Animated Solution for Mathematics - Differential Equations: The degree and order of the differential equation of the family of all parabolas whose axis is $x$-axis, are respectively.

Imagine you are standing on a vast, empty coordinate plane. You are not just looking at one parabola; you are looking at an entire family of them. These parabolas are special because their axis of symmetry is locked onto the $x$-axis.

Mathematically, we capture this entire infinite family with a single, elegant equation:

Here, $h$ acts as the horizontal shifter, moving the vertex to any point $(h,0)$, while $a$ acts as the sculptor, determining the focal length and the direction of the opening. These two variables, $a$ and $h$, are the arbitrary constants that define every single member of this family.

Now, we want to find the differential equation that governs this entire family. There is a golden rule in the world of differential equations: the order of the differential equation is equal to the number of essential arbitrary constants in its general solution.

Since we have two constants, $a$ and $h$, we know immediately that we are looking for a second-order differential equation. This means we will need to differentiate our general equation twice to eliminate these constants.

Let us begin the elimination. We start with our general equation: $y^2=4a(x-h)$.

Our first goal is to eliminate $h$. We differentiate both sides with respect to $x$. Using the chain rule on the left, the derivative of $y^2$ becomes $2y\cdot y^{\prime }$. On the right, the derivative of $4a(x-h)$ is simply $4a$, because $h$ is a constant.

So, we have $2y{y}^{\prime }=4a$. We can simplify this by dividing both sides by 2, giving us:

Look at that! The constant $h$ has vanished. We are halfway there.

Now, we have one constant left: $a$. To eliminate it, we must differentiate one more time. We take the derivative of $y{y}^{\prime }=2a$ with respect to $x$.

On the left side, we must use the product rule: the first function $y$ times the derivative of the second $y^{\prime \prime }$, plus the second function $y^{\prime }$ times the derivative of the first $y^{\prime }$. On the right side, the derivative of the constant $2a$ is zero. This gives us:

We have successfully eliminated all arbitrary constants. We are left with the differential equation $y{y}^{\prime \prime }+(y^{\prime }{)}^2=0$.

The highest order derivative present is $y^{\prime \prime }$, which confirms our order is 2. The power to which this highest derivative is raised is 1, which means the degree is 1.

The question asks for the degree and order respectively. Therefore, the degree is 1 and the order is 2.