# ordinary differential equations – finding terms in ODE

Consider the equation

$$frac{partial{y}}{partial{t}}+yfrac{partial{y}}{partial{x}}=Gamma frac{partial^2y}{partial{x}^2}$$

where $$Gamma$$ is a constant term.

Given that $$y=frac{y_0}{2}-frac{y_0}{2}tanh(frac{x-ct}{L})$$ satisfies the equation above, find $$L$$ and $$c$$ in terms of the constants $$y_0$$ or $$Gamma$$

My attempt:

$$frac{partial{y}}{partial{x}}=frac{-y_0}{2L}sech^2(Phi)$$ where $$Phi=frac{x-ct}{L}$$.

$$frac{partial{y}}{partial{t}}=frac{y_0c}{2L}sech^2(Phi)$$

$$frac{partial^2{y}}{partial{x}^2}=frac{y_0}{L^2}sech^2(Phi)tanh(Phi)$$

Hence plugging them into the equation above we get

$$1-frac{y}{c}=Gamma frac{2}{cL}tanh(Phi)$$, so $$1-frac{y_0}{2c}=frac{4Gamma-y_0L}{2cL}tanh(Phi)$$

What else should I do?