Which variables to choose in a change of variables for a PDE

Question

I want to solve the following PDE: $$\frac{\partial f}{\partial x}-3 \frac{\partial f}{\partial y}=x+y$$ with boundary condition: $$f(x,-2 x)=\sin \left(x^{2}\right)$$ Using a change of variables. How can I determine the correct variables in such a transformation?

Answer 1

$$\frac{\partial f}{\partial x}-3 \frac{\partial f}{\partial y}=x+y$$ System of characteristic ODEs (Charpit-Lagrange) : $$\frac{dx}{1}=\frac{dy}{-3}=\frac{df}{x+y}$$ A first characteristic equation from solving $\frac{dx}{1}=\frac{dy}{-3}$ : $$3x+y=c_1$$ A second characteristic equation from : $$\frac{dx}{1}=\frac{dy}{-3}=\frac{(x)dx+(-\frac13 y)dy}{1(x)-3(-\frac13 y)}=\frac{xdx-\frac13 ydy}{x+y}=\frac{df}{x+y}$$ $$xdx-\frac13 ydy=df$$ $$f-\frac12 x^2+\frac16 y^2=c_2$$ General solution of the PDE expressed on implicit form $c_2=\Phi(c_1)$ : $$f-\frac12 x^2+\frac16 y^2=\Phi(3x+y)$$ $\Phi$ is an arbitrary function, to be determined according to the boundary condition. $$f(x,y)=\frac12 x^2-\frac16 y^2+\Phi(3x+y)$$ Condition : $$f(x,-2x)=\sin(x^2)=\frac12 x^2-\frac16 (-2x)^2+\Phi(3x+(-2x))$$ $$\Phi(x)=\sin(x^2)+\frac16 x^2$$ Now the function $\Phi$ is known. We put it into the above general solution where the argument is $(3x+y)$ : $$f(x,y)=\frac12 x^2-\frac16 y^2+\sin\big((3x+y)^2\big)+\frac16 (3x+y)^2$$ After simplification : $$f(x,y)=2x^2+xy+\sin\big((3x+y)^2\big)$$