, c < y < d. Then a necessary and
sufficient condition that M(x, y) dx + N(x, y) dy be an
exact differential is(4)
THEOREM 2.1Criterion for an Extra Differential
xN
yM
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Proof of Necessity for Theorem 2.1
If M(x, y) dx + N(x, y) dy is exact, there exists some function f such that for all x in R
M(x, y) dx + N(x, y) dy = (f/x) dx + (f/y) dyTherefore
M(x, y) = , N(x, y) = and
The sufficient part consists of showing that there is a function f for which
= M(x, y) and = N(x, y)
xN
yf
xxyf
xf
yyM
2
xf
yf
xf
yf
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Since f/x = M(x, y), we have (5)
Differentiating (5) with respect to y and assume f/y = N(x, y)
Then
and (6)
dxyxMy
yxNxg ) ,() ,()('
) ,()(') ,( yxNygdxyxMyy
f
)() () ,( ygdxyx,Myxf
Method of Solution
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Integrate (6) with respect to y to get g(y), and substitute the result into (5) to obtain the implicit solution f(x, y) = c.
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Example 1
Solve 2xy dx + (x2 – 1) dy = 0.
Solution:With M(x, y) = 2xy, N(x, y) = x2 – 1, we have M/y = 2x = N/xThus it is exact. There exists a function f such that
f/x = 2xy, f/y = x2 – 1Then
f(x, y) = x2y + g(y)f/y = x2 + g’(y) = x2 – 1g’(y) = -1, g(y) = -y
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Example 1 (2)
Hence f(x, y) = x2y – y, and the solution isx2y – y = c, y = c/(1 – x2)
The interval of definition is any interval not containing x = 1 or x = -1.
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Example 2
Solve (e2y – y cos xy)dx+(2xe2y – x cos xy + 2y)dy = 0.
Solution:This DE is exact because
M/y = 2e2y + xy sin xy – cos xy = N/xHence a function f exists, and
f/y = 2xe2y – x cos xy + 2ythat is,
xyyexhxyyexf
xhyxyxe
ydyxydyxdyexyxf
yy
y
y
cos)('cos
)(sin
2cos2) ,(
22
22
2
CH2_73
Example 2 (2)
Thus h’(x) = 0, h(x) = c. The solution isxe2y – sin xy + y2 + c = 0
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Example 3
Solve
Solution:Rewrite the DE in the form
(cos x sin x – xy2) dx + y(1 – x2) dy = 0 Since
M/y = – 2xy = N/x (This DE is exact)Now
f/y = y(1 – x2)f(x, y) = ½y2(1 – x2) + h(x)f/x = – xy2 + h’(x) = cos x sin x – xy2
2)0( ,)1(
sincos2
2
yxy
xxxydxdy
CH2_75
Example 3 (2)
We have h(x) = cos x sin x h(x) = -½ cos2 x
Thus ½y2(1 – x2) – ½ cos2 x = c1 or
y2(1 – x2) – cos2 x = c (7)
where c = 2c1. Now y(0) = 2, so c = 3.The solution is
y2(1 – x2) – cos2 x = 3
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Fig 2.28
Fig 2.28 shows the family curves of the above example and the curve of the specialized VIP is drawn in color.
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It is sometimes possible to find an integrating factor(x, y), such that
(x, y)M(x, y)dx + (x, y)N(x, y)dy = 0 (8)is an exact differential.Equation (8) is exact if and only if
(M)y = (N)x
Then My + yM = Nx + xN, or
xN – yM = (My – Nx) (9)
Integrating Factors
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Suppose is a function of one variable, say x, then x = d /dx
(9) becomes(10)
If we have (My – Nx) / N depends only on x, then (10) is a first-order ODE and is separable. Similarly, if is a function of y only, then
(11)
In this case, if (Nx – My) / M is a function of y only, then we can solve (11) for .
