This is a second order nonlinear differential equation. By rearranging the terms, we can write it in a form that is easier to work with:
yy'' = (y')^2
y'' = (y')^2 / y
Let u = y',
Then y'' = du/dx = d(u)/dx
So the equation becomes:
du/dx = (u)^2 / y
Separating variables, we have:
y dy = u du
Integrating both sides, we get:
1/2 y^2 = 1/2 u^2 + C
Where C is the constant of integration.
Therefore, the general solution to the differential equation yy'' = (y')^2 is given by:
y^2 = u^2 + C
or
y^2 = (dy/dx)^2 + C
This is a second order nonlinear differential equation. By rearranging the terms, we can write it in a form that is easier to work with:
yy'' = (y')^2
y'' = (y')^2 / y
Let u = y',
Then y'' = du/dx = d(u)/dx
So the equation becomes:
du/dx = (u)^2 / y
Separating variables, we have:
y dy = u du
Integrating both sides, we get:
1/2 y^2 = 1/2 u^2 + C
Where C is the constant of integration.
Therefore, the general solution to the differential equation yy'' = (y')^2 is given by:
y^2 = u^2 + C
or
y^2 = (dy/dx)^2 + C