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Math Help - first order ordinary differential equation

  1. #1
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    first order ordinary differential equation

    How many solutions does the differential equation \frac{dy}{dx}=60(y^2)^{1/5}; \quad x>0, y(0)=0 have?
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  2. #2
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    Re: first order ordinary differential equation

    First, (y^2)^{1/5} is a very strange way to write y^{2/5}. Is that what you meant?

    If it is, then this is a separable equation that is easily solvable: \dfrac{dy}{y^{2/5}}= y^{-2/5}dy= 60 dx.

    What do you get when you integrate both sides?
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  3. #3
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    Re: first order ordinary differential equation

    Ya this is easily solvable. But how many solutions does it have? The answer was given as 2. But I cant understand how? It was set in an entrance exam.
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    Re: first order ordinary differential equation

    Start by solving it! That should help you see what solutions there are!

    Have you considered what would happen if y were identically 0?
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    Re: first order ordinary differential equation

    I have arrived at the solution 5y^{3/5}=3x. This is the only solution I got. Then how the no of solution is 2?
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    Re: first order ordinary differential equation

    Did you even read HallsofIvy's last post? You can only solve the equation through dividing, which means you're making the assumption that \displaystyle \begin{align*} y \neq 0 \end{align*}. WHAT IF IT WAS?!?!
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    Re: first order ordinary differential equation

    So the two solutions are 5y^{3/5}=180x (\mbox{when} \quad y \neq 0)\quad \mbox{and}\quad y=0?
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  8. #8
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    Re: first order ordinary differential equation

    Yes
    Thanks from Suvadip and HallsofIvy
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