Question

$y=a{e}^{mx}+b{e}^{-mx}$ satisfies which of the following differential equation?

• A. $\frac{dy}{dx}+my=0$
• B. $\frac{dy}{dx}-my=0$
• C. $\frac{d^{2}y}{d{x}^2}-m^{2}y=0$
• D. $\frac{d^{2}y}{d{x}^2}+m^{2}y=0$

Answer 1

Answer: (C)

$y=a{e}^{mx}+b{e}^{-mx}\dots \left(i\right)$
Differentiating w.r.t. $x$, we get
$\frac{dy}{dx}=ma{e}^{mx}-mb{e}^{-mx}=m\left(a{e}^{mx}-b{e}^{-mx}\right)$
Again differentiating w.r.t. $x$, we get
$\frac{d^{2}y}{d{x}^2}=m\left(am{e}^{mx}+bm{e}^{-mx}\right)$
$=m^2\left(a{e}^{mx}+b{e}^{-mx}\right)=m^{2}y$ [using $(i)$]
$\Rightarrow \frac{d^{2}y}{d{x}^2}-m^{2}y=0$