Solve the following differential equations:

Given

This is a first order linear differential equation of the form

Here, and Q = cos x

The integrating factor (I.F) of this differential equation is,

We have

[∵ m log a = log a^m]

∴ I.F = x² [∵ e^{log x} = x]

Hence, the solution of the differential equation is,

Recall

⇒ yx² = x²sin x – 2{–x cos x + sin x} + c

⇒ yx² = x²sin x + 2x cos x – 2 sin x + c

Thus, the solution of the given differential equation is