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Notes and Formulae
N.l TRIGONOMETRICAL AND HYPERBOLIC FUNCTIONS
sin2 x + eos2 x = 1; see2 x = 1 + tan2 x ;
eosee2 x = 1 + eot2 x
sin (x ± y) = sin x eosy ± eos x siny
eos (x ± y) = eos x eosy + sin x siny
tan (x + y) = tan x ± tan y - 1 + tan x tan y
sin 2x = 2 sin x cos x
cos 2x = cos2 X - sin2 x = 2 cos2 x-I = 1 - 2 sin2 x
tan 2x = 2 tan x/(1 - tan2 x)
sinh x = t(eX - e- X ); cosh x = t(eX + e- X )
cosh2 x - sinh2 x = 1
sinh (x ± y) = sinh x cosh y ± cosh x sinh y
cosh (x ± y) = cosh x cosh y ± sinh x sinh y
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh2 X + sinh2 x = 2 cosh2 x-I = 1 + 2 sinh2 x
sinh- 1 x = In {x + y(x2 + I)};
COSh-lX = ±ln{x + y(x2 - I)}
206 DIFFERENTIAL EQUATIONS
N.2 DIFFERENTIATION
Standard Forms
y = xn dy = nxn - l dx
y = lnx dy 1 -=-dx x
y = eX dy x - = e dx
y = sinx dy - = cosx dx
y = cosx dy
-sinx dx=
Y = tan x dy = sec2 x dx
y = cot x dy
-cosec2 x dx=
y = sec x ~ = sec x tanx x
y = cosec x dy
- cosec x cot x dx=
Y = sinh x dy dx = cosh x
Y = coshx ~~ = sinh x
y=sin-lx dy 1 dx = y(1 - x2 )
y=cos-lx dy 1 dx= y(l - x2 )
y = tan-lx dy 1 dx=1+x2
Y = sinh- l X dy 1 dx = y(x2 + 1)
Y = cosh- l X dy 1 dx = y(x2 - 1)
(-i ~ y ~ ~) (0 ~ Y ~ 7T)
NOTES AND FORlIIULAE
Partial Differentiation
If z = f(x, y) and x, y are functions of u and v
then 8z = 8z 8x + 8z 8y 8u 8x 8u 8y 8u
and 8z 8z 8x 8z 8y -=--+--8v 8x 8v 8y 8v
If z = f(x, y) and x, y are functions of a single variable t
then dz 8z dx 8z dy dt = 8x dt + 8y dt
The total differential 8z 8z
dz = -dx + -dy 8x 8y
N.3 INTEGRATION
Standard forms
J xn+l
xn dx = n + 1 (n '# - 1)
Jexdx = eX
J sin x dx = - cos x
J sec x dx = In (sec x + tan x)
J sinh x dx = cosh x
J~ = lnx
J cos x dx = sin x
J sec2 x dx = tan x
J cosh x dx = sinh x
207
208 DIFFERE~TIAL EQUATIONS
Differentiation under the Integral Sign Let f(x, a) be a continuous function of x and a. If a and bare
constants let f: f(x, a) dx denote the integral of the function assuming that a is constant.
Then ! f f(x, a) dx = Ib 8~ f(x, a) dx
EXAMPLE
I1l dC\: ----,;--
o a + b cos x
Differentiate with respect to a using the above result,
then 5011
(a + ~:os X)2 = (a2 :~2r'" and differentiating with respect to b
I1l cos x dx o (a + b cos X)2
NA LAPLACE TRANSFORMS
The Laplace transform x(s) of x(t) is defined by
x(s) = Lx> e- st x(t) dt
The table on p. 209 can be extended by using the following theorem.
If x(s) is the Laplace transform of x(t) then x(s + a) is the Laplace transform of e- at x(t).
N.S PARTIAL FRACTIONS
The following are examples of the forms to be assumed for partial fractions:
ax+b A B (x + c)(x + d) = x + c + x + d
ax2 + bx + c Ax + B C (x2 + dx + e) (x + f) = x2 + dx + e + x + f ax2 + bx + c ABC
(x + d)2(X + e) = (x + d)2 + X + d + x + e
NOTES AND FORMULAE 209
SHORT TABLE OF LAPLACE TRANsFoRMs
x(t) x(s)
1 1 s
tn n! sn+l
eat 1
s - a
cos at s
S2 + a2
sin at a
S2 + a2
cosh at s
S2 _ a2
sinh at a
S2 _ a2
t sin at s -----za (S2 + a2)2
2~3 (sin at - at cos at) 1
(S2 + a2)2
N.S cont. Other cases can easily be deduced from these examples.
aoxm + a1xm - 1 + ... + am
boxn + b1xn- 1 + ... + bn If (m ~ n)
then before expressing in partial fractions the denominator must be divided into the numerator, giving
R Q + boxn + b1xn 1 + ... + bn
Q being the quotient and R the remainder. To determine the partial fractions the following methods
given without proof are useful.
