<s>[INST] Solve first order partial diferential equation Consider the Cauchy problem:
$$\left\{\begin{array}{lll}
x^2\partial_x u+y^2\partial_yu=u^2\\
u(x,2x)=1
\end{array}\right.$$
It is easy to show that the characteristic equations are given by:
$$\frac{dx}{x^2}=\frac{dy}{y^2}=\frac{dz}{px^2+qy^2}=\frac{dp}{2pu-2xp}=\frac{dq}{2qu-2yq}=dt$$
By the Lagrange-Charpit's method, we need a first integral $\Psi$ different to $\Phi=x^2p+y^2q-u^2$.
How I can find  $\Psi$? Maybe we should get directly the characteristic curves of the previous system?
Many thanks!
 [/INST] $$\left\{\begin{array}{lll}
x^2\partial_x u+y^2\partial_yu=u^2\\
u(x,2x)=1
\end{array}\right.$$
FIRST METHOD :
The characteristics equations are :
$$\frac{dx}{x^2}=\frac{dy}{y^2}=\frac{du}{u^2}$$
First characteristic curve, from $\frac{dx}{x^2}=\frac{dy}{y^2} \quad\to\quad \frac{1}{x}-\frac{1}{y}=c_1$
Second characteristic curve, from $\frac{dy}{y^2}=\frac{du}{u^2} \quad\to\quad \frac{1}{y}-\frac{1}{u}=c_2$
General solution expressed on implicit form :
$$\Phi\left(c_1\:,\:c_2\right)= \Phi\left(\frac{1}{x}-\frac{1}{y}\:,\:\frac{1}{y}-\frac{1}{u}\right)$$
where $\Phi$ is any differentiable function of two variables.
Solving the implicit equation for its second variable leads to :
$$\frac{1}{y}-\frac{1}{u}=F\left(\frac{1}{x}-\frac{1}{y} \right)$$
where $F$ is any differentiable function.
$$u(x,y)=\frac{1}{\frac{1}{y}-F\left(\frac{1}{x}-\frac{1}{y} \right)}$$
With condition $u(x,2x)=1 \quad\to\quad \frac{1}{\frac{1}{2x}-F\left(\frac{1}{x}-\frac{1}{2x} \right)}=1 \quad\to\quad F\left(\frac{1}{2x} \right)= \frac{1}{2x}-1$
This determines the function $F(Z)=Z-1$ and so, $F\left(\frac{1}{x}-\frac{1}{y} \right)= (\frac{1}{x}-\frac{1}{y})-1$
$$u(x,y)=\frac{1}{\frac{1}{y}-\left(\frac{1}{x}-\frac{1}{y}-1 \right)}= \frac{1}{\frac{2}{y}-\frac{1}{x}+1}= $$
$$u(x,y)=\frac{xy}{2x-y+xy} $$
SECOND METHOD :
Change : $\begin{cases} x=\frac{1}{X}\\ y=\frac{1}{Y}\\ u=\frac{1}{U}\end{cases}$ transforms the PDE : $\frac{x^2}{u^2}\frac{\partial u}{\partial x}+\frac{y^2}{u^2}\frac{\partial u}{\partial y}=1$ to :
$$\quad U_X+U_Y=1$$
Solving this elementary linear PDE leads to :
$$U=Y+f(X-Y)$$
any differentiable function $f$.
The condition $u(x,2x)=1$ is transformed to $\frac{1}{U(\frac{1}{x},\frac{1}{2x})}=\frac{1}{U(X,\frac{X}{2})}=1 \quad\to\quad U(X,\frac{X}{2} )=1$
$1=\frac{X}{2}+f(X-\frac{X}{2}) \quad\to\quad f(\frac{X}{2})=1-\frac{X}{2} $ which determines $f(Z)=1-Z$
$f(X-Y)=1-(X-Y)=1-X+Y $
$U=Y+f(X-Y)=Y+(1-X+Y)=1-X+2Y$
$\frac{1}{u}=1-\frac{1}{x}+2\frac{1}{y} \quad\to\quad u=\frac{xy}{2x-y+xy}$
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