2.1.27 $$x u_x+y u_y=n u$$ Example 3.5.2 in Lokenath Debnath

problem number 27

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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