Question

Using the method of variation of parameters find the solution of d²y/dx² - 2dy/dx + y = e^x/x.

Answer 1

We have (D² + 2D + 1)y = e^x/x

A.E. is m² – 2m + 1 = 0 ⇒ (m – 1)² = 0

m = 1, 1 are the roots of A.E.

 C.F. = (C₁ + C₂x)e^x

= (A + Bx) e^x ...(1)

where A = A(x), B = B(x)

be the complete solution of the d.e. and we shall find A, B, we have

y₁ = e^x, y₂ = xe^x