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Find the general solution of the PDE $ xu_x-xyu_y-y=0 $

for all $u(x,y)$

and find the parametric form of the solution of the PDE which follows the side condition $ **u(s^2,s)=s^3** $

I got part (a) of the solution. The general solution is $ u(x,y)=-xf(ye^x) $

I have the solution that the parametric form of the PDE is $ x(s,t)=s^2e^t $ but i am not sure on how to solve it.

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From this we have $dx=-\frac{dy}{y}$ and so $x=-\ln(y)+C_1$, or in other words $\psi_1(x,y,u)=x+\ln(y)=C_1$

We also have $-\frac{dy}{x}=du$ and so $-\frac{y}{x}=u+C_2$ or in other words $\psi_2(x,y,u)=u+\frac{y}{x}=C_2$.

The general solution is then $\Phi(\psi_1(x,y,u),\psi_2(x,y,u))$ where $\Phi$ is any continuously differentiable function of $2$ variables.

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You found wrongly about the general solution.




Follow the method in

$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$

$\dfrac{dy}{dt}=-y$ , letting $y(0)=y_0$ , we have $y=y_0e^{-t}=y_0e^{-x}$

$\dfrac{du}{dt}=\dfrac{y}{x}=\dfrac{y_0e^{-t}}{t}$ , we have $u(x,y)=y_0\int^t\dfrac{e^{-t}}{t}dt+f(y_0)=ye^x\int^x\dfrac{e^{-t}}{t}dt+f(ye^x)$

$u(s^2,s)=s^3$ :



$\therefore u(x,y)=ye^x\int^x\dfrac{e^{-t}}{t}dt+f(ye^x)$ , where $f(s)$ satisfy $f(se^{s^2})=s^3-se^{s^2}\int^{s^2}\dfrac{e^{-t}}{t}dt$

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