I have just started learning partial differential equations. I am trying to solve this question, could anyone check if my calculations are correct?
Question: Classify the PDE: $$ \sin(y) \frac{\partial \phi}{\partial x}+x \frac{\partial \phi}{\partial y}=\frac{x}{\phi} $$ and find its general solution. Find also the solution obeying $\phi=x^2$ on the line $y=3$.
Attempt: This PDE is first order, semi-linear.
The characteristic equations are: $$\frac{dx}{\phi \sin(y)}=\frac{dy}{\phi x}=\frac{d \phi}{x}$$ The latter pair $\frac{dy}{d \phi}=\frac{d \phi}{x}$ gives: $$y=\frac{x^2}{2}+C_1$$ The first and last then give: $$\frac{d\phi}{dx}=\frac{x}{\phi \sin(y)}=\frac{x}{\phi \sin(\frac{x^2}{2}+C_1)}\Longrightarrow \phi \phi'=x\csc(\frac{x^2}{2}+C_1) \Longrightarrow \frac{1}{2}\phi^2=\ln(\tan(\frac{x^2}{4}+\frac{C_1}{2}))+C_2$$ Hence, $$\phi=\pm \sqrt{2\ln(\tan(\frac{x^2}{4}+\frac{C_1}{2}))+A}$$ Each characteristic equation has a unique value of $C$, so we take $A=f(C)$, and the general solution is: $$\phi=\pm \sqrt{2\ln(\tan(\frac{x^2}{4}+\frac{y}{2}-\frac{x^2}{4}))+f(y-\frac{x^2}{2})}=\pm \sqrt{2\ln(\tan(\frac{y}{2}))+f(y-\frac{x^2}{2})}$$ Finally, when $\phi=x^2$ on the line $y=3$, we have $$x^2=\pm \sqrt{2\ln(\tan(\frac{3}{2}))+f(3-\frac{x^2}{2}}) \Longrightarrow f(3-\frac{x^2}{2})=x^4-2\ln(\tan(\frac{3}{2}))$$ So, the solution is $$\phi=\pm \sqrt{2 \ln(tan(\frac{y}{2}))+x^4-2\ln(\tan(\frac{3}{2}))}$$ Please let me know if you find any mistake in my calculations. Thanks for your attention. I am looking forward to your reply.
