y" +a^2y = Cos ax

Question

Find:

y" + a²y = Cos ax

Answer 1

y¹¹ + a²y = cos ax

It's complementary equation is:

m² + a² = 0

⇒ m = \(\pm ac\) 

\(\therefore\) C. F. = C₁ cos ax + C₂ sin ax

P.I. = \(\frac{1}{D^2+a^2}cos\,ax=\frac{x}{2a}sin\,ax\) 

\(\therefore\) Complete solution of given differential equation is

y = C.F. + P.I.

 = C₁ cos ax + C₂ sin ax + x/2a sin ax