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

Options

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

Answer

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

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

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

This gives us:

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

### Step 2: Calculate the integrals
Now, we compute the integrals on the right-hand side:

1. The integral of \(e^{2x}\) is \(\frac{e^{2x}}{2}\).
2. The integral of \(e^{-x}\) is \(-e^{-x}\).

Thus, we have:

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

### Step 3: Integrate again to find \(y\)
Next, we integrate \(\frac{dy}{dx}\) to find \(y\):

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

This gives us:

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

Calculating each integral:

1. \(\int \frac{e^{2x}}{2} \, dx = \frac{e^{2x}}{4}\)
2. \(\int e^{-x} \, dx = -e^{-x}\)
3. \(\int C_1 \, dx = C_1 x\)

Putting it all together, we get:

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

This simplifies to:

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

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

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