2.1.27 \(x u_x+y u_y=n u\) Example 3.5.2 in Lokenath Debnath

problem number 27

Added June 2, 2019.

From example 3.5.2, page 211 nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for \(u(x,y)\) \begin {align*} x u_x+y u_y&=n u \end {align*}

Mathematica


\[\left \{\left \{u(x,y)\to x^n c_1\left (\frac {y}{x}\right )\right \}\right \}\]

Maple


\[u \left (x , y\right ) = x^{n} \mathit {\_F1} \left (\frac {y}{x}\right )\]

Hand solution

Solve \[ xu_{x}+yu_{y}=nu \] Using the Lagrange-charpit method\[ \frac {dx}{x}=\frac {dy}{y}=\frac {du}{nu}\] The first pair of equations gives \[ x=C_{1}y \] And \(\frac {dx}{x}=\frac {du}{nu}\) gives\begin {align*} \ln x & =\frac {1}{n}\ln u+C_{2}\\ x & =C_{2}u^{\frac {1}{n}}\\ x^{n} & =C_{3}u \end {align*}

Since \(C_{3}=G\left ( C_{1}\right ) \) then \(\frac {x^{n}}{u}=G\left ( \frac {x}{y}\right ) \) or \(u=x^{n}G^{-1}\left ( \frac {x}{y}\right ) \). Let \(G^{-1}=F\). Then the solution \[ u\left ( x,y\right ) =x^{n}F\left ( \frac {x}{y}\right ) \]

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