# Solve differential equation with 3 variables and plotting contour

Suppose we have this equation:

$$2 - x g(z) f \left(x, \frac{\partial y}{\partial x}\right) = 3y$$

using initial condition $y = 2$ where $g = \left| \frac{3}{2-iz} \right|$ and $f = \frac{2x}{1-3\frac{\partial y}{\partial x}}$

I want to plot the contours of $\frac{\partial y}{\partial x}$ within axes $x$ and $z$. How do I go about doing it?

I tried using $NDSolve$ but it doesn't quite work.

g := Abs[3 / (2 - iz)]
f := 2x / ( 1 - 3D[y[x,z],x] )
NDSolve [ {2-x*g*f == 3y[x,z] ,y[x,z]==2}, D[y[x,z],x], {x,0,1}, {z,0,1}  ]


Too long for a comment - Your answer works, but I improvised this to my current problem and it doesn't seem to work.. I want to find the contours of $\frac{\partial T_e}{\partial P_{local}}$ in axes $P_{probe}$ against $w$ in the limit $P_{local} \rightarrow 0, T_e \rightarrow 0.2$.

Clear["Global*"]
C1 = 10^(-10);
C2 = 0.1*C1;
R = 50;
Tb = 0.1;
Geb = 5*10^-15;
Z0 = 50;
L[Te_] := 10^-9 + 10^-9*(Te - 0.1);
Zlcr[Te_, w_] := (1/R + 1/(I*L[Te]*w) + I*C1*w)^-1;
Zload[Te_, w_] := -I*w*C2 + Zlcr[Te, w];
\[CapitalGamma][Te_, w_] := (Zload[Te, w] - Z0)/(Zload[Te, w] + Z0);
x[Te_, w_] := Abs[\[CapitalGamma][Te, w]];
y[Te_, w_] := (Abs[\[CapitalGamma][Te, w]])^2;

eqn1 [Te_, Pprobe_, w_, Plocal_] := 1 - 2 Pprobe*D[x[Te, w], Plocal] ==  Geb*D[Te, Plocal]
sol = First@NDSolve [{eqn1[0.2, Pprobe, w, 0]},{D[Te, Plocal]},{Pprobe,0,10^(-15)}, {w,0,10^9}]

Plot3D[Evaluate@D[Te, Plocal] /.  sol, {Pprobe,0,10^(-15)}, {w,0,10^9} ]


## 2 Answers

Too small to post as comment, as I am not sure about your equations.

First, You have few syntax errors. 1) it is I not i, 2) Need space, as in I z and not iz 3) You are solving for y[x,z] and not D[y[x,z],x 4) You are using y[x, z] == 2 but y[x,z] 5) do not know what y=2 is initial conditions mean. You mean y[0,z]=2 or y[x,0]=2? Also, why write g as you did, is this the same as g=3/Sqrt[4+z^2] ?

If I understand what you have, here is an attempt:

Clear[y, x, z]
f = 2 x/(1 - 3 D[y[x, z], x]);
g = 3/Sqrt[4 + z^2];
eq = 2 - x g f == 3 y[x, z];
sol = First@NDSolve[{eq, y[0, z] == 2}, {y[x, z]}, {x, 0, 1}, {z, 0, 1}]
Plot3D[Evaluate@y[x, z] /. sol, {x, 0, 1}, {z, 0, 1}] • Actually I realised it's simpler than it looks. What I meant was $y=2$ for all x and z. So The task would simply be to rearrange, and plot $\frac{\partial y}{\partial x}$ as a contour using x-z axes. – user44840 Jul 11 '14 at 9:46
• What I meant was y=2 for all x and z how could this be? y[x,z] is the solution we are after? If you know that y[x,z]=2 for all x,y, then what are we solving for? You already have the solution? I must be missing something here. – Nasser Jul 11 '14 at 9:51
• Ok this equation was derived earlier by differentiating implicitly with respect to x earlier. I want to find the contours of dy/dx given y =2 – user44840 Jul 11 '14 at 9:58

Try:

NDSolveValue[{2 - x*Abs[3/(2 - I z)]*2 x/(1 - 3 D[y[x, z], x]) ==
3 y[x, z], y[0, z] == 2}, D[y[x, z], x], {x, 0, 1}, {z, 0, 1}]


Initial conditions need to be set at a specific point, the complex is I not i.

Then plot the derivative:

Plot3D[%, {x, 0, 1}, {z, 0, 1}]
`