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Thread: IVP and interval of validity

  1. #1
    Senior Member Vinod's Avatar
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    IVP and interval of validity

    I want to solve the following IVP and also want to find interval of validity of the solution.
    $y'=\frac{xy^3}{\sqrt{1+x^2}} y(0)=-1$

    Solution:
    First separate and then integrate both sides.$y^{-3}dy=x(1+x^2)^{0.5}dx$

    $\int y^{-3}dy=\int x(1+x^2)^{0.5} dx$

    $\frac{-1}{2y^2}=\frac{\sqrt{(1+x^2)^3}}{3} +c$

    Applying initial condition to get the value of c.

    $\frac{-1}{2}=\frac{\sqrt{1}}{3}+c. c=\frac{-5}{6}$

    The implicit solution is then

    $\frac{-1}{2y^2}=\frac{\sqrt{(1+x^2)^3}}{3}-\frac56$

    Now let's solve for y(x).

    $\frac{1}{y^2}=\frac{-5}{3}-\frac23 \sqrt{(1+x^2)^3}$

    $y^2=\frac{3}{5-2\sqrt{(1+x^2)^3}}$

    $y(x)=\pm\frac{\sqrt{3}}{\sqrt{5-2\sqrt{(1+x^2)^3}}}$

    Reapplying the initial condition shows us that the "-" is the correct sign. The explicit solution is then,$y(x)=\frac{-\sqrt{3}}{\sqrt{5-2\sqrt{(1+x^2)^3}}}$

    I am not giving full computation for the interval of validity.
    $x^2=(\frac{25}{4})^{0.3333333}-1$

    Hence -0.917614<x<0.917614 is the interval of validity of the solution.

    But answer provided to me do not match with my calculated answer. Would any one find here what is wrong with my calculations?
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  2. #2
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    Re: IVP and interval of validity

    Looks correct to me. What answer was provided to you?
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  3. #3
    Senior Member Vinod's Avatar
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    Re: IVP and interval of validity

    Quote Originally Posted by SlipEternal View Post
    Looks correct to me. What answer was provided to you?
    Hello,
    The explicit solution is $y(x)=\frac{1}{\sqrt{3-2\sqrt{(1+x^2)}}}$

    And the interval of validity of solution is $\frac{-\sqrt{5}}{2}<x<\frac{\sqrt{5}}{2}$

    Now I am putting this problem in wolfram alpha to verify the answer.
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  4. #4
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    Re: IVP and interval of validity

    Oh, you forgot a negative exponent before integrating $x(1+x^2)^{-1/2}$
    Thanks from Vinod
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