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 2 xy y x y ′= − Solution: 2 2 x y Cy + = : Differentiate: ( ) 2 2 2 2 2 2 2 2 x yy Cy x Cy yy x C yy x y C y ′ ′ + = ′ ′ = − ′ = − ′= − Check in diff. equ: 2 2 2 2 2 x xy C y x y = − − get left to match 2 2 2 xy Cy y = − mult by y y replace Cy 2 2 2 2 2 xy x y y = + − simplify 2 2 2 xy x y = − checks! 5. Differential equation: 0 y y ′′ + = Solution: 1 2 cos sin y C x C x = + : Differentiate: 1 2 sin cos y C x C x ′=− + 1 2 cos sin y C x C x ′′ = − − Check in diff. equ: 1 2 1 2 cos sin cos sin 0 C x C x C x C x − − + − = 2 2 sin 0 C x − = checks! Matches! See if (0) y gives you –2. Works! 11. 2 2 6 x y e − = 4 y xy ′=− (0) 6 y = See if (0) y gives you 6. Works! Show that the derivative = 4 xy − .
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6.1 Slope Fields (skip Euler’s Method) Apply & Practice 6.1 · Ch 06 Homework Complete Solutions: S. Stirling Name _____ Calculus: Early Transcendental Functions, 4e Larson S. Stirling
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Ch 06 Homework Complete Solutions: S. Stirling Name _________________________________ Calculus: Early Transcendental Functions, 4e Larson
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: