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.
The left-hand side does simplify to 2.
After you have expanded the terms where it looks messy, you would use the identity
eln(a)+ln(b)
=eln(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.