2.1.21 Clairaut equation. \(x u_x + y u_y + \frac {1}{2} ( (u_x)^2+ (u_y)^2 ) = 0\) with \(u(x,0)= \frac {1}{2} (1-x^2)\)

problem number 21

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


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

Maple


\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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