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Step-by-step solutions to various differential equations using the trial-solution method. The equations are homogeneous and nohomogeneous, and the solutions involve finding the values of constants and trigonometric functions. The document also includes initial conditions.
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HW #1: Differential Equations
Solve the following differential equations using the Trial-Solution method.
x && + 5 x &+ 4 x = 0
Let
st x = e , then we get ( 5 4 ) 0
2
st s s e Î 5 4 0
2 s + s + =
2 s + s + = s + s + = Î s =− 1 or− 4 Î
t t x e e
4 or
t t x xc Ae Be
− − 4 = = +
x A B
x A B
t t x t e e
4 ( ) 3
− − = +.
x && + 6 x &+ 9 x = 0
Let
st x = e , then we get ( 6 9 ) 0
2
st s s e Î 6 9 0
2 s + s + =
2 2 s + s + = s + = Î s =− 3 or− 3 Î
t t x e te
3 3 or
t t x xc Ae Bte
− 3 − 3 = = +
x A B
x A
t t x e te
3 3 6
− − = +.
x &&+ 16 x = 0
Let
st x = e , then we get ( 16 ) 0
2
st s e Î 16 0
2 s + =
2 s + = Î s = ± 4 j Î
j t jt x e e
4 4 or
− = Î x =cos( 4 t )orsin( 4 t )
x (^) c = A cos( 4 t )+ B sin( 4 t )
x &&+ 16 x = 8
Try a constant solution, x = C , then 16 C = 8 Î 2
xp = 0. 5
Therefore, x = xc + xp = A cos( 4 t )+ B sin( 4 t )+ 0. 5
x B
x A
sin( 4 ) 4
cos( 4 ) 2
x ( t )= − t + t +.
x && + 6 x &+ 34 x = 0
Let
st x = e , then we get ( 6 34 ) 0
2
st s s e Î 6 34 0
2 s + s + =
2 s + s + = Î s = − 3 ± j 5 Î
t jt t jt x e e e e
3 5 3 5 or
− − − = Î
cos( 5 )or sin( 5 )
3 3 x e t e t
cos( 5 ) sin( 5 )
3 3 x Ae t Be t
t t c
− − = +
x && + 6 x &+ 34 x = 68
Try a constant solution, x = C , then 34 C = 68 Î C = 2
xp = 2
Therefore, cos( 5 ) sin( 5 ) 2
3 3 = + = + +
− − x x x Ae t Be t
t t c p
x A B
x A
( ) cos( 5 )
3 3 = − − +
− − x t e t e t
t t .
2 & x & + 5 x &+ 2 x = 5 x ( 0 )= x & =
2 x && + 5 x &+ 2 x = 0
Let
st x = e , then we get ( 2 5 2 ) 0
2
st s s e Î 2 5 2 0
2 s + s + =
2 s + s + = Î ( 2 s + 1 )( s + 2 )= 0 Î s =− 1 2 or− 2 Î
t
t x e e 2 2
1
or
−
t
t xc Ae Be 2 2
1 −
− = +
2 x && + 5 x &+ 2 x = 5
Try a constant solution, x = C , then 2 C = 5 Î C = 52
xp =
Therefore, 2
1
= + = + +
−
− t
t x xc xp Ae Be
xp =
Therefore, 4
= + = cos( )+ sin()+
− − x x x Ae t Be t
t t c p
x A B
x A
4
sin() 4
cos() 4
− − x t e t e t
t t .