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Math Help - Substituting

  1. #1
    Senior Member I-Think's Avatar
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    Substituting

    Show that, with a suitable value of the constant \alpha, the substitution y=x^{\alpha}w reduces the differential equation

    2x^{2}\frac{d^{2}y}{dx^2}+(3x^2+8x)\frac{dy}{dx}+(  x^2+6x+4)=f(x)

    to

    2\frac{d^{2}w}{dx^2}+3\frac{dw}{dx}+w=f(x)

    Preparing for an exam,so all help is greatly appreciated

    Edit
    Ooops.Forgot the y
    The expression is
    2x^{2}\frac{d^{2}y}{dx^2}+(3x^2+8x)\frac{dy}{dx}+(  x^2+6x+4)y=f(x)
    Last edited by I-Think; October 6th 2009 at 05:50 AM.
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  2. #2
    Flow Master
    mr fantastic's Avatar
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    Quote Originally Posted by I-Think View Post
    Show that, with a suitable value of the constant \alpha, the substitution y=x^{\alpha}w reduces the differential equation

    2x^{2}\frac{d^{2}y}{dx^2}+(3x^2+8x)\frac{dy}{dx}+(  x^2+6x+4)=f(x)

    to

    2\frac{d^{2}w}{dx^2}+3\frac{dw}{dx}+w=f(x)

    Preparing for an exam,so all help is greatly appreciated
    Get \frac{dy}{dx} using the product rule. Get \frac{d^2y}{dx^2} by using the product rule to differentiate your expression for \frac{dy}{dx}.

    Substitute your expressions for \frac{dy}{dx} and \frac{d^2y}{dx^2} into the DE and simplify.

    If you need more help please show what you've done and say where you get stuck.
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  3. #3
    Senior Member I-Think's Avatar
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    Still in a spot of bother over the differentiation process.

    If  y=x^{\alpha}w
    Using the product rule

    \frac{dy}{dx}=\alpha{wx^{\alpha-1}}+x^{\alpha}{\frac{dw}{dx}}

    And

    \frac{d^{2}y}{dx^2}=\alpha{x^{\alpha-1}}\frac{dw}{dx}+w(\alpha^2-\alpha)x^{\alpha-2}+\alpha{x^{\alpha-1}}\frac{dw}{dx}+x^\alpha{\frac{d^2w}{dx^2}}

    When I substitute this into the equation, I'm not getting the correct answer. May I have more guidance please?
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  4. #4
    Flow Master
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    Quote Originally Posted by I-Think View Post
    Still in a spot of bother over the differentiation process.

    If  y=x^{\alpha}w
    Using the product rule

    \frac{dy}{dx}=\alpha{wx^{\alpha-1}}+x^{\alpha}{\frac{dw}{dx}}

    And

    \frac{d^{2}y}{dx^2}=\alpha{x^{\alpha-1}}\frac{dw}{dx}+w(\alpha^2-\alpha)x^{\alpha-2}+\alpha{x^{\alpha-1}}\frac{dw}{dx}+x^\alpha{\frac{d^2w}{dx^2}}

    When I substitute this into the equation, I'm not getting the correct answer. May I have more guidance please?
    Have you checked for careless mistakes?
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