Question

\[ y\ddot{y}-\dot{y}^2=1\]

Answer 1

\[ y\ddot{y}-\dot{y}^2=1\]

Answer 2

The boundary conditions are \[ y(a)=y(-a)=1\] .

\[ \large \ddot{y}=\frac{ \dot{y}^2+1}{y} \]

Let \[ p=\dot{y}\] .

\[ \large \dot{p}=\frac{dp}{dy}p=\frac{ p^2+1}{y}\]

\[ \large\int \frac{p}{p^2+1}dp= \int \frac{1}{y}dy\] Let \[ u=p^2\]

\[ \large \frac{1}{2}ln|u+1| =ln|\sqrt{u+1}|=ln|y|+c\]

\[ \large \sqrt{u+1} =ky\]

\[ \large \dot{y}^2+1 =by^2\]

What do I do now?

Answer 3

after this dy/dx = sqrt(by2-1) and then proceed ...