# Solving a quasi- or nonlinear PDE

Is the following PDE solvable in mathematica 9? When i solve it, the DSolve command does not do anything.

 eqn = y*D[u[x, y], x] + (x^3 + x - u[x, y])*D[u[x, y], y] == u[x, y]^2 + u[x, y];
sol = DSolve[eqn, u[x, y], {x, y}]

• Please include the Mathematica code that you are using. – Mr.Wizard Feb 1 '16 at 10:16
• z := u[x, y] p := D[u[x, y], x] q := D[u[x, y], y] eqn = y*p + (x^3 +x-z)*q == z^2 + z; sol = DSolve[eqn, z, {x, y}] – Salman Zaffar Feb 1 '16 at 10:44
• Please edit your question accordingly, and take your time to learn how to format your code (short version: four spaces indent, more: mathematica.stackexchange.com/editing-help). – Yves Klett Feb 1 '16 at 10:53
• I think the equation and its Mathematica code are both ok now....please help..... – Salman Zaffar Feb 1 '16 at 10:55
• MMA 10.3 also DSolve can not solve it. – Mariusz Iwaniuk Feb 2 '16 at 18:28

Only numerically,and that under certain circumstances.

eqn = y*D[u[x, y], x] + (x^3 + x - u[x, y])*D[u[x, y], y] == u[x, y]^2 + u[x, y];
sol = NDSolve[{eqn, u[x, 1] == -1/2, u[1, y] == -1/2}, u[x, y], {x, 1, 4}, {y, 1, 4}, PrecisionGoal -> 10,
MaxStepSize -> 0.001];

Plot3D[Evaluate[u[x, y] /. sol], {x, 1, 4}, {y, 1, 4},PlotRange -> All] Example2:

sol2 = With[{eps = 0.01},
NDSolve[{y*D[u[x, y], x] + (x^3 + x - u[x, y])*D[u[x, y], y] ==
u[x, y]^2 + u[x, y], u[x, 2] == eps, u[eps, y] == eps},
u, {x, eps, 2}, {y, eps, 2}]];
With[{eps = 0.01},
Plot3D[Evaluate[u[x, y] /. sol2], {x, eps, 2}, {y, eps, 2},
PlotRange -> All]] Example3:

 sol3 = NDSolve[{y*D[u[x, y], x] + (x^3 + x - u[x, y])*D[u[x, y], y] ==
u[x, y]^2 + u[x, y], u[1, y] == 1, u[x, 1] == 1},
u, {x, 1, 2}, {y, 1, 2}, Method -> "Shooting",
PrecisionGoal -> 2]; Plot3D[u[x, y] /. sol3, {x, 1, 2}, {y, 1, 2},
PlotRange -> All, PlotPoints -> 200] Maple a little more able to.  • Thank you for the answer...How did you come up with the initial (Cauchy) data? – Salman Zaffar Feb 3 '16 at 3:33
• I throw random initial data until only works.You can say a brute-force search. – Mariusz Iwaniuk Feb 3 '16 at 11:58

You need to specify at least two reasonable boundary conditions, such as equations for u(x,0) and u(0,y) or u(0,0) and u(L,L).

• It does not work now either. – Salman Zaffar Feb 2 '16 at 3:50