form a differential equation for the equation y = asin2x+bcos2x

the given equation is: $y=a\sin 2x+b\cos 2x ..........\left(1\right)$
differentiating eq(1) wrt x:
$$\begin{gathered} \frac{dy}{dx}=2a\cos 2x-2b\sin 2x ........\left(2\right) \\ again differentiating the eq\left(2\right): \\ \frac{d^{2}y}{d{x}^2}=2a.(-2\sin 2x)-2b.\left(2\cos 2x\right) \\ \frac{d^{2}y}{d{x}^2}=-4a\sin 2x-4b\cos 2x \\ \frac{d^{2}y}{d{x}^2}=-4.(a\sin 2x+b\cos 2x) \\ \frac{d^{2}y}{d{x}^2}=-4y [by eq(1) \\ \Rightarrow \frac{d^{2}y}{d{x}^2}+4y=0 \end{gathered}$$

hope this helps you