2.1.69 \((x+y) u_x + (x-y)u_y =0\). Problem 3.17(c) Lokenath Debnath

problem number 69

Added June 3, 2019.

Problem 3.17(c) nonlinear pde’s by Lokenath Debnath, 3rd edition.

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

\[ (x+y) u_x + (x-y)u_y =0 \]

Mathematica ✓

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

\begin{align*}& \left \{u(x,y)\to c_1\left (\log \left (-\sqrt {-x^2+2 x y+y^2}\right )\right )\right \}\\& \left \{u(x,y)\to c_1\left (\frac {1}{2} \log \left (-x^2+2 x y+y^2\right )\right )\right \}\\\end{align*}

Maple ✓

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

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

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