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

problem number 56

Added June 3, 2019.

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

Solve for \(u(x,y,z)\) \[ x(y-z)u_x+y(z-x)u_y+z(x-y)u_z=0 \]

Mathematica

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

\[\{\{u(x,y,z)\to c_1(-x y z,x+y+z)\}\}\]

Maple

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

\[u \left ( x,y,z \right ) ={\frac {{x}^{{\it \_C2}}{{\rm e}^{{\it \_C2}}}{\it \_C5}\,{y}^{{\it \_C2}}{\it \_C4}\,{z}^{{\it \_C2}}}{{\it \_C3}\,{{\rm e}^{{\it \_C1}\,x}}{{\rm e}^{{\it \_C1}\,y}}{{\rm e}^{{\it \_C1}\,z}}}}\]

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