Question

Solve d²y/dx² + y = tan x by the method of variation of parameters.

Answer 1

Rewrite the given equation as (D² + 1)y = tanx

Here A.E D² + 1 = 0 or D = ±i

Thus y_C.F.(x) = c₁y₁ + c₂y₂ = c₁cosx + c₂ sinx

Assume y_P.I.(x) = u₁(x) y₁ + u₂(x) y₂ = u₁cosx + u₂ sinx

By method of variation of parameter, we have

On integration,

u₁(x) = ∫-sin xtanxdx = -∫sin xsinx/cos² x - 1/cos x  dx

= ∫(cos x - sec x)dx = sinx - log(sec x + tan x)

and u₁(x) = ∫cos xtan xdx = ∫sin xdx = -cos x

Whence yP.I. (x) = cosx[sinx - log(sec x + tan x)] + sin x(-cos x)

= -cosx log(sec x + tan x)

And therefore the complete solution becomes,

y = (c₁cosx + c₂sinx) - (cosx) log(sec x + tan x)