Solve each of the following initial value problems: dy/dx + y cot x = 2 cos x,

Question

Solve each of the following initial value problems:

\(\frac{dy}{dx}+y\,cot\,x=2\,cos\,x,\,y(\frac{\pi}{2})=0\)

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,

⇒ cosxdx = dt [Differentiating both sides]

By substituting this in the above integral, we get

However, when x = \(\frac{\pi}{2}, \) we have y = 0

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

Thus, the solution of the given initial value problem is y = –cosec x cot x