How do you solve this differential equation: f''(x) =c (f'(x) /f(x)) +a?

Answer 1

There were no initial conditions given? Assuming f(0) = A and f'(0) = B

1) You could use numerical analysis methods to solve

2) You could obtain a series solution using Taylor's series for example.

f(x) = f(0) + f'(0)x + f"(0)x²/2! + f"'(0)x³/3! + ..............................

or f(x) = A + Bx + ((aA+cB)/A)x²/2 + ((acA+c²B-c²B²)/A²)x³/6 + .....................

the higher order derivatives may be cumbersome and the series may converge slowly but it is a valid solution and Taylor's series do converge!