Solve each of the following initial value problems: x(dy/dx) + y = x cos x + sin x,

Question

Solve each of the following initial value problems:

\(x\frac{dy}{dx}+y=x\,cos\,x\) \(+\,sin\,x, y(\frac{\pi}{2})=1\)

Answer 1

This is a first order linear differential equation of the form

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

Hence, the solution of the differential equation is,

However, when \(x=\frac{\pi}{2},\) we have y = 1

By substituting the value of c in the equation for y, we get

Thus, the solution of the given initial value problem is y = sin x