N
NM
dyd xy
M
MN
dyd yx
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We summarize the results forM(x, y) dx + N(x, y) dy = 0 (12)
If (My – Nx) / N depends only on x, then
(13)
If (Nx – My) / M depends only on y, then
(14)
dxN
NM xy
ex
)(
dyM
MN yx
ey
)(
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Example 4
The nonlinear DE: xy dx + (2x2 + 3y2 – 20) dy = 0 is not exact. With M = xy, N = 2x2 + 3y2 – 20, we find My = x, Nx = 4x. Since
depends on both x and y.
depends only on y.The integrating factor is
e 3dy/y = e3lny = y3 = (y)
2032
3
2032
42222
yx
x
yx
xxN
NM xy
yM
MN yx 3
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Example 4 (2)
then the resulting equation isxy4 dx + (2x2y3 + 3y5 – 20y3) dy = 0
It is left to you to verify the solution is½ x2y4 + ½ y6 – 5y4 = c
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2.5 Solutions by Substitutions
IntroductionIf we want to transform the first-order DE:
dx/dy = f(x, y)by the substitution y = g(x, u), where u is a function of x, then
Since dy/dx = f(x, y), y = g(x, u),
Solving for du/dx, we have the form du/dx = F(x, u). If we can get u = (x), a solution is y = g(x, (x)).
dxdu
uxguxgdxdy
ux ),() ,(
dxdu
uxguxguxgxf ux ) ,() ,()) ,( ,(
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If a function f has the property f(tx, ty) = tf(x, y), then f is called a homogeneous function of degree .eg: f(x, y) = x3 + y3 is homogeneous of degree 3,
f(tx, ty) = (tx)3 + (ty)3 = t3f(x, y)A first-order DE:
M(x, y) dx + N(x, y) dy = 0(1)is said to be homogeneous, if both M and N are homogeneous of the same degree, that is, if
M(tx, ty) = tM(x, y), N(tx, ty) = tN(x, y)
Homogeneous Equations
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Note: Here the word “homogeneous” is not the same as in Sec 2.3.
If M and N are homogeneous of degree , M(x, y)=x M(1, u), N(x, y)=xN(1, u), u=y/x (2)M(x, y)=y M(v, 1), N(x, y)=yN(v, 1), v=x/y (3)Then (1) becomes
x M(1, u) dx + x N(1, u) dy = 0, or
M(1, u) dx + N(1, u) dy = 0where u = y/x or y = ux and dy = udx + xdu,
CH2_85
then M(1, u) dx + N(1, u)(u dx + x du) = 0,
and[M(1, u) + u N(1, u)] dx + xN(1, u) du = 0
or
0) ,1() ,1(
) ,1(
uuNuM
dxuNx
dx
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Example 1
Solve (x2 + y2) dx + (x2 – xy) dy = 0.
Solution: We have M = x2 + y2, N = x2 – xy are homogeneous of degree 2. Let y = ux, dy = u dx + x du, then
(x2 + u2x2) dx + (x2 - ux2)(u dx + x du) = 0
011
xdx
duuu
01
21
xdx
duu
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Example 1 (2)
Then
Simplify to get
Note: We may also try x = vy.
cxxy
xy
cxuu
lnln1ln2
lnln1ln2
xycxeyxxy
csyx /2
2
)(or )(
ln
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The DE: dy/dx + P(x)y = f(x)yn (4)where n is any real number, is called Bernoulli’s Equation.
Note for n = 0 and n = 1, (4) is linear, otherwise, letu = y1-n
to reduce (4) to a linear equation.
Bernoulli’s Equation
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Example 2
Solve x dy/dx + y = x2y2.
Solution:Rewrite the DE as
dy/dx + (1/x)y = xy2
With n = 2, then y = u-1, anddy/dx = -u-2(du/dx)
From the substitution and simplification, du/dx – (1/x)u = -x
The integrating factor on (0, ) is
1lnln/ 1
xeee xxxdx
CH2_90
Example 2 (2)
Integrating
gives x-1u = -x + c, or u = -x2 + cx.
Since u = y-1, we have y = 1/u and a solution of the DE is y = 1/(−x2 + cx).
xuxdx
du 1
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Reduction to Separation of Variables
A DE of the form
dy/dx = f(Ax + By + C)(5)
can always be reduced to a separable equation by means of substitution u = Ax + By + C.
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Example 3
Solve dy/dx = (-2x + y)2 – 7, y(0) = 0.
Solution:Let u = -2x + y, then du/dx = -2 + dy/dx,
du/dx + 2 = u2 – 7 or du/dx = u2 – 9This is separable. Using partial fractions,
or
dxuu
du )3)(3(
dxduuu
31
31
61
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Example 3 (2)
then we have
or
Solving the equation for u and the solution is
or (6)
Applying y(0) = 0 gives c = -1.
