# Thread: Solve the differential equation (show steps!)

1. ## Solve the differential equation (show steps!)

I came across this differential equation while trying to solve a specific parametric equation for a parabola. The equation is:

y*y''+(y')^2=0

Wolfram|Alpha suggests the square root function, but I want to know the mathematics that derive that result. The furthest I've gotten in my personal study of differential equations is the Wronskian Method for Non-homogeneous Linear Differential Equations of the Second Order.

2. Originally Posted by pikalax1
I came across this differential equation while trying to solve a specific parametric equation for a parabola. The equation is:

y*y''+(y')^2=0

Wolfram|Alpha suggests the square root function, but I want to know the mathematics that derive that result. The furthest I've gotten in my personal study of differential equations is the Wronskian Method for Non-homogeneous Linear Differential Equations of the Second Order.
The left side of that equation is just the derivative of $\displaystyle yy'$. So you can integrate the equation once to get $\displaystyle yy' =$ const. Then integrate again to get $\displaystyle y^2 = ax+b$. So the general solution is $\displaystyle y = \pm\sqrt{ax+b}$.

3. Thank you.