Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
S. Stirling 2011-2012 Page 1 of 13
6.1 Slope Fields (skip Euler’s Method)
Apply & Practice 6.1: P 391-294 #1, 3, 5, 10, 11, 13, 15, 19, 21, 23, 27, 31, 33 #37, 39, 43, 45, 49, 51, 53 – 56, 57
3. Differential equation: 2 2
2xyyx y
′ =−
Solution: 2 2x y Cy+ = : Differentiate:
( )
2 22 22 2
22
x yy Cyx Cy yyx C y y
xyC y
′ ′+ =′ ′= −
′= −
′ =−
Check in diff. equ: 2 2
2 22x xy
C y x y=
− −
get left to match 2
22
xyCy y
=−
mult by yy
replace Cy 2 2 2
22
xyx y y
=+ −
simplify 2 2
2xyx y
=−
checks!
5. Differential equation: 0y y′′ + = Solution: 1 2cos siny C x C x= + : Differentiate: 1 2sin cosy C x C x′ = − + 1 2cos siny C x C x′′ = − − Check in diff. equ:
1 2 1 2cos sin cos sin 0C x C x C x C x− − + − = 22 sin 0C x− = checks!
Matches!
See if (0)y gives you –2. Works!
11. 226 xy e−=
4y xy′ = − (0) 6y =
See if (0)y gives you 6. Works!
Show that the derivative = 4xy− .
Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
S. Stirling 2011-2012 Page 2 of 13
13. Find derivatives:
(4)
3cos3sin3cos
3sin3cos
y xy xy xy x
y x
=′ = −′′ = −′′′ =
=
.
Test:
( )
(4) 16 03cos 16 3cos 0
45cos 0
y yx x
x
− =
− =
− ≠
3cosy x= is NOT a solution.
15. Find derivatives: 2
2
2
2
(4) 2
24
816
x
x
x
x
x
y ey ey ey e
y e
−
−
−
−
−
=
′ = −
′′ =
′′′ = −
=
.
Test:
( )(4)
2 2
16 0
16 16 0
0 0
x x
y y
e e− −
− =
− =
=
2xy e−= IS a solution.
19. Given: 2y x= 2y x′ =
( ) ( )3
2 3
3
2
2 2
0
x
x
x
xy y x e
x x x x e
x e
′ − =
− =
≠
Not a solution.
21. Given: ( )2 2 xy x e= +
( ) ( )2
2
2 2
4 2
x x
x x
y x e x e
y x xe x e
′ = + +
′ = + +
( ) ( )( )2 2 3
2 2 3 2 2 3
3 3
4 2 2 2
4 2 4 2
x x x x
x x x x
x x
x x xe x e x e x e
x x e x e x x e x ex e x e
+ + − + =
+ + − − =
=
Is a solution.
23. Given: lny x=
1y x′ =
( ) ( ) 31 2 ln xx x x ex − ≠
Not a solution.
→ 22 xy Ce−′ = −
Works!
Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
S. Stirling 2011-2012 Page 3 of 13
33. Differential equation: 9 0y y′′ + = General Solution: 1 2sin3 cos3y C x C x= + : Differentiate:
1 2
1 2
1 2
sin3 cos33 cos3 3 sin3
9 sin3 9 cos3
y C x C xy C x C xy C x C x
= +′ = −′′ = − −
Check in diff. equ: ( ) ( )1 2 1 2
1 2 1 2
9 sin3 9 cos3 9 sin3 cos3 09 sin3 9 cos3 9 sin3 9 cos3 0
0 0
C x C x C x C xC x C x C x C x
− − + + =
− − + + ==
checks!
