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Thread: How to solve this ODE?

  1. #1
    Senior Member Vinod's Avatar
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    How to solve this ODE?

    Hello,
    $y'+x\sqrt{y}=x^2$

    Solution:- This ODE is already in standard form.Thus P(x)=x. And we have$\int P(x)dx=\int x dx=\frac{x^2}{2}$ So integrating factor is $e^{\int P(x)dx}=e^{\frac{x^2}{2}}$

    Therefore,multiplying both sides of standard form by $e^{\frac{x^2}{2}}$ yields

    $e^{\frac{x^2}{2}}y'+x*\sqrt{y}e^{\frac{x^2}{2}}=x ^2*e^{\frac{x^2}{2}}$

    $\frac{d[\sqrt{y}*e^{\frac{x^2}{2}}]}{dx}=x^2*e^{\frac{x^2}{2}}$

    Now, integrating both sides we get

    $\sqrt{y}*e^{\frac{x^2}{2}}=\int x^2*e^{\frac{x^2}{2}}$

    How to proceed further?

    I think integral of right side of this equation contains error function. How to solve that integral?
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: How to solve this ODE?

    Are you sure you've presented the ODE correctly? You're treating it as if it is linear, when it's not, at least as given.
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  3. #3
    Senior Member Vinod's Avatar
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    Re: How to solve this ODE?

    Quote Originally Posted by MarkFL View Post
    Are you sure you've presented the ODE correctly? You're treating it as if it is linear, when it's not, at least as given.
    Hello,
    You mean to say the given equation is not in a standard form and my standard form is incorrect. So shall i square the whole euqtion? and try to solve it.
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: How to solve this ODE?

    It's not a linear ODE. Best I can tell, it's some kind of Chiri's equation.
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  5. #5
    Senior Member Vinod's Avatar
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    Re: How to solve this ODE?

    Quote Originally Posted by MarkFL View Post
    It's not a linear ODE. Best I can tell, it's some kind of Chiri's equation.
    Hello,
    What is that? Would you explain something more about that?
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  6. #6
    MHF Contributor MarkFL's Avatar
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    Re: How to solve this ODE?

    Quote Originally Posted by Vinod View Post
    Hello,
    What is that? Would you explain something more about that?
    It's a type of equation I know nothing about. We'll have to wait for someone more knowledgeable to come along and show us how to solve it.
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