If `(dy)/(dx)+y=2e^(2x)`, then y is equal to

Options

A. `ce^(x)+(2)/(3)e^(2x)`
B. `(1+x)e^(-x)+(2)/(3)e^(2x)+c`
C. `ce^(-x)+(2)/(3)e^(2x)`
D. `e^(-2x)+(2)/(3)e^(2x)+c`

Explanation

To solve the differential equation \(\frac{dy}{dx} + y = 2e^{2x}\), we will follow the steps for solving a first-order linear ordinary differential equation.

### Step 1: Identify p and q
The given equation is in the form \(\frac{dy}{dx} + py = q\), where:
- \(p = 1\)
- \(q = 2e^{2x}\)

### Step 2: Find the integrating factor
The integrating factor (IF) is given by:
\[ IF = e^{\int p \, dx} = e^{\int 1 \, dx} = e^{x} \]

### Step 3: Multiply the entire equation by the integrating factor
Multiply both sides of the differential equation by the integrating factor \(e^{x}\):
\[ e^{x} \frac{dy}{dx} + e^{x} y = 2e^{3x} \]

### Step 4: Rewrite the left side as a derivative
The left side can be rewritten as the derivative of a product:
\[ \frac{d}{dx}(e^{x} y) = 2e^{3x} \]

### Step 5: Integrate both sides
Now, integrate both sides with respect to \(x\):
\[ \int \frac{d}{dx}(e^{x} y) \, dx = \int 2e^{3x} \, dx \]
This gives us:
\[ e^{x} y = \frac{2}{3} e^{3x} + C \]
where \(C\) is the constant of integration.

### Step 6: Solve for y
Now, solve for \(y\):
\[ y = \frac{2}{3} e^{3x} e^{-x} + Ce^{-x} \]
This simplifies to:
\[ y = \frac{2}{3} e^{2x} + Ce^{-x} \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = \frac{2}{3} e^{2x} + Ce^{-x} \]

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