Solve each of the following initial value problems: dy/dx - 3y cot x = sin 2x, y = 2

Question

Solve each of the following initial value problems:

\(\frac{dy}{dx}-3y\,cot\,x=sin\,2x,\,y=2\) when x = \(\frac{\pi}{2}\)

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,

∴ y = –2sin²x + csin³x

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

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

y = –2sin²x + (4)sin³x

∴ y = –2sin²x + 4sin³x

Thus, the solution of the given initial value problem is y = –2sin²x + 4sin³x