Animated Solution for Mathematics - Differential Equations: The differential equation of all circles passing through the origin and having their centres on the $x$-axis is

Imagine you are standing on the coordinate plane, looking at a collection of circles. Every single one of these circles is anchored to the origin, passing through the point $(0,0)$.

Their centers are not floating randomly; they are strictly constrained to the $x$-axis. As you slide the center along the $x$-axis, the circle breathes—expanding or shrinking to maintain its connection to the origin.

To describe this family, we need a mathematical blueprint. Let the center of any circle in this family be $C(a,0)$, where $a$ is an arbitrary constant that defines the specific circle.

Because the circle must pass through the origin $(0,0)$, the distance from the center $C(a,0)$ to the origin must be the radius $r$. Using the distance formula:

Thus, the radius is simply $r=|a|$. Now, we invoke the standard equation of a circle: $(x-h{)}^2+(y-k{)}^2=r^2$.

Substituting our center $(h,k)=(a,0)$ and radius $r=a$, we get the equation:

Simplifying this, we have $(x-a{)}^2+y^2=a^2$.

Now, let's expand the term $(x-a{)}^2$. Using the identity $(A-B{)}^2=A^2-2AB+B^2$, we get:

Notice the beauty of the symmetry here: we have an $a^2$ on both sides of the equation. They cancel out perfectly, leaving us with:

This is our fundamental equation, Equation 1. It elegantly captures the relationship between $x$, $y$, and the parameter $a$.

Our goal is to find the differential equation, which means we must eliminate the arbitrary constant $a$. Since there is only one constant, we differentiate Equation 1 with respect to $x$ exactly once.

Differentiating $x^2$ gives $2x$, and differentiating $y^2$ requires the chain rule, giving $2y\frac{dy}{dx}$. On the right side, the derivative of $2ax$ is simply $2a$.

So, we have:

This is Equation 2. Now, we have a direct expression for $2a$. We can substitute this back into our original Equation 1, which was $x^2+y^2=(2a)x$.

Replacing $2a$ with $(2x+2y\frac{dy}{dx})$, we get:

Distributing the $x$ on the right side, we obtain:

Finally, rearranging the terms to isolate $y^2$ on the left, we subtract $x^2$ from both sides:

This simplifies to the final differential equation:

This is the differential equation that governs our entire family of circles. You have successfully translated a visual family of curves into a precise mathematical law.