The solution of the differential equation `(d^(2)y)/(dx^(2))=e^(-2x)`, is

Options

A. `(1)/(4)e^(-2x)`
B. `(1)/(4)e^(-2x)+cx+d`
C. `(1)/(4)e^(-2x)+cx^(2)+d`
D. `(1)/(4)e^(-2x)+c+d`

Answer

To solve the differential equation \(\frac{d^2y}{dx^2} = e^{-2x}\), we will follow these steps:

### Step 1: Integrate the equation once
We start by integrating both sides of the equation with respect to \(x\).

\[ \int \frac{d^2y}{dx^2} \, dx = \int e^{-2x} \, dx \]

The left side becomes \(\frac{dy}{dx}\) and the right side can be integrated:

\[ \frac{dy}{dx} = \int e^{-2x} \, dx = -\frac{1}{2} e^{-2x} + C_1 \]

Where \(C_1\) is the constant of integration.

### Step 2: Integrate the first derivative
Next, we integrate \(\frac{dy}{dx}\) to find \(y\).

\[ \int \frac{dy}{dx} \, dx = \int \left(-\frac{1}{2} e^{-2x} + C_1\right) \, dx \]

This gives us:

\[ y = -\frac{1}{2} \int e^{-2x} \, dx + \int C_1 \, dx \]

Calculating the integrals:

\[ y = -\frac{1}{2} \left(-\frac{1}{2} e^{-2x}\right) + C_1 x + C_2 \]

Where \(C_2\) is another constant of integration. Simplifying this:

\[ y = \frac{1}{4} e^{-2x} + C_1 x + C_2 \]

### Final Solution
Thus, the general solution of the differential equation is:

\[ y = \frac{1}{4} e^{-2x} + C_1 x + C_2 \]