Compute wronskian and variation of parameters for

Question

The differential equation y′′ + b(x)y′ + c(x)y  =  0, x  >  0, has a fundamental solution set {x, xe^(3x)}

(a) Compute the Wronskian.

(b) Use variation of parameters to compute a particular solution to

y′′ + b(x)y′ + c(x)y  =  4*x*e^(3x)

For the final answer, make sure that your yp does not contain any terms present in the complementary solution.

Answer 1

(a) We have fundamental solution: y₁ = x, y₂ = xe^3x.

Wronskian W(x) = det[(x xe^3x) (1 e^3x+ 3xe^3x)] = 3x²e^3x.

By variation of parameters:

y = -x∫xe^3x·4xe^3x/(3x²e^3x)dx + xe^3x∫x·4xe^3x/(3x²e^3x)dx = - 4/3x∫e^3xdx + 4/3xe^3x∫dx =

-4/9xe^3x + 4/3x²e^3x= 4/9(3x² - x)e^3x