#### 2.1.26 $$x u_x+y u_y=u$$ (Example 3.5.1 in Lokenath Debnath)

problem number 26

From example 3.5.1, page 210 nonlinear pde’s by Lokenath Debnath, 3rd edition.

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

Mathematica

$\left \{\left \{u(x,y)\to x c_1\left (\frac {y}{x}\right )\right \}\right \}$

Maple

$u \left (x , y\right ) = x \mathit {\_F1} \left (\frac {y}{x}\right )$

Hand solution

Solve $xu_{x}+yu_{y}=u$ Using the Lagrange-charpit method$\frac {dx}{x}=\frac {dy}{y}=\frac {du}{u}$ The ﬁrst pair of equations gives $x=C_{1}y$ And $$\frac {dx}{x}=\frac {du}{u}$$ gives$x=C_{2}u$ Since $$C_{2}=G\left ( C_{1}\right )$$ then $$\frac {x}{u}=G\left ( \frac {x}{y}\right )$$ or $$u=xG^{-1}\left ( \frac {x}{y}\right )$$. Let $$G^{-1}=F$$. Then the solution $u\left ( x,y\right ) =xF\left ( \frac {x}{y}\right )$

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