Question

Which of the following differential equations has y = C₁ e^x + C₂ e^−x as the general solution?
(a) $\frac{d^{2}y}{d{x}^2}+y=0$

(b) $\frac{d^{2}y}{d{x}^2}-y=0$

(c) $\frac{d^{2}y}{d{x}^2}+1=0$

(d) $\frac{d^{2}y}{d{x}^2}-1=0$

Answer 1

$\left(\mathrm{b}\right)\frac{d^{2}y}{d{x}^2}-y=0$

We have,
$y=C_1{e}^x+C_2{e}^{-x}.....\left(1\right)$
Differentiating both sides of (1) with respect to x, we get
$\frac{dy}{dx}=C_1{e}^x-C_2{e}^{-x}.....\left(2\right)$
Differentiating both sides of (2) with respect to x, we get
$$\begin{gathered} \frac{d^{2}y}{d{x}^2}=C_1{e}^x+C_2{e}^{-x} \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=y\left[\text{Using}\left(1\right)\text{and}\left(2\right)\right] \\ \Rightarrow \frac{d^{2}y}{d{x}^2}-y=0 \end{gathered}$$