133
ln61
cxuu
x
x
ce
cexy 6
6
1
)1(32
x
x
ce
ceu 6
6
1
)1(3
x
x
e
exuxy 6
6
1
)1(322
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Example 3 (3)
The graph of the particular solution
is shown in Fig 2.30 in solid color.
x
x
e
exuxy 6
6
1
)1(322
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Fig 2.30
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Using the Tangent LineLet us assume
y’ = f(x, y), y(x0) = y0 (1)possess a solution. For example, the resulting graph is shown in Fig 2.31.
2.6 A Numerical Method
4)2( ,4.01.0' 2 yxyy
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Fig 2.31
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Using the linearization of the unknown solution y(x) of (1) at x0,
L(x) = f(x0, y0)(x - x0) + y0 (2)Replacing x by x1 = x0 + h, we have
L(x1) = f(x0, y0)(x0 + h - x0) + y0
or y1 = y0 + h f(x0, y0)and yn+1 = yn + h f(xn, yn) (3)where xn = x0 + nh. See Fig 2.32
Euler’s Method
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Fig 2.32
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Example 1
Consider Use Euler’s method to obtain y(2.5) using h = 0.1 and then h = 0.05.
Solution:Let the results step by step are shown in Table 2.1 and table 2.2.
.4)2( ,4.01.0' 2 yxyy
24.01.0) ,( xyyxf
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Table 2.1 Table 2.2
Table 2.1 h = 0.1
xn yn
2.00 4.0000
2.10 4.1800
2.20 4.3768
2.30 4.5914
2.40 4.8244
2.50 5.0768
Table 2.2 h = 0.05
xn yn
2.00 4.0000
2.05 4.0900
2.10 4.1842
2.15 4.2826
2.20 4.3854
2.25 4.4927
2.30 4.6045
2.35 4.7210
2.40 4.8423
2.45 4.9686
2.50 5.0997
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Example 2
Consider y’ = 0.2xy, y(1) = 1. Use Euler’s method to obtain y(1.5) using h = 0.1 and then h = 0.05.
Solution:We have f(x, y) = 0.2xy, the results step by step are shown in Table 2.3 and table 2.4.
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Table 2.3
Table 2.3 h = 0.1
xn yn ActualValue
AbsoluteError
% Rel.Error
1.00 1.0000 1.0000 0.0000 0.00
1.10 1.0200 1.0212 0.0012 0.12
1.20 1.0424 1.0450 0.0025 0.24
1.30 1.0675 1.0714 0.0040 0.37
1.40 1.0952 1.1008 0.0055 0.50
1.50 1.1259 1.1331 0.0073 0.64
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Table 2.4
Table 2.4 h = 0.05
xn yn ActualValue
AbsoluteError
% Rel.Error
1.00 1.0000 1.0000 0.0000 0.00
1.05 1.0100 1.0103 0.0003 0.03
1.10 1.0206 1.0212 0.0006 0.06
1.15 1.0318 1.0328 0.0009 0.09
1.20 1.0437 1.0450 0.0013 0.12
1.25 1.0562 1.0579 0.0016 0.16
1.30 1.0694 1.0714 0.0020 0.19
1.35 1.0833 1.0857 0.0024 0.22
1.40 1.0980 1.1008 0.0028 0.25
1.45 1.1133 1.1166 0.0032 0.29
1.50 1.1295 1.1331 0.0037 0.32
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Numerical Solver
See Fig 2.33 to know the comparisons of numerical methods.
Fig 2.33
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When the result is not helpful by numerical solvers, as in Fig 2.34, we may decrease the step size, use another method, or use another solver.
Using a Numerical Solver
Fig 2.34
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2.7 Linear Models
Growth and Decay
(1)00)( , xtxkx
dtdx
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Example 1: Bacterial Growth
P0 : initial number of bacterial = P(0)P(1) = 3/2 P(0)Find the time necessary for triple number.
Solution:Since dP/dt = kt, dP/dt – kt = 0, we have P(t) = cekt, using P(0) = P0
then c = P0 and P(t) = P0ekt Since P(1) = 3/2 P(0), then P(1) = P0ek = 3/2 P(0)So, k = ln(3/2) = 0.4055.Now P(t) = P0e0.4055t = 3P0 , t = ln3/0.4055 = 2.71.See Fig 2.35.
