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.