Question

The differential equation representing the family of curves y = A sin x + B cos x is _____________.

Answer 1

Given: y = A sinx + B cosx


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

Hence, the differential equation representing the family of curves y = A sinx + B cosx is $\underline{\frac{d^{2}y}{d{x}^2}+y=0}$.