Animated Solution for Mathematics - Differential Equations: The differential equation representing the family of curves $y^2=2c(x+\sqrt{c})$, where $c$ is a positive parameter, is of

Welcome, fellow traveler, to the fascinating world of differential equations. Today, we are going to peel back the layers of a seemingly simple family of curves:

Imagine these curves as a family of parabolas, all opening to the right, shifting and stretching as we vary the parameter $c$. Our goal is to find the 'DNA' of this family—the differential equation that governs every single one of them.

The Memory of the System

Before we touch a single derivative, let us pause and look at the structure. In the realm of differential equations, the 'order' is a measure of the system's memory.

It is defined by the number of essential arbitrary constants. Here, we have only one parameter, $c$.

Because there is only one constant to eliminate, we know, with absolute certainty, that we only need to differentiate once. This tells us immediately that the order of our differential equation is 1.

Now, let us prepare for the differentiation. We have $y^2=2c(x+\sqrt{c})$. To make our lives easier, let us expand this:

Now, we differentiate both sides with respect to $x$. Using the chain rule on the left, the derivative of $y^2$ is $2y\frac{dy}{dx}$.

On the right, the derivative of $2cx$ is simply $2c$, and since $c$ is a constant, the derivative of $2c^{3/2}$ is zero. We are left with:

Canceling the 2, we find a bridge between our parameter and our variables: $c=y{y}^{\prime }$, where $y^{\prime }$ represents $\frac{dy}{dx}$.

We have our bridge, $c=y{y}^{\prime }$. Now, we must substitute this back into our original equation to eliminate $c$ entirely.

Substituting $c=y{y}^{\prime }$ into $y^2=2cx+2c^{3/2}$, we get:

We have successfully eliminated the parameter, but we are not done yet. We have a fractional power, $(y{y}^{\prime }{)}^{3/2}$, which prevents us from defining the degree. The degree of a differential equation is only defined when it is a polynomial in its derivatives.

To clear the fractional power, we isolate the radical term:

Now, we square both sides. The left side becomes $(y^2-2xy{y}^{\prime }{)}^2$, and the right side becomes $4(y{y}^{\prime }{)}^3$.

Expanding the left side would give us terms involving $(y^{\prime }{)}^2$, but the right side clearly contains $(y^{\prime }{)}^3$. Since the degree is the highest power of the highest derivative, and our highest derivative is $y^{\prime }$, we look at the powers of $y^{\prime }$.

The highest power is 3. Thus, we have arrived at our destination: an equation of order 1 and degree 3.