Question

Can someone please work this algebra.

verify that y(x) = -ln (x - cx^2) satisfies the ODE x(e^y - y') = 2 where c is constant.

I have been working through this and it is way too messy.

Answer 1

The left-hand side does simplify to 2.

After you have expanded the terms where it looks messy, you would use the identity
e^ln(a)+ln(b)
=e^ln(ab)
=ab
which will greatly simplify the exponential expression to a simple algebraic expression.

One of the expressions in question is:
e^{(log(1-cx)+log(x))}
=e^{(log(x(1-cx)))}
=x(1-cx)

It takes a little patience but it can be done.