Question

The differential equation whose solution is $y=c_{1}cosax+c_{2}sinax$ is
(Where $c_1,c_2$ are arbitrary constants)

[MP PET 1996]

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

Answer 1

The correct option is B $\frac{d^{2}y}{d{x}^2}+a^{2}y=0$

$y=c_{1}cosax+c_{2}sinax$
Differentitate it w.r.t.x, we get
$\frac{dy}{dx}=-c_{1}asinax+c_{2}acosax$
Again $\frac{d^{2}y}{d{x}^2}=-c_1{a}^{2}cosax-c_2{a}^{2}sinax$
$\frac{d^{2}y}{d{x}^2}=-a^2(c_{1}cosax+c_{2}sinax)\Rightarrow \frac{d^{2}y}{d{x}^2}=-a^{2}y$
or $\frac{d^{2}y}{d{x}^2}+a^{2}y=0.$