210 DIFFERENTIAL EQUATIONS
(a) The Cover Up Rule
Express (x + a;(x + b) in partial fractions.
To find the term in IJ(x + a) cover up (x + a) in the original expression and put x = - a in the remainder of the expression. This gives 1/(b - a) as the coefficient and the term is
1 (b - a)(x + a)
Similarly the other term is
1 (a - b)(x + b)
(b) Express x(x _ 1 ~ (x + 2) in partial fractions.
The coefficient of I/x is obtained by putting x = 0 in
(x _ l)I(X + 2)' i.e. -1; the coefficient of 1/(x - 1) is ob-
tained by putting x = 1 in x(x ~ 2) giving t and the coefficient
of 1/(x + 2) is obtained by putting x = -2 in x(x ~ 1) giving
-l The whole partial fraction is
1 1 1 - 2x + 3(x - 1) + 6(x + 2)
(c) If we apply the cover up rule to (x + 1)~X _ 3)2 which
bk ' ABC P l' rea s up mto --1 + --3 + ( 3)2' ut x = - m x+ x- x-1/(x - 3)2 to give A and put x = 3 in 1/(x + 1) to give C. B must be determined independently.
1 (d) To express (x _ 2)(x2 _ X + 1) in partial fractions the
quiekest method is as follows:
NOTES AND FORMULAE 211
By the cover up rule the coeffieient of 1/(x - 2) is t. Then
1 1 _ 3 - (x2 - X + 1) (x - 2)(x2 - X + 1) 3(x - 2) - 3(x - 2)(x2 - X + 1)
_ _(x2 - X - 2) - 3 (x - 2)(x2 - X + 1)
Hence
1 1 (x - 2)(x2 - X + 1) 3(x - 2)
N.6 COORDINATE GEOMETRY
Two Dimensional
- (x + 1) 3(x2 - X + 1)
x + 1 3(x2 - X + 1)
The Line. The length of the perpendicular from (h, k) to the line ax + by + c = 0 is
(alt + bk + c) y'(a2 + b2 )
The Circle. The equation x2 + y2 + 2gx + 2fy + c = 0 represents a eircle centre (-g, - f) radius y'(g2 + J2 - cl.
The Parabola. y2 = 4ax represents a parabola whose axis is the x axis, vertex (0,0), focus (a, 0) and directrix x + a = O.
x2 y2 The Ellipse. a2 + b2 = 1 represents an ellipse centre the
origin and axes of lengths 2a and 2b. If a > b
then
where e is the eccentricity. The foei are at (± ae, 0) and the equations of the directrices x = ± ale.
x2 y2 The Hyperbola. a2 - b2 = 1 represents a hyperbola centre
the origin, b2 = a2(e2 - 1), where eis the eccentricity. The foei are at (± ae, 0) and the equations of the directrices are x = ± ale. The equations of the asymptotes are y = ± (bla)x. The equations x2 - y2 = a2 and xy = c2 represent rectangular hyperbolas.
212 DIFFERENTIAL EQUATIONS
Curvature. The radius of curvature = { I + (~r}% /~~. Orthogonal Trajectories. Two families of curves are orthogonal
if at a point of intersection of any curve of one family with any curve of the other the tangents to the curves are at right angles.
If dyjdx = f(x) is the differential equation of a family of curves then dyjdx = -l/f(x) is the differential equation of the orthogonal trajectories.
Envelopes. If a family of curves is defined by f(x, y, c) = 0, where c is a parameter, then the envelope of the family is obtained by eliminating c between the equationf(x, y, c) = 0 and of oe (x,y, c) = O.
Three Dimensional
The Plane. The equation of a plane is
Ax + By + Cz + D = 0
The Line. If (Xl> Yl> Zl) and (x2, Y2' Z2) are two points on a line the equation is
X - Xl Y - YI Z - Zl -"'----'::....::... = --=-
x2 - Xl Y2 - YI Z2 - Zl
x2 - Xl' Y2 - Yl' Z2 - Zl are direction ratios of the line. Qttadric Surfaces. The general equation of the second degree
in three variables represents a quadric surface. Any plane section of a quadric is a conic or limiting form of a conic.
In particular x2 + y 2 + Z2 = a2
represents a sphere centre the origin and radius a and
x2 + y 2 = a2
represents a cylinder whose axis is the axis of Z.
Intersecting Surfaces. The intersection of any two surfaces expressed by a pair of simultaneous equations represents a curve.
If f(x, y, z) = 0 and g(x, y, z) = 0 are a pair of surfaces
then f(x, y, z) + Ag(X, y, z) = 0
represents a surface passing through a curve of intersection of f(x, y, z) = 0 and g(x, y, z) = O.
