Solution to LDE

Question

Why some linear differential equations have different solutions by different methods? I have come across $$0.25y" + ~y =1.5\cos(0.5x) -7\sin(7.5x)$$ This LDE have two solutions by undetermined coefficient method and variation of parameters method, so why?

Answer 1

Hint: At first you must solve the equation $$\frac{1}{4}y''+y=0$$ by the ansatz $$y=e^{\lambda x}$$

Answer 2

I get same solution with both methods: $$y=\frac{560 \sin{\left( \frac{15 x}{2}\right) }+1672 \cos{\left( \frac{x}{2}\right) }}{1045}+\mathit{C_2} \sin{\left( 2 x\right) }+\mathit{C_1} \cos{\left( 2 x\right) }$$