Question

If y = e^x (sin x + cos x) prove that $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y=0$.

Answer 1

Here,
$$\begin{gathered} y=e^x\left(\sin x+\cos x\right) \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.x\mathit{,}\mathit{}\mathrm{we}\mathrm{get} \\ \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=e^x\left(\sin x+\cos x\right)+e^x\left(\cos x-\sin x\right)=2e^x\cos x \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.x\mathit{,}\mathit{}\mathrm{we}\mathrm{get} \\ \frac{{\mathit{d}}^{\mathit{2}}\mathit{y}}{\mathit{d}{\mathit{x}}^{\mathit{2}}}=2e^x\cos x-2e^x\sin x \\ \mathrm{Now}, \\ \mathrm{LHS}=\frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}-2\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}+2y \\ =2e^x\cos x-2e^x\sin x-4e^x\cos x+2e^x\left(\sin x+\cos x\right) \\ =0=\mathrm{RHS} \end{gathered}$$

Hence proved.