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Thread: Parabolic Linear 2nd Order PDE with Trig terms

  1. #1
    Senior Member bugatti79's Avatar
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    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?..
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  2. #2
    MHF Contributor
    Jester's Avatar
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    Looking good - keep going :-)
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  3. #3
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by Danny View Post
    Looking good - keep going :-)
    I had no idea what I was letting myself in for...
    The derivatives are turning out to be a monster of a task! Particualrly t_xx

    $\displaystyle \displaystyle u_x=w_s s_x+w_t t_x=w_s+\overbrace{y \left[\frac{csc^2(x)+csc(x)cot(x)}{(cot(x)+csc(x))^2}\ri ght ]}^{A}w_t$

    $\displaystyle u_{xx}=s_x(w_{ss}s_x+w_{ts}t_x)+t_x(w_{st}s_x+w_{t t}t_x)+w_tt_{xx}$

    The $\displaystyle s_{xx}$ term is 0

    $\displaystyle =w_{ss}+2Aw_{ts}+A^2w_t+w_t t_{xx}$

    $\displaystyle \displaystyle u_y=w_t t_y=w_t\left [\frac{1}{cot(x)+csc(x)}\right ]$

    $\displaystyle \displaystyle u_{yy}=t_y(w_{tt} t_y)= \frac{w_{tt}}{(cot(x)+csc(x))^2}$

    $\displaystyle u_{xy}=w_{st}s_xt_y+w_{tt}t_xt_y+w_{tt}t_{xy}$

    I have attached the derivation of t_xx

    I have no idea where to start in order to simplify it. I am not sure if I will continue on with this unless I get some encouragement. I think its beyond the scope of a possible exam question!
    Attached Thumbnails Attached Thumbnails Parabolic Linear 2nd Order PDE with Trig terms-imag0181.jpg  
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