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Solve the initial value problem using the method of undetermined coefficients. y"+4y'+4y=(3+x)e-2x, y(0)=2, y'(0)=5

Question

Solve the initial value problem using the method of undetermined coefficients. 

y"+4y'+4y=(3+x)e-2x, y(0)=2, y'(0)=5

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Step 1

The given intial value problem is given by,

y4+4y3+x)e, y(0) = 2. y'(0) 5
The characteristic equation of above differential equation is m2 +4m+ 4=0
Solve the characteristic equation m2 +4m+4 = 0
m2 2m 2m+ 4 = 0
m(m+2)+2(m+2) = 0
(m+2)(m+2)= 0
The characteristic equation has repeated roots m m, = -2
The complementary function is given by, y. = ce'
+С хе.
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y4+4y3+x)e, y(0) = 2. y'(0) 5 The characteristic equation of above differential equation is m2 +4m+ 4=0 Solve the characteristic equation m2 +4m+4 = 0 m2 2m 2m+ 4 = 0 m(m+2)+2(m+2) = 0 (m+2)(m+2)= 0 The characteristic equation has repeated roots m m, = -2 The complementary function is given by, y. = ce' +С хе.

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Step 2

Find particular solution.

Suppose assume y,=(Ax+ B)e, it becomes fail because it is apparent from
complementary function such that a constant term is a solution of associated homogenous
equation y" 4y 4y 0
Assume y, (Ax +B )e
Differentiate y (Ax' +Bx2)e2* with respect to x
(Axe +Bxe
dx
Ур
—-2 Ах е?
+ ЗАх'е
— 2Вх е* + 2Вхе
y,-2(4x + Bx)(3.4x? +2Bx)e*:
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Suppose assume y,=(Ax+ B)e, it becomes fail because it is apparent from complementary function such that a constant term is a solution of associated homogenous equation y" 4y 4y 0 Assume y, (Ax +B )e Differentiate y (Ax' +Bx2)e2* with respect to x (Axe +Bxe dx Ур —-2 Ах е? + ЗАх'е — 2Вх е* + 2Вхе y,-2(4x + Bx)(3.4x? +2Bx)e*:

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Step 3

Use substitution to find p...

Substitute -2(Ax Bx+(34x +2 Bx)e2
2x
for x in given differential equation
4(Ax Bx)e-4(3.4x +2Bx)e +6 Axe
4(-2(4x
-2x
-2x
+2Be
)e*) = (3 + x)c*
x* )e + (3,4x +2 Bx)e2)+4((4x + Bx
= (3+ x)e
-2x
0 6Axe22Be
-2 x
Equate the coefficient of x
-2x
6.Ae
6.A 1
1
A =
6
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Substitute -2(Ax Bx+(34x +2 Bx)e2 2x for x in given differential equation 4(Ax Bx)e-4(3.4x +2Bx)e +6 Axe 4(-2(4x -2x -2x +2Be )e*) = (3 + x)c* x* )e + (3,4x +2 Bx)e2)+4((4x + Bx = (3+ x)e -2x 0 6Axe22Be -2 x Equate the coefficient of x -2x 6.Ae 6.A 1 1 A = 6

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