# How to solve following system of partial differential equations?

Let's have a system of equations $$\begin{matrix} p_{1} - 2p_{2} - p_{4}x^{2}_{4}(x_{1} + 1) = 0 \\ p_{1} + p_{2}x_{1} + p_{3}x_{1}x_{3}^{2} = 0 \end{matrix}, \quad p_{i} = \frac{\partial u}{\partial x_{i}}.$$ How to solve it? I can't simply express $p_{i}$ from first equation and substitute it to second.

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Why can't you just substitute $p_2$ from the first into the second? – vonbrand Feb 23 '13 at 14:27
The answer is $$u = f(x_{1}^2 - x_{2} + \frac{1}{x_{3}} - \frac{2}{x_{4}}).$$ By substituting $p_{2}$ I only decrease the number of dimensions of equation's variables by one. So I don't know what to do. – John Taylor Feb 23 '13 at 15:30
So you know the answer? What do you want then? Prove all solutions are of this form, ...? – vonbrand Feb 23 '13 at 16:14
I want to understand the solution. After substitution you write I can get an answer $$u = f(g(x_{i}), \varphi (x_{i})),$$ but not the answer $$u = f(\psi (x_{i})).$$ – John Taylor Feb 23 '13 at 16:20