The differential equation whose solution is $y=c_{1} \operatorname{cosax}+c_{2}…
The differential equation whose solution is $y=c_{1} \operatorname{cosax}+c_{2} \operatorname{sinax}\left(\right.$ Where $c_{1}$
and $c_{2}$ are arbitrary constants) is
$\frac{d^{2} y}{d x^{2}}-a^{2} y=0$
$\frac{d^{2} y}{d x^{2}}+a^{2} y=0$
$\frac{d^{2} y}{d x^{2}}+a y^{2}=0$
$\frac{d^{2} y}{d x^{2}}+y^{2}=0$
Solution
The correct option is \(B \frac{d^2 y}{d x^2}+a^2 y=0\) \(\mathrm{y}=\mathrm{c}_1 \cos \mathrm{ax}+\mathrm{c}_2 \sin \mathrm{ax}\)
Differentitate it w.r.t.x, we get \(\frac{d y}{d x}=-c_1 a \sin a x+c_2 a \cos a x\)
Again \(\frac{d^2 y}{d x^2}=-c_1 a^2 \cos a x-c_2 a^2 \sin a x\)
\(\frac{d^2 y}{d x^2}=-a^2\left(c_1 \cos a x+c_2 \sin a x\right)\)
\(\Rightarrow \frac{d^2 y}{d x^2}=-a^2 y\) or \(\frac{d^2 y}{d x^2}+a^2 y=0\)