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Thread: Constant solution

  1. #1
    woo
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    Constant solution

    I want to find constant solutions of the differential equation $\frac{dy}{dx}=\frac{y^3}{x^2}-2\frac{y}{x}$. I set $y=c$, where $c$ is a constant. Then $\frac{dy}{dx}=0$ and the differential equation becomes

    $0=\frac{c^3}{x^2}-2\frac{c}{x}$
    $\frac{c}{x}(\frac{c^2}{x}-2)=0$
    $c/x=0$ or $c^2/x-2=0$
    $c=0$ or $c^2=2x$

    How to explain in words the equation $c^2=2x$ does not give any constant solution of the differential equation?
    Last edited by woo; Nov 12th 2017 at 03:57 AM.
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: Constant solution

    Quote Originally Posted by woo View Post
    I want to find constant solutions of the differential equation $\frac{dy}{dx}=\frac{y^3}{x^2}-2\frac{y}{x}$. I set $y=c$, where $c$ is a constant. Then $\frac{dy}{dx}=0$ and the differential equation becomes

    $0=\frac{c^3}{x^2}-2\frac{c}{x}$
    $\frac{c}{x}(\frac{c^2}{x}-2)=0$
    $c/x=0$ or $c^2/x-2=0$
    $c=0$ or $c^2=2x$

    How to explain in words the equation $c^2=2x$ does not give any constant solution of the differential equation?
    c^2 = 2x \rightarrow x = \frac{c^2}{2}

    Looks like a constant solution to me.

    -Dan
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  3. #3
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    Re: Constant solution

    No it's not. It's a solution that works for only one value of $x$!

    I would say that the left hand side of c^2=2x is constant and has only one value while the right hand side is variable and can take many values.
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