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Math Help - Can anyone help solve this NL 2ODE?

  1. #1
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    Can anyone help solve this NL 2ODE?

    Can anyone help me solve this non linear second order ODE?
    I'm sure i know how but i'm suffering from a bit of a mental block

    y''(x)=-(1/y)*(y'(x))^2

    Thanks in advance
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  2. #2
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    Re: Can anyone help solve this NL 2ODE?

    Quote Originally Posted by Moonysgirl159 View Post
    Can anyone help me solve this non linear second order ODE?
    I'm sure i know how but i'm suffering from a bit of a mental block

    y''(x)=-(1/y)*(y'(x))^2

    Thanks in advance
    Substitute :

    y'_x=v \Rightarrow y''_x=v'_y \cdot v

    hence :

    v'_y \cdot v = \frac{-1}{y} \cdot v^2 \Rightarrow y \cdot v'_y=-v \Rightarrow \int \frac{dv}{v}=-\int \frac{dy}{y}
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  3. #3
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    Re: Can anyone help solve this NL 2ODE?

    Quote Originally Posted by Moonysgirl159 View Post
    Can anyone help me solve this non linear second order ODE?
    I'm sure i know how but i'm suffering from a bit of a mental block

    y''(x)=-(1/y)*(y'(x))^2

    Thanks in advance
    To use some notation that's easier to read... Your DE is \displaystyle \frac{d^2y}{dx^2} = -\frac{1}{y}\left(\frac{dy}{dx}\right)^2

    Let \displaystyle v = \frac{dy}{dx} \implies \frac{dv}{dx} = \frac{d^2y}{dx^2} . Then your DE becomes

    \displaystyle \begin{align*} \frac{dv}{dx} &= -\frac{1}{y}\,\frac{dy}{dx}\,v \\ \frac{1}{v}\,\frac{dv}{dx} &= -\frac{1}{y}\,\frac{dy}{dx} \\ \int{\frac{1}{v}\,\frac{dv}{dx}\,dx} &= \int{-\frac{1}{y}\,\frac{dy}{dx}\,dx} \\ \int{\frac{1}{v}\,dv} &= \int{-\frac{1}{y}\,dy} \end{align*}

    Can you go from here?
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  4. #4
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    Re: Can anyone help solve this NL 2ODE?

    I need help in this question; it’s a first order nonhomogenous differential equation. I know how to do the rest of the question but I just don’t know how the book got from this part to the other. thanks!!!

    (T – Tinfinity – (b/a)) / (Ti – Tinfinity – (b/a)) = exp (-at)

    Hence

    (T – Tinfinity) / (Ti – Tinfinity ) = exp (-at) + (b/a) / (Ti – Tinfinity) (1- exp (-at))
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