Find the differential equation of all circles with fixed radius and the centers are on the x-axis.

Please po :'(

# Differential Equations (PLEASE HELP PO)

## New forum topics

- I need guidance in designing a beam supporting specified ultimate moment of 1100 kN.m (doubly reinforced beam)
- I need guidance in solving the ultimate moment capacity (doubly reinforced beam)
- I need guidance in solving the balance steel area
- Please help me solve this problem using WSD (Working Stress Design) Method
- Physics: Uniform Motion
- Rescue at Sea
- Using two pumps
- Emptying a Tank
- Speed of a Plane
- Range of an Airplane

The center of the circle is $(a, 0)$ and the fixed radius is, say, $r$.

Thus the equation of the circle is $(x-a)^2+y^2 = r^2$. Since $a$ is the arbitrary constant here, get the derivative once.

\begin{eqnarray*}

(x-a)^2+y^2 &=& r^2\\

2(x-a) + 2y y' &=& 0\\

(x-a) + yy' &=& 0\\

x-a &=& -yy'

\end{eqnarray*}

Then substitute to the original equation.

\begin{eqnarray*}

(x-a)^2 + y^2 &=& r^2\\

(yy')^2 + y^2 &=& r^2\\

y^2((y')^2 + 1) &=& r^2

\end{eqnarray*}

Thus the differential equation is $\boxed{y^2((y')^2 + 1) = r^2}$