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Fig 2.35
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Fig 2.36
k > 0 is called a growth constant, and k > 0 is called a decay constant. See Fig 2.36.
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Example 2: Half-Life of Plutonium
A reactor converts U-238 into the isotope plutonium-239. After 15 years, there is 0.043% of the initial amount A0 of the plutonium has disintegrated. Find the half-life of this isotope.
Solution:Let A(t) denote the amount of plutonium remaining at time t. The DE is as
(2)
The solution is A(t) = A0ekt. If 0.043% of A0 has disintegrated, then 99.957% remains.
0)0( , AAkAdtdA
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Example 2 (2)
Then, 0.99957A0 = A(15) = A0e15k, thenk = (ln 0.99957) / 15 =-0.00002867
Let A(t) = A0e-0.00002867t = ½ A0
Then years. 24180
00002867.02ln T
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Example 3: Carbon Dating
A fossilized bone contains 1/1000 the original amount C-14. Determine the age of the fossil.
Solution:We know the half-life of C-14 is 5600 years.Then A0 /2 = A0e5600k, k = −(ln 2)/5600 = −0.00012378.And
A(t) = A0 /1000 = A0e -0.00012378t
years. 558000.00012378
1000ln T
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(3)
where Tm is the temperature of the medium around the object.
)( mTTkdxdT
Newton’s Law of Cooling
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Example 4
A cake’s temperature is 300F. Three minutes later its temperature is 200F. How long will it for this cake to cool off to a room temperature of 70F?
Solution:We identify Tm = 70, then
(4)and T(3) = 200. From (4), we have
300)0( ),70( TTkdxdT
ktectT
cktTkdtT
dT
2
1
70)(
70ln ,70
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Example 4 (2)
Using T(0) = 300 then c2 = 230Using T(3) = 200 then e3k = 13/23, k = -0.19018Thus T(t) = 70 + 230e-0.19018t (5)From (5), only t = , T(t) = 70. It means we need a reasonably long time to get T = 70. See Fig 2.37.
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Fig 2.37
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Mixtures
(6)outin RR
dtdx
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Example 5
Recall from example 5 of Sec 1.3, we have
How much salt is in the tank after along time?
Solution:Since
Using x(0) = 50, we have x(t) = 600 - 550e-t/100 (7)
When t is large enough, x(t) = 600.
50)0( ,6100
1 xxdtdx
100/100/100/ 600)( ,6][ ttt cetxexedtd
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Fig 2.38
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Series Circuits
See Fig 2.39. (8)
See Fig 2.40.
(9)
(10)
)(tERidtdi
L
)(1
tEqC
Ri
)(1
tEqCdt
dqR
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Fig 2.39
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Fig2.40
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Example 6
Refer to Fig 2.39, where E(t) = 12 Volt, L = ½ HenryR = 10 Ohms. Determine i(t) where i(0) = 0.
Solution:From (8),
Then
Using i(0) = 0, c = -6/5, then i(t) = (6/5) – (6/5)e-20t.
0)0( ,121021 ii
dtdi
t
t
ceti
eiedtd
20
2020
56
)(
24][
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Example 6 (2)
A general solution of (8) is
(11)
When E(t) = E0 is a constant, (11) becomes
(12)
where the first term is called a steady-state part, and the second term is a transient term.
tLRtLRtLR
cedttEeL
eti )/()/(
)/(
)()(
tLRo ceRE
ti )/()(
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Note:
Referring to example 1, P(t) is a continuous function. However, it should be discrete. Keeping in mind, a mathematical model is not reality. See Fig 2.41.
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Fig 2.41
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2.8 Nonlinear Models
Population DynamicsIf P(t) denotes the size of population at t, the relative (or specific), growth rate is defined by
(1)
When a population growth rate depends on the present number , the DE is
(2)
which is called density-dependent hypothesis.
PdtdP /
)(or )(/
PPfdtdP
PfP
dtdP
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Logistic Equation
If K is the carrying capacity, from (2) we have f(K) = 0, and simply set f(0) = r. Fig 2.46 shows three functions that satisfy these two conditions.
