Question

The differential equation for which y = a cos x + b sin x is a solution is _______________.

Answer 1

Given: y = a cosx + b sinx


$$\begin{gathered} y=a\cos x+b\sin x...\left(1\right) \\ \mathrm{Differentiating}\mathrm{both}\mathrm{sides}\mathrm{with}\mathrm{respect}\mathrm{to}x \\ \Rightarrow \frac{dy}{dx}=-a\sin x+b\cos x \\ \mathrm{Differentiating}\mathrm{both}\mathrm{sides}\mathrm{with}\mathrm{respect}\mathrm{to}x \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=-a\cos x-b\sin x \\ =-\left(a\cos x+b\sin x\right) \\ =-y\left(\mathrm{from}\left(1\right)\right) \\ \Rightarrow \frac{d^{2}y}{d{x}^2}+y=0 \end{gathered}$$

Hence, the differential equation for which y = a cosx + b sinx is a solution is $\underline{\frac{d^{2}y}{d{x}^2}+y=0}$.