Answer to Question #200107 in Differential Equations for Kevin

Question

1)form a pde by eliminating the arbitrary function from z=exp(2x+3y)f(2x-3y)

Answer 1

We multiply by $exp(-(2x+3y))$ and receive: $z\cdot exp(-(2x+3y))=f(2x+3y)$. After denoting $w=z\cdot exp(-(2x+3y))$ we receive: $w_x=2f'$ , $w_y=3f'$. We receive the equation: $3w_x=2w_y$, $w_x=z_x\cdot exp(-(2x+3y))-2z\cdot exp(-(2x+3y))$ and $w_y=z_y\cdot exp(-(2x+3y))-3z\cdot exp(-(2x+3y))$. Finally, we receive: $3z_x\cdot exp(-(2x+3y))-6z\cdot exp(-(2x+3y))=2z_y\cdot exp(-(2x+3y))-6z\cdot exp(-(2x+3y))$

Thus, finally, we get the equation: $3z_x=2z_y$