Animated Solution for Mathematics - Differential Equations: The differential equation which represents the family of curves $y=c_1{e}^{c_{2}x}$, where $c_1$ and $c_2$ are arbitrary constants, is

Welcome, future engineer! Today, we are going to peel back the curtain on one of the most elegant topics in calculus: the construction of differential equations.

You might look at the equation $y=c_1{e}^{c_{2}x}$ and see just a bunch of letters and exponents. But I want you to see it as a blueprint.

This isn't just a curve; it's a family of possibilities. By changing $c_1$ and $c_2$, you are shifting, stretching, and transforming the curve. Our mission is to find the 'DNA' of this family—the differential equation that governs every single curve in this set, regardless of the constants.

Before we touch a pen to paper, let's talk strategy. In the realm of differential equations, there is a golden rule: the order of the differential equation is determined by the number of independent arbitrary constants.

Here, we have two constants: $c_1$ and $c_2$. This means we have two degrees of freedom.

To 'lock down' this family and eliminate these constants, we must differentiate exactly twice. It is a non-negotiable requirement. If you differentiate only once, you will still have a constant left over. If you differentiate three times, you are doing more work than necessary. Two is the magic number.

Let's begin. We start with our original equation:

We differentiate this with respect to $x$. Using the chain rule, the derivative of $e^{c_{2}x}$ is $c_2{e}^{c_{2}x}$. So, we get:

Now, here is where the 'pro' move comes in. Instead of just moving on, look at the expression $c_1{e}^{c_{2}x}$. Does it look familiar? It is exactly our original $y$!

This is the beauty of substitution. We can rewrite our first derivative as:

This is much cleaner, isn't it? We have successfully linked $y^{\prime }$ to $y$ and $c_2$.

We aren't done yet. We still have $c_2$ in our equation, and our goal is to eliminate all arbitrary constants.

From our simplified equation $y^{\prime }=c_{2}y$, we can easily isolate $c_2$:

Keep this in your back pocket; it is going to be our secret weapon in the final step.

We need to differentiate one more time to satisfy our 'two constants, two derivatives' rule. Let's take $y^{\prime }=c_{2}y$ and differentiate it with respect to $x$.

The derivative of $y^{\prime }$ is $y^{\prime \prime }$. On the right side, $c_2$ is a constant, so the derivative of $c_{2}y$ is simply $c_2{y}^{\prime }$. Thus, we have:

Now, the moment of truth. We have $y^{\prime \prime }=c_2{y}^{\prime }$, but we still have that pesky $c_2$. Remember our secret weapon? We know that $c_2=\frac{y^{\prime }}{y}$.

Let's substitute that into our equation:

Look at what we have:

To make this look like a standard differential equation, we multiply both sides by $y$. We get:

And there it is! A beautiful, clean differential equation, completely free of $c_1$ and $c_2$. You have successfully captured the essence of the entire family of curves. This is the power of calculus—taking a complex family and distilling it down to its fundamental governing law. Well done!