Initial conditions: 26
y π⎛ ⎞ =⎜ ⎟⎝ ⎠
, 16
y π⎛ ⎞′ =⎜ ⎟⎝ ⎠
Find 1C :
1 2
1 2
1 2
1 2
1
sin3 cos3
2 sin3 cos36 6
2 sin cos2 2
2 1 02
y C x C x
C C
C C
C CC
π π
π π
= +
⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= +=
i i
Find 2C :
( )
1 2
2
2
2
1 3 cos3 3 sin6 6
1 3 2 cos 3 sin2 2
1 6 0 3 113
C C
C
C
C
π π
π π
⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= −
− =
i i
Particular Solution: 12sin3 cos33
y x x= −
Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
S. Stirling 2011-2012 Page 4 of 13
Oscillates. Same vertically & when x = 0, dy/dx = 1. When x = π, dy/dx = 1. Matches (b).
Oscillates. Same vertically & when x = 0, dy/dx = 0. Matches (c).
Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
S. Stirling 2011-2012 Page 5 of 13
6.2 Differential Equations: Growth and Decay
Apply & Practice 6.2 Set 1: P 400-402 #1, 3, 6, 9, 11, 13, 15, 18, 19 Separation of variables. #31, 32 Think slope fields!
9. ( )21 2 0x y xy′+ − =
( )2
2
2
1 2 0
21
21
x y xy
dy xydx xdy x dxy x
′+ − =
=+
=+∫ ∫
21u x= + , 2 du x dx= , 2du dx
x=
2ln 2
x duyu x
= ∫ i
( )( )
( )
21
2
2
ln ln 1
ln ln 1 ln
ln ln 1
y x C
y x C
y C x
= + +
= + +
⎡ ⎤= +⎣ ⎦
( )
( )( )
( )
22ln 1ln
22
22
2
1
1
1
C xye e
y C x
y C x
y C x
⎡ ⎤+⎣ ⎦=
= +
= ± +
= +
6. 3
xyy
′ =
12
32 21
32 2
3
3
3 22 3
9 4
dy xdx y
y dy x dx
y x C
y x C
=
=
= +
= +
∫ ∫
3. 2dy ydx
= +
1
1
1
ln 2
21
2
2ln 2
22
y x C
C x
x
dy ydxdy dx
ydy dx
yy x C
e ey e ey Ce
+ +
+=
=+
=+
+ = +
=
+ = ±
= −
∫ ∫
i
1.
2
2
2
1 22
dy xdxdy x dx
y x x C
= +
= +
= + +
∫ ∫
11.
( )
2
2
1
1
dQ kdt tdQ k t dt
Q k t C
kQ Ct
−
−
=
=
= − +
= − +
∫ ∫
13.
( )
( )
( )
2
2
250
250
12502
5002
dN k sdsdN k s ds
N k s s C
kN s s C
= −
= −
⎛ ⎞= − +⎜ ⎟⎝ ⎠
= − − + +
∫ ∫
Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
S. Stirling 2011-2012 Page 6 of 13
15b.
( )
( )
22
2
2
2
21
1ln 6 2
12
12
6
6
1ln 62
6
6
x Cy
xC
x
dy x ydx
dy x dxy
y x C
e e
y e e
y Ce
− +−
−
−
= −
=−
− − = +
=
− =
= +
∫ ∫
Initial Condition: ( )21 0
20 66
CeC
−= += −
Soln. 21
26 6x
y e−
= −
Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
S. Stirling 2011-2012 Page 7 of 13
→ When 0y′ > .
Apply & Practice 6.2 Set 2: P 400-402 #23, 26 Thm 6.1 #33, 39, 43, 47, 57 Thm 6.1 apps.
Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
S. Stirling 2011-2012 Page 8 of 13
$1822.12≈
Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
S. Stirling 2011-2012 Page 9 of 13
Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
S. Stirling 2011-2012 Page 10 of 13
6.3 Differential Equations: Separation of Variables
Apply & Practice 6.3: P 413 – 414 #1, 3, 5, 6, 7, 10, 14, 16, 18, 21, 23, 24, 45 – 47, 55 – 58
→ 22 xy Ce−′ = −
Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
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Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
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