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Math Help - 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...

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

     <br />
A= sin^{2}(x), B=-y sin(x), C=y^2<br />

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

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

     <br />
\implies ln y = ln(cot(x)+cosec(x))+C'<br />
therefore y = A(cot(x)+cosec(x)) where A=e^{C'}

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

    Check the determinant is not = 0 ie,

     <br />
\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<br />

    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 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

    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 s_{xx} term is 0

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

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

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

    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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