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

problem number 55

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'));
 

\[u \left (x , y , z\right ) = \frac {c_{4} c_{5} x^{c_{2}} y^{c_{2}} z^{c_{2}} {\mathrm e}^{\left (-x -y -z \right ) c_{1} +c_{2}}}{c_{3}}\]

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