# Solution to the differential equation $$\frac{dy}{dx} + 2y = e^{-x}$$, $$y(0) = 1$$

### Solution to the differential equation $$\frac{dy}{dx} + 2y = e^{-x}$$, $$y(0) = 1$$

A: $$y = e^{-x}$$

### Worksheet

Step 1: Identify the Form

This is a first-order linear non-homogeneous differential equation of the form:
$\frac{dy}{dx} + P(x)y = Q(x)$
Here, $$P(x) = 2$$ and $$Q(x) = e^{-x}$$.

Step 2: Find the Integrating Factor

The integrating factor $$\mu(x)$$ is given by:
$\mu(x) = e^{\int P(x) \, dx}$
Since $$P(x) = 2$$:
$\mu(x) = e^{\int 2 \, dx} = e^{2x}$

Step 3: Multiply the Differential Equation by the Integrating Factor

Multiply both sides of the differential equation by the integrating factor $$e^{2x}$$:
$e^{2x} \frac{dy}{dx} + 2e^{2x} y = e^{2x} e^{-x}$
$e^{2x} \frac{dy}{dx} + 2e^{2x} y = e^{x}$

Step 4: Recognize the Left Side as a Product Rule

The left side of the equation is the derivative of $$y \cdot e^{2x}$$:
$\frac{d}{dx} \left( y \cdot e^{2x} \right) = e^x$

Step 5: Integrate Both Sides

Integrate both sides with respect to $$x$$:
$\int \frac{d}{dx} \left( y \cdot e^{2x} \right) \, dx = \int e^x \, dx$
$y \cdot e^{2x} = \int e^x \, dx$
$y \cdot e^{2x} = e^x + C$

Step 6: Solve for $$y$$

Divide both sides by $$e^{2x}$$:
$y = \frac{e^x + C}{e^{2x}}$
$y = e^{-x} + Ce^{-2x}$

Step 7: Apply the Initial Condition

Use the initial condition $$y(0) = 1$$ to find $$C$$:
$y(0) = e^{-0} + Ce^{-2 \cdot 0} = 1 + C = 1$
$1 + C = 1$
$C = 0$

Step 8: Write the Particular Solution

Substitute $$C = 0$$ back into the general solution:
$y = e^{-x} + 0 \cdot e^{-2x}$
$y = e^{-x}$

Final Solution

The particular solution to the differential equation $$\frac{dy}{dx} + 2y = e^{-x}$$ with the initial condition $$y(0) = 1$$ is:

### $y = e^{-x}$

#### Recap

1. Identify the Form: Recognize the equation as a first-order linear non-homogeneous differential equation.
2. Find the Integrating Factor: Compute the integrating factor $$\mu(x) = e^{2x}$$.
3. Multiply by Integrating Factor: Multiply both sides of the equation by $$e^{2x}$$.
4. Recognize Product Rule: Identify the left side as the derivative of $$y \cdot e^{2x}$$.
5. Integrate Both Sides: Integrate to find $$y \cdot e^{2x} = e^x + C$$.
6. Solve for $$y$$: Isolate $$y$$ and use the initial condition to solve for $$C$$.
7. Substitute $$C$$: Substitute the value of $$C$$ back into the solution to get the particular solution.