Parabolic Linear 2nd Order PDE with Trig terms

Hi Mathematicians!,

Make a suitable change of variables and hence transform to its canonical form. Heres goes...

$\displaystyle

Sin^{2}(x)u_{xx} -2ysin(x)u_{xy}+y^2u_{yy}=0

$

$\displaystyle

A= sin^{2}(x), B=-y sin(x), C=y^2

$

The discrimant B^2-AC is = $\displaystyle y^2 sin^{2}(x)-sin^{2}(x)(y^2)=0$ making it parabolic.

$\displaystyle

\displaystyle \implies \frac{dy}{dx}=\frac{B\pm\sqrt{B^2-AC}}{A}=\frac{-y}{sin(x)}

$

$\displaystyle

\implies ln y = ln(cot(x)+cosec(x))+C'

$ therefore $\displaystyle y = A(cot(x)+cosec(x)) $where $\displaystyle A=e^{C'}$

Let $\displaystyle \displaystyle t(x,y)= \frac{y}{cot(x)+cosec(x)} $and $\displaystyle s(x,y)=x$

Check the determinant is not = 0 ie,

$\displaystyle

\displaystyle det\begin{bmatrix}\frac{\partial s}{\partial x} &\frac{\partial s}{\partial y}\\ \frac{\partial t}{\partial x} &\frac{\partial t}{\partial y}\end{bmatrix}=\frac{1}{cot(x)+cosec(x)}\neq 0

$

How am I doing so far?..