I need to transform my equation into two first-order equation. But I don't know how to do that. My equation is the Gibbons-Tsarev equation : \begin{equation} \frac{\partial u}{\partial t}\frac{\partial^{2} u}{\partial x\partial t} - \frac{\partial u}{\partial x}\frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial^{2} u}{\partial x^{2}} +1 = 0 \end{equation}
by posing $y_1=u$ et $y_2=∂u/∂t$
We have the analytical solution : \begin{equation} u(x,t) = \frac{c_{1}}{c_{2}} e^{c_{2}x} + \frac{x}{c_{2}} + {c_{2}} + 0.5{c_{2}} t^{2} + {c_{3}}t + {c_{4}} \end{equation}
with: 0 $\le$ x $\le$ 2 and 0 $\le$ t $\le$ 1 and c1=c2=c3=c4=1
I need to solve this problem with matlab by using ode45.
Thank you