Find a particular solution of the following ODE

Question

Use the method of Variation of Parameters to find a particular solution of the following ODE:

Answer 1

Let found fundamental solutions of homoginius equationsoluyion: y'' - y = 0,

Characteristic equation: r² - 1 = 0, r = ±1, y₁ = e^x, y₂ = e^-x.

Wronscian W = det[(e^x, e^x) (e^-x, - e^-x) = - 2,

So, y_p = - e^x·∫e^-x(e^x/(e^x + e^-x))/(-2)dx + e^-x·∫e^x(e^x/(e^x + e^-x))/(-2)dx = 1/2e^xtan^-1(e^x) -1/2e^-x(e^x - tan^-1(e^x)) =

1/2(e^x + e^-x)tan^-1(e^x) - 1/2