Question
The general solution of the differential equation $$\frac{{{d^2}y}}{{d{x^2}}} = \cos \,nx$$ is :
[Where $$C$$ and $$D$$ are arbitrary constants]
A.
$${n^2}y + \cos \,nx = {n^2}\left( {Cx + D} \right)$$
B.
$${n^2}y - \sin \,nx = {n^2}\left( { - Cx + D} \right)$$
C.
$${n^2}y + \cos \,nx = \frac{{Cx + D}}{{{n^2}}}$$
D.
none of these
Answer :
$${n^2}y + \cos \,nx = {n^2}\left( {Cx + D} \right)$$
Solution :
The differential equation is $$\frac{{{d^2}y}}{{d{x^2}}} = \cos \,nx$$
Integrating we get
$$\frac{{dy}}{{dx}} = \frac{{\sin \,nx}}{n} + C.....\left( {\text{i}} \right)$$
Integrating again
$$\eqalign{
& y = - \frac{{\cos \,nx}}{{{n^2}}} + Cx + D \cr
& \Rightarrow {n^2}y + \cos \,nx = {n^2}\left( {Cx + D} \right) \cr} $$