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 $\displaystyle \displaystyle \frac{d^2y}{dx^2} = -\frac{1}{y}\left(\frac{dy}{dx}\right)^2$
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?
I need help in this question; its a first order nonhomogenous differential equation. I know how to do the rest of the question but I just dont 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))