Question

If y = 3 e^2x + 2 e^3x, prove that $\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}+6y=0$

Answer 1

Here,
$$\begin{gathered} y=3e^{2x}+2e^{3x} \\ \mathrm{Differentiating}w.\mathrm{r}.\mathrm{t}.x\mathit{,}\mathit{}\mathrm{we}\mathrm{get} \\ \frac{\mathit{d}y}{\mathit{d}x}=6e^{2x}+6e^{3x} \\ \mathrm{Differentiating}\mathrm{again}\mathrm{w}.\mathrm{r}.\mathrm{t}.x\mathit{,}\mathit{}\mathrm{we}\mathrm{get} \\ \frac{{\mathit{d}}^{2}y}{\mathit{d}{x}^2}=12e^{2x}+18e^{3x} \\ \Rightarrow \frac{{\mathit{d}}^{2}y}{\mathit{d}{x}^2}=5\left(6e^{2x}+6e^{3x}\right)-6\left(3e^{2x}+2e^{3x}\right) \\ \Rightarrow \frac{{\mathit{d}}^{2}y}{\mathit{d}{x}^2}=5\left(\frac{dy}{dx}\right)-6y \\ \Rightarrow \frac{{\mathit{d}}^{2}y}{\mathit{d}{x}^2}-5\left(\frac{dy}{dx}\right)+6y=0 \\ \mathrm{} \end{gathered}$$