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Fig 2.46
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Suppose f (P) = c1P + c2. Using the conditions, we have c2 = r, c1 = −r/K. Then (2) becomes
(3)
Relabel (3), then
(4)
which is known as a logistic equation, its solution is called the logistic function and its graph is called a logistic curve.
P
Kr
rPdtdP
)( bPaPdtdP
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Solution of the Logistic Equation
From
After simplification, we have
dtbPaP
dP )(
acatPba
P
dtdPbPa
b/aP
a
)(ln
lnln
1
atat
at
ebc
ac
ebc
eactP
1
1
1
1
1)(
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If P(0) = P0 a/b, then c1 = P0/(a – bP0)
(5) atebPabP
aPtP
)(
)(00
0
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Graph of P(t)
Form (5), we have the graph as in Fig 2.47. When 0 < P0 < a/2b, see Fig 2.47(a).When a/2b < P0 < a/b, see Fig 2.47(b).
Fig 2.47
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Example 1
Form the previous discussion, assume an isolated campus of 1000 students, then we have the DE
Determine x(6).
Solution:Identify a = 1000k, b = k, from (5)
1)0( ,)1000( xxkxdtdx
ktetx 10009991
1000)(
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Example 1 (2)
Since x(4) = 50, then -1000k = -0.9906, Thusx(t) = 1000/(1 + 999e-0.9906t)
See Fig 2.48.
students 2769991
1000)6( 9436.5
e
x
tetx 9906.09991
1000)(
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Fig 2.48
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Modification of the Logistic Equation
or
(6)
or
(7)
which is known as the Gompertz DE.
hbPaPdtdP )(
hbPaPdtdP )(
)ln( PbaPdtdP
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Chemical Reactions
(8)
or
(9)
X
NMN
bXNM
Ma
dtdX
))(( XXkdtdX
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Example 2
The chemical reaction is described as
Then
By separation of variables and partial fractions,
(10)
Using X(10) = 30, 210k = 0.1258, finally (11)
See Fig 2.49.
X
XdtdX
54
325
50
)40)(250( XXkdtdX
ktecXX
cktXX 210
21 40250
or 21040250
ln
t
t
e
etX 1258.0
1258.0
425
11000)(
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Fig 2.49
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2.9 Modeling with Systems of First-Order DEs
Systems
(1)
where g1 and g2 are linear in x and y.
Radioactive Decay Series
(2)
) , ,(1 yxtgdtdx ) , ,(2 yxtg
dtdy
ydtdz
yxdtdy
xdtdx
2
21
1
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From Fig 2.52, we have
(3)21
2
211
252
252
501
252
xxdt
dx
xxdtdx
Mixtures
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Fig 2.52
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Let x, y denote the fox and rabbit populations at t.When lacking of food,
dx/dt = – ax, a > 0 (4)When rabbits are present,
dx/dt = – ax + bxy (5)When lacking of foxes,
dy/dt = dy, d > 0 (6) When foxes are present, dy/dt = dy – cxy (7)
A Predator-Prey Model
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Then
(8)
which is known as the Lotka-Volterra predator-prey model.
)(
)(
cxdycxydydtdy
byaxbxyaxdtdx
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Example 1
Suppose
Figure 2.53 shows the graph of the solution.
4)0( ,4)0( ,9.05.4
08.016.0
yxxyydtdy
xyxdtdx
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Fig 2.53
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Competition Models
dx/dt = ax, dy/dt = cy (9)Two species compete, then
dx/dt = ax – bydy/dt = cy – dx (10)
or dx/dt = ax – bxydy/dt = cy – dxy (11)
or dx/dt = a1x – b1x2
dy/dt = a2y – b2y2 (12)
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ordx/dt = a1x – b1x2 – c1xydy/dt = a2y – b2y2 – c2xy (1
3)
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Network
Referring to Fig 2.54, we havei1(t) = i2(t) + i3(t)
(14)
(15)
(16)
dtdi
LRitE
Ridtdi
LRitE
3211
222
111
)(
)(
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Using (14) to eliminate i1, then
(17)
Referring to Fig 2.55, please verify
(18)
)(
)()(
31213
2
332212
1
tEiRiRdtdi
L
tEiRiRRdtdi
L
0
)(
122
21
iidtdi
RC
tERidtdi
L
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Fig 2.54
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Fig 2.55