Finding solutions to the equation y''=(y'^2-1)/y

May 2019
1
0
France
Hello, it's actually my first time posting a thread on this site so I hope you'll forgive me if I do not post as I should. Here is the problem I've got :
Solve the differential equation y''=(y'^2-1)/y with y(1) = 1 and y(2) = 7. I tried different methods but none seemed to be a help in the case. I hope you'll help me solve it. I thank you in advance for your answers.
 
Dec 2013
2,000
757
Colombia
I would start as follows: \begin{align}\frac{\mathrm d}{\mathrm dx}\frac{y'}{y} &= \frac{yy'' - y'^2}{y^2} \\[12pt] \text{and so} \qquad y'' &= \frac{y'^2-1}{y} \\ yy'' &= y'^2 -1 \\ yy'' -y'^2 &= -1 \\ \frac{yy'' - y'^2}{y^2} &= -\frac{1}{y^2} \\ \frac{\mathrm d}{\mathrm dx}\frac{y'}{y} &= -\frac{1}{y^2} \end{align}
 
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