How to solve the following system of partial differential equations?

Question

I have a system of partial differential equations: \begin{align} & u(a,b,c) \frac{\partial y}{\partial c} = \frac{4}{3} ab, \\ & u(a,b,c) \frac{\partial y}{\partial b} = \frac{2}{3} ac + 2 b^2, \\ & u(a,b,c) \frac{\partial y}{\partial a} = \frac{4}{3} bc. \end{align} I tried to solve it using maple as follows. First I define \begin{align} pde := u(a, b, c)*(diff(y(a, b, c), c)) = (4/3)*a*b, \\ u(a, b, c)*(diff(y(a, b, c), b)) = (2/3)*a*c+2*b^2, \\ u(a, b, c)*(diff(y(a, b, c), a)) = (4/3)*b*c \end{align} Then I use the command: pdsolve(pde). But there is an error. How to solve this system of equations using maple? Any help would be greatly appreciated!

Answer 1

This is an atempt to solve the PDE system, leading to : $$y(a,b,c)=F\left(a\,c\,b^{\frac{1}{2}}+\frac{3}{5}b^{\frac{5}{2}} \right)$$ with $F$ arbitrary derivable function until no bounding condition is specified.

Answer 2

This approach is heuristic, but it gets you a solution.

The first and third equations, derivatives with respect to $a$ and $c$ integrate out the same, so it makes sense to let $u(a,b,c)=u(b)$.

You can get a solution by assuming that $u(b)=b^n$. Then you have to solve for the right value of $n$ to make the derivatives match an exact differential.

Answer 3

The command "pdsolve([pde]);" can solve this system of partial differential equations.