$e^c = c$ differential equation

Question

is $e^c=c$? link of differential equation : https://scontent-vie1-1.xx.fbcdn.net/v/t34.0-12/14658402_1227936723934107_1724693471_n.jpg?oh=bd47ff94556ad5200117ae40751929f1&oe=58020475 my friends are telling me that I made a mistake because $e^x=c$, so I need another opinion.

Answer 1

No, your solution is correct for any $c \in \mathbb R$. I guess they want to tell you the following: You can rewrite the solution as $$y(x) = \frac{e^{2x} + c}{2} - 1$$ where $c > 0$. This is a common trick when you solve differential equations. You just rename constants simplify the solutions. Please take note that you have to assume $c > 0$ because $e^{2c} > 0$. If you have $c < 0$ then $y(x) = \frac{e^{2x} + c}{2} - 1$ is no solution (!) of your differential equation.

I hope it help you :)