# Can anyone help solve this NL 2ODE?

• Mar 18th 2012, 08:21 PM
Moonysgirl159
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

• Mar 18th 2012, 09:37 PM
princeps
Re: Can anyone help solve this NL 2ODE?
Quote:

Originally Posted by Moonysgirl159
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

Substitute :

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

hence :

$\displaystyle v'_y \cdot v = \frac{-1}{y} \cdot v^2 \Rightarrow y \cdot v'_y=-v \Rightarrow \int \frac{dv}{v}=-\int \frac{dy}{y}$
• Mar 19th 2012, 06:00 AM
Prove It
Re: Can anyone help solve this NL 2ODE?
Quote:

Originally Posted by Moonysgirl159
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

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

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

\displaystyle \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?
• Mar 21st 2012, 05:34 AM
Lida
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))