Can anyone help solve this NL 2ODE?

**Moonysgirl159** · March 18th 2012, 08:21 PM

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

**princeps** · March 18th 2012, 09:37 PM

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

Thanks in advance

Substitute :

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

hence :

$v'_y \cdot v = \frac{-1}{y} \cdot v^2 \Rightarrow y \cdot v'_y=-v \Rightarrow \int \frac{dv}{v}=-\int \frac{dy}{y}$

**Prove It** · March 19th 2012, 06:00 AM

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

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?

**Lida** · March 21st 2012, 05:34 AM

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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