Here is my solution.

**Solution. **Let

, then by the chain rule we have

.

Hence the given differential equation reduces to

........(1)

Clearly, this equation is not exact since

, where

and

.

The integrating factor for this equation is

Grouping the terms after multiplying through (1) by

, we get

or equivalently

From this, we are lead to

which gives us

by using the relation

.

This is the desired solution to (1).

....
It seems not possible to solve the last integral.

I think there may be other solutions for this equation which may give explicit result.