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Thread: 2nd order DE (non linear I think but homogeneous)

  1. #1
    MHF Contributor arbolis's Avatar
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    2nd order DE (non linear I think but homogeneous)

    Find the general solution of $\displaystyle x^2y''-5xy'+9y=0$. Hint: Once known a solution $\displaystyle y_1(x)$, another solution is given by $\displaystyle y(x)=v(x)y_1(x)$, with $\displaystyle v(x)$ to determinate.
    Attempt: I just don't know what method to use. I don't think variation of parameters work here since it's non linear.
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    Rhymes with Orange Chris L T521's Avatar
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    Its a Cauchy-Euler equation.

    So suppose that the equation $\displaystyle ax^2y^{\prime\prime}+bxy^{\prime}+cy=0$ has a solution of the form $\displaystyle y=x^r$. Plugging this back into the ODE yields the characteristic equation $\displaystyle ar^2+(b-a)r+c=0$.

    In this particular problem, $\displaystyle x^2y^{\prime\prime}-5xy^{\prime}+9y=0$ has the characteristic equation $\displaystyle r^2-6r+9=0$, which yields repeated roots.

    So once you have $\displaystyle y_1(x)$, you suppose $\displaystyle y_2(x)=u(x)y_1(x)$. It turns out that once you write the ODE in the form $\displaystyle y^{\prime\prime}+P(x)y^{\prime}+Q(x)y=0$ and substitute $\displaystyle y_2(x)=u(x)y_1(x)$ into it, we see that $\displaystyle u(x)=\displaystyle\int\left[\frac{e^{-\int P(x)dx}}{[y_1(x)]^2}\right]dx$ (this is the reduction of order technique as seen in my tutorial here.)

    Then at this point, we see that $\displaystyle y_2(x)=\left(\displaystyle\int\left[\frac{e^{-\int P(x)dx}}{[y_1(x)]^2}\right]dx\right)y_1(x)$ is the second solution.

    Can you continue with the problem?
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    MHF Contributor arbolis's Avatar
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    Thanks for the help, I will try. However isn't the characteristic equation $\displaystyle r^2-5r+9=0$ instead of $\displaystyle r^2-6r+9=0$? So that the roots aren't repeated. I should solve both problems anyway.
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    Rhymes with Orange Chris L T521's Avatar
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    Let $\displaystyle a=1$, $\displaystyle b=-5$ and $\displaystyle c=9$. In this particular case, it has to be $\displaystyle r^2+(-5-1)r+9=r^2-6r+9=0$. Observe that $\displaystyle ax^2y^{\prime\prime}+bxy^{\prime}+cy=0$ and $\displaystyle ay^{\prime\prime}+by^{\prime}+cy=0$ have different characteristic equations since they assume different solutions (the former assumes solutions of the form $\displaystyle y=x^r$ and the latter assumes solutions of the form $\displaystyle y=e^{rx}$)!

    The former type of DE has the characteristic equation $\displaystyle ar^2+(b-a)r+c=0$, where as the latter type has the equation $\displaystyle ar^2+br+c=0$.
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  5. #5
    MHF Contributor arbolis's Avatar
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    I should have read better your first reply.
    Ok, I get $\displaystyle y_1 (x)=x^3$.
    With $\displaystyle P(x)=-\frac{5}{x}$, I reach that $\displaystyle u(x)=k \log |x| +C$.
    Thus the general solution would be of the form $\displaystyle y(x)=x^3 (c_1 \log |x| + c_2)+x^3=x^3(c_1 \log |x| + c_3)$.
    I hope it's right. And sorry, I didn't recognize it was a Cauchy-Euler equation.
    Last edited by arbolis; Aug 1st 2010 at 06:30 PM. Reason: I forgot to add the first solution
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  6. #6
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by arbolis View Post
    I should have read better your first reply.
    Ok, I get $\displaystyle y_1 (x)=x^3$.
    With $\displaystyle P(x)=-\frac{5}{x}$, I reach that $\displaystyle u(x)=k \log |x| +C$.
    Thus the general solution would be of the form $\displaystyle y(x)=x^3 (c_1 \log |x| + c_2)+x^3=x^3(c_1 \log |x| + c_3)$.
    I hope it's right. And sorry, I didn't recognize it was a Cauchy-Euler equation.
    No need to be sorry! Its just something that is useful to know down the road. Now, when we go ahead and find $\displaystyle u(x)$, we don't care much for the constants (they should drop out in the end anyways); so what where really after is that $\displaystyle u(x)=\log x$ (or $\displaystyle u(x)=\ln x$, depending on what notation is used where you're at). Thus, we see that the general solution is $\displaystyle y=c_1x^3+c_2x^3\log x$, which is what you have for your solution (but written in a slightly different way).

    Hope that clarifies things!
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