NOTES AND FORMULAE 213
N.7 SERIES
The Binomial Theorem
n(n - 1) n(n - 1)(n - 2) (1 + x)n = ] + nx + 2! x2 + 3! x3 + ...
Case 1. If n is a positive integer the series terminates and x can have any value.
Case 2. If n is not a positive integer the series is infinite and - 1 < x < 1 for convergence.
Also if n is a positive integer we can use the form
n(n - l)(n - 2) n- 3b3 + 3! a + ...
1l1aclaurin's Theorem
2 3
f(x) = f(O) + xf'(O) + ~!j"(0) + ~!r(O) + ...
Taylor's Theorem
x 2 x 3
f(a + x) = f(a) + xf'(a) + 2! j"(a) + TI r(a) + ...
Fourier Series
If f(x) is defined in the range -7T to 7T and
f(x) = ao + a1 cos x + a2 cos 2x + .. . + b1 sin x + b2 sin 2x + . . .
then 1 fn ao = -2 f(x) dx 7T -n
1 fn an = - f(x) cos nx dx 7T -n
= - f(x) sin nx dx 1 fn 7T -n
214 DIFFERENTIAL EQUATIONS
If the function is symmetrie (even) f(x) = f( -x) and the series is
f(x) = ao + a1 eos x + a2 eos 2x + ... If the funetion is skew-symmetrie (odd) f(x) = -f(-x) and
the series is f(x) = b1 sm x + b2 sin 2x + ...
and we have ao = .! fn f(x) dx 7T J 0
an = - f(x) eos ux dx 2 in 7T 0
2 in bn = - f(x) sin nx dx 7T 0
If a function is defined only in the half range 0 to 7T we ean either expand into a half range eosine series or a half range sine series.
If f(x) is defined in the range -l to 1
then 7TX 27TX
f(x) = ao + a 1 eos T + a2 eos -l- + ...
b . 7TX b . 27TX + lsmT + 2 sm -l- + ...
where ao = ~l f~/(X) dx
1 fl n7TX an = y f(x) eos -l- dx -I
2fZ . n7TX bn = y _/(x) sm -l- dx
Results ean be easily written down for expansion into half range eosine and sine series for the range 0 to l.
Convergence 1. If U o - U 1 + U 2 - U 3 + ... is aseries where U n is positive
and deereasing and if lim 1In = 0
n .... oo
then the series is eonvergent.
NOTES AND FORMULAE 215
2 .. The series 111 IP + 2p + 3P + ...
is convergent if p > 1 and divergent if p ~ 1.
3. If Uo + U 1 + U 2 + U 3 + ... is aseries of positive tenns and
lim U n = 0 and also n-+ 00
then the series is convergent.
4. If ao + a1x + a2x2 + ... is apower series then the series is absolutely convergent if
lim lan + 1xl < 1 11.-+ 00 an
N.S ELECTRIC CIRCUITS
The following symbols are used:
C the capacitance of a capacitor in farads L the coefficient of self inductance of an inductor in
henrys R the resistance of a resistor in ohms i the current in amperes q the charge on a capacitor in coulombs
The voltage drop across an inductor = L ~; The voltage drop across a resistor = Ri
1 The voltage drop across a capacitor = C q
If a capacitor is charging then i = dq/dt and if it is discharging, i = - (dqJdt).
N.9 THE OPERATOR D
We put Dy = dyJdx, D2y = d2y/dx2 , etc., where D is considered to be an operator d/dx.
216 DIFFERENTIAL EQUATIONS
Whilst D, D2, D3, ... are not ordinary algebraic symbols, in many cases it is possible to treat them as if they were, e.g.
D2(D3y ) = ~ (d3y) = d5y = D5y dx2 dx3 dx5
J)3(y + z) = D3y + J)3Z
(D2 - 3D + 2)y = (D - l)(D - 2)y
which means
~~ - 3 ix + 2y = (! - l)(ix - 2y)
Also, since 1 -·Dy = y D
it appears that liD means integrate once and I/D2 means integrate twice, etc.
To attach a meaning to such expressions as D ~ 1 x 2
1 __ x 2 = 1l
D + 1 let
Then 1 D + 1. D + 1 x 2 = (D + l)~t
l.e. (D + l)tt = x 2
This is a first-order linear equation and the integrating factor is eX •
Hence 1teX = J x 2ex dx
= (x2 - 2x + 2)eX + C
1/' = Ce- x + x2 - 2x + 2
the particular integral being x2 - 2x + 2. This result could have been obtained as follows:
1 -- x2 = (1 + D)-lX 2 l+D
= (1 - D + D2 - .. . )x2
= x2 - 2x + .. 2
NOTES AND FORMULAE 217
In general if F(D) is a polynomial in D andf(x) is a polynomial
in x it can be verified that F!D) {f(x)} can be obtained by
expanding 1/F(D) by the binomial theorem. 1 ePX
To show that F(D){ePX} = ePXF(p) and F(D) {ePX} = F(P)
Let F(D) = ao + alD + a2D2 + ... + anDn.
