Taken from Mathematica Symbolic PDE document
Clairaut equation with initial value
Solve for \(u(x,y)\) \[ x u_x + y u_y + \frac {1}{2} ( (u_x)^2+ (u_y)^2 ) = 0 \] With \(u(x,0)= \frac {1}{2} (1-x^2)\)
Mathematica ✓
ClearAll["Global`*"]; pde = u[x, y] == x*D[u[x, y], {x}] + y*D[u[x, y], {y}] + (1/2)*(D[u[x, y], {x}]^2 + D[u[x, y], {y}]^2); ic = u[x, 0] == (1*(1 - x^2))/2; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, ic}, u[x, y], {x, y}], 60*10]];
\[\left \{\left \{u(x,y)\to -\frac {x^2}{2}+y+\frac {1}{2}\right \}\right \}\]
Maple ✓
restart; interface(showassumed=0); pde := x*diff(u(x, y), x) + y*diff(u(x, y),y) + 1/2 * ( diff(u(x, y), x)^2 + diff(u(x, y), y)^2)=0; ic := u(x,0)=1/2*(1-x^2); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic],u(x,y))),output='realtime'));
\begin {align*} & u \left ( x,y \right ) =-{\frac { \left ( x-y+1 \right ) \left ( x-y-1 \right ) }{2}}\\& u \left ( x,y \right ) =-{\frac { \left ( x+y+1 \right ) \left ( x+y-1 \right ) }{2}}\\ \end {align*}
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