#### 2.5.1 Inviscid Burgers $$u_x + u u_y = 0$$

problem number 91

Taken from Mathematica Symbolic PDE document

quasilinear ﬁrst-order PDE, scalar conservation law

Solve for $$u(x,y)$$ $u_x + u u_y = 0$

Mathematica

$\left \{\left \{u(x,t)\to x^4-12 t^2\right \}\right \}$ Implicit solution

Maple

$x u \left (x , y\right )-y +\mathit {\_F1} \left (u \left (x , y\right )\right ) = 0$

Hand solution

Solve for $$u\left ( x,y\right )$$ in $$u_{x}+u\ u_{y}=0.$$ Using the Lagrange-Charpit method, the characteristic equations are$\frac {dx}{1}=\frac {dy}{u}=\frac {du}{0}$ From the ﬁrst pair of equation we obtain$u=\frac {dy}{dx}$ But $$du=0$$ or $$u=C_{2}$$. Hence the above becomes

\begin {align*} \frac {dy}{dx} & =C_{2}\\ y & =xC_{2}+C_{1}\\ C_{1} & =y-xC_{2} \end {align*}

Since $$C_{2}=F\left ( C_{1}\right )$$ where $$F$$ is arbitrary function, then $u\left ( x,y\right ) =F\left ( y-ux\right )$

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