Then F(D){ePX} = ePX(ao + alP + a2P + ... + anpn) = ePXF(p)
Also F(D) {;~;)} = ePX
Hence
provided F(P) =f: O.
To show that
F(D){ePXV} = ePX{F(D + P)V}
and FtD) {ePXV} = ePX {F(D 1+ P) v}
where V is a function of x. Leibnitz theorem for the nth derivative of a product states
that n(n - 1)
Dn(ttv) = Dnu.v + nDn-1u.Dv + 2! Dn-2D2v + ...
Hence
Dn{ePXV} = ePX {pnv + npn-l DV + n(n2~ 1) pn-2D2V + ... }
= ePX {pn + npn- l D + n(n 2~ 1) pn-2 D2 + ... } V
= ePX(D + p)nv
by the binomial theorem.
F(D){ePXV} = (ao + alD + a2D2 + ... + anDn){ePXV} = ePX{ao + al(D + P) + a2(D + P)2
+ . .. + an(D + p)n}v = ePX{F(D + P)V}
218 DIFFERENTIAL EQU A TIONS
Since
F(D) {epx 1 } V = ePX {F(D + a) V} = ePxV F(D + a) F(D + a)
then _1_ { pxV' _ px { 1 V} F(D) e f - e F(D + a)
The above treatment of the operator D is not intended to be rigorous. The Laplace transform provides an alternative method much easier to justify mathematically.
N.lO DYNAMICS
If x is the displacement, v the velo city and t the time, thc acceleration can be expressed as
dv - or dt
dv v
dx
For most purposes it is sufficiently accurate to take g = 32 ft/sec2 • Terms in differential equations involving g should be expressed in lb, ft, sec units.
Newton's Second Law of motion states that
d - (mv) = F dt
m being the mass and F for applied force. Only if m is constant can this be written
dv m-=F
dt
Horse power is the rate of doing work and a horse power of His equal to 550H ft.lb.wt/sec.
The applied force due to this horse power is 550H/v lb.wt, where v is the velocity in ft/sec.
Index
The abbreviation D.E. will be used for differential equations throughout this index.
Arbitrary constants and order of D.E., 4
Auxiliary equation, 65
Beams, 138 Bending moment, 138 Bernoulli's equation, 24 Binomial theorem, 213
Clairaut's form, 36 Complementary function, 71
one term known, 97 Complete primative, 5 Convergence of series, 214 Coordinate geometry, 211 Curvature, 212
Damped oscillations, 129 Definition of Laplace transform,
113 Deflection of beams, 138 Degree of D.E., 1 Differentiation, 206
under the integral sign, 208 Dynamics, 218
Electric circuits, 215 Electrical applications of first
order D.E., 56 of second-order D.E., 150
Elimination of constants, 2 of functions, 182
Envelopes, 212 Euler's homogeneous D.E., 92 Exact D.E., 18, 27
Forced oscillations, 134, 135 Formation of D.E., 2 Frobenius' method, 166 Fourier series, 213
solution of partial D.E., 190
Geometrical applications of firstorder D.E., 58
Higher degree first-order D.E., 31 second-order D.E., 105
Higher-order linear D.E., 69, 84 Homogeneous first-order D.E., 13 Homogeneous partial D.E. of
second order, 196 Hyperbolic functions, 205
Indicial equation, 167 Integration, 207 Integrating factor, 18 Inversion of Laplace transform,
116
Lagrange's linear partial D.E., 194 Laplace transform. Standard
forms, 209 Linear D.E., 2 Linear first-order D.E., 18
Mac1aurin's theorem, 213 Miscellaneous substitutes for
second-order D.E., 102 Motion under resistance, 46
Non-linear D.E., 2
220 INDEX
Operator D., 77 Order of D.E., 1 Ordinary D.E., 1 Orthogonal trajeetories, 212
Partial D.E., 1 differentiation, 207 fraetions, 208
Partieular integrals, 6, 71 exponentials, sines and eosines,
72 polynomials, 77 produets, 80
Power series solution oi D.E., 161
Resonanee, 136, 137 Roots of auxiliary equation, 68 Roots of indicial equation differing
by integer, 170, 176
equal, 173 unequal, 166
Separation of variables solution of partial D.E., 187
Sign of bending moment, 138 Simple harmonie motion, 127 Simultaneous D.E., 90, 122 Singular solutions, 36 Struts, 143
Taylor's theorem, 213 Trigonometrieal functions, 205
Variables separable, 8 Variation of parameters, 99
Whirling shafts, 146, 147
