Animated Solution for Mathematics - Differential Equations: The solution of the equation $\frac{d^{2}y}{d{x}^2}=e^{-2x}$

Unraveling the Second-Order Differential Equation

Welcome, future engineers! Today, we are going to embark on a journey through the heart of calculus. We are looking at a second-order differential equation:

Now, I know that seeing a second derivative can feel intimidating. It looks like a complex, abstract wall standing between you and the solution.

But let's reframe this. Think of differentiation as a process of 'peeling' away layers of information. If the second derivative is the acceleration, the first derivative is the velocity, and the function $y(x)$ is the position. To find the position, we simply need to walk back up the staircase.

The First Step Down the Staircase

Our goal is to find $y(x)$. Since we are given the second derivative, we must integrate once to find the first derivative, $\frac{dy}{dx}$.

We start by applying the integral operator to both sides of our equation:

This is the moment where many students stumble. When you integrate the left side, you are essentially 'undoing' the derivative, leaving you with $\frac{dy}{dx}$.

On the right side, we use the standard exponential integration rule: $\int e^{ax}dx=\frac{e^{ax}}{a}+C$. Here, our $a$ is $-2$.

So, the integral becomes:

Notice that constant $C$? Never forget it! It represents the information lost during the first differentiation.

The Final Leap to the Function

Now that we have $\frac{dy}{dx}$, we are only one step away from our destination. We need to integrate one more time to isolate $y$.

We set up the integral:

Using the sum rule, we can split this into two manageable parts: $y=\int -\frac{1}{2}{e}^{-2x}dx+\int Cdx$.

Let's tackle the first term. We pull the constant $-\frac{1}{2}$ outside, and we are left with the integral of $e^{-2x}$ again. As we saw before, this yields $\frac{e^{-2x}}{-2}$.

When we multiply this by the $-\frac{1}{2}$ already outside, the negatives cancel out, and we get:

This is the beauty of the math—the signs resolve themselves if you are patient.

The Final Synthesis

Finally, we integrate the second term, $\int Cdx$, which simply gives us $Cx$. And, because this is our second integration, we must introduce a second constant of integration, $D$.

Putting it all together, we arrive at the general solution:

This is the complete picture! We started with a second-order equation and ended with two constants, $C$ and $D$.

This is not a coincidence; it is the fundamental law of differential equations. You have successfully navigated the path from the second derivative back to the original function.

Take a moment to appreciate the elegance of this result. You didn't just solve an equation; you reconstructed a function from its curvature. Keep this mindset, and no differential equation will ever be able to stop you!