Question

Solve following ODE by method of undetermined coefficients.

(𝑫^𝟐―𝟖𝑫+𝟏𝟓)𝒚=𝟗𝒙𝒆^𝟐𝒙, 𝒚(𝟎)= 𝟓,𝒚′(𝟎)=𝟏𝟎

Answer 1

Solution

First let’s find fundamental solutions as solution of homogeneous equation.

Y’’ - 8Y’ + 15 = 0

Characteristic equation for it is λ²-8 λ +15=0.  =>  λ₁=3, λ₂=5

Therefore solution for Y(x) is Y = C₁e^3x + C₂e^5x  where C₁ and C₂ are arbitrary constants.

Solution of nonhomogeneous equation let’s find as Y_p(x)=(Ax+B)e^2x  

Substitution in equation gives

Y_p‘=(A+2Ax+2B)e^2x , Y_p‘’=(2A+2A+4Ax+4B)e^2x = 4(A+Ax+B)e^2x  =>  

[4(A+Ax+B)-8(A+2Ax+2B)+15(Ax+B)] e^2x = 9xe^2x

(-4A+3B+3Ax) e^2x = 9xe^2x

3A = 9, 3B-4A = 0  => A = 3, B = 4  => Y_p(x)=(3x+4)e^2x

So y(x) = Y(x)+Y_p(x) = C₁e^3x + C₂e^5x +(3x+4)e^2x 

From initial conditions C₁ + C₂ + 4 = 5, 3C₁ + 5C₂ +11 =10  =>  C₁ + C₂ = 1, 3C₁ + 5C₂ = -1  =>   

2C₁ = 6, C₁ = 3, C₂ = 1 - C₁ = -2 =>

y(x) = 3e^3x - 2e^5x +(3x+4)e^2x 

Answer

y(x) = 3e^3x - 2e^5x +(3x+4)e^2x