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Thread: 1.6 Verify the differential equation is = to the solution

  1. #1
    Super Member bigwave's Avatar
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    1.6 Verify the differential equation is = to the solution

    Verify for each of the following differential equations that given function or functions are solutions:

    $\textbf{(a)}$
    \begin{align*}\displaystyle
    u_{xx}+u_{yy}&=0\\
    u_1(x,y)&=x^2-y^2\\
    u_2(x,y)&=\cos{x}cosh{y}
    \end{align*}
    $\textbf{(b) $\lambda$ a real constant }$
    \begin{align*}\displaystyle
    a^2u_{xx}&=u_t\\
    u_1(x,t)&=e^{-\alpha^2t}\sin{x}\\
    u_2(x,t)&=e^{-\alpha^2\lambda^2t}\sin{\lambda x}
    \end{align*}
    $\textbf{(c) $\lambda$ a real constant }$
    \begin{align*}\displaystyle
    a^2u_{xx}&=u_{tt}\\
    u_1(x,t)&=\sin{\lambda x}\sin{\lambda \alpha t}\\
    u_2(x,t)&=\sin{(x-\alpha t)}
    \end{align*}

    ok I have never done this before and just trying to learn as much as I can before I take the class
    I read the chapter but it was kinda ???

    much thank you ahead..
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  2. #2
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    Re: 1.6 Verify the differential equation is = to the solution

    Quote Originally Posted by bigwave View Post
    Verify for each of the following differential equations that given function or functions are solutions:

    $\textbf{(a)}$
    \begin{align*}\displaystyle
    u_{xx}+u_{yy}&=0\\
    u_1(x,y)&=x^2-y^2\\
    u_2(x,y)&=\cos{x}cosh{y}
    \end{align*}

    much thank you ahead..
    ${u_1}_x =2x;~{u_1}_{xx} = 2$

    ${u_1}_y = -2y;~{u_1}_{yy} = -2$

    ${u_1}_{xx} + {u_1}_{yy} = 2+(-2) = 0$

    similarly

    ${u_2}_{xx} = -\cos(x)\cosh(y)$

    ${u_2}_{yy} = \cos(x)\cosh(y)$

    ${u_2}_{xx} + {u_2}_{yy} = 0$

    you can figure out the rest
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  3. #3
    Super Member bigwave's Avatar
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    Re: 1.6 Verify the differential equation is = to the solution

    so what is $a^2$ and $u_t$

    in (b) and (c)
    Last edited by bigwave; Mar 2nd 2018 at 10:36 AM.
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    Re: 1.6 Verify the differential equation is = to the solution

    Quote Originally Posted by bigwave View Post
    so what is $a^2$ and $u_t$

    in (b) and (c)
    $a^2$ is just some constant

    oh I see the confusion. I bet they meant to use either just $a$ or just $\alpha$. Work the problem assuming $a=\alpha$ and see if it works out.

    $u_t = \dfrac{du}{dt}$
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  5. #5
    Super Member bigwave's Avatar
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    Re: 1.6 Verify the differential equation is = to the solution

    ok i think alpha in all cases
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