2.1.54 \(x^2 u_x+y^2 u_y+z(x+y)u_z=0\) Problem 3.8(b) Lokenath Debnath

problem number 54

Added June 3, 2019.

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

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x^2*D[u[x, y,z], x] +y^2*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]];
 

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

Maple ✓

restart; 
pde :=x^2*diff(u(x,y,z),x)+y^2*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))),output='realtime'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________