2.1.56 \(y z u_x - x z u_y+ x y (x^2+y^2) u_z=0\) Problem 3.8(d) Lokenath Debnath

problem number 56

Added June 3, 2019.

Problem 3.8(d) nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for \(u(x,y,z)\)

\[ y z u_x - x z u_y+ x y (x^2+y^2) u_z=0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  y*z*D[u[x, y,z], x] - x*z*D[u[x, y,z], y]+x*y*(x^2+y^2)*D[u[x, y,z], z]== 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde ,u[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{u(x,y,z)\to c_1\left (\frac {1}{2} \left (x^2+y^2\right ),\frac {1}{2} \left (-x^4-x^2 y^2+z^2\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=y*z*diff(u(x,y,z),x) - x*z*diff(u(x,y,z),y)+x*y*(x^2+y^2)*diff(u(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y,z))),output='realtime'));
 

\[u \left (x , y , z\right ) = f_{1} \left (x^{2}+y^{2}, -x^{4}-x^{2} y^{2}+z^{2}\right )\]

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