Solve the initial value problem dy/dx+2y-3e^(-x)=0 where y(0)=6

We are given the first-order linear differential equation:

\[
\frac{dy}{dx} + 2y - 3e^{-x} = 0
\]
with the initial condition \( y(0) = 6 \).

### Step 1: Rewrite the equation
We first rewrite the differential equation in standard linear form:

\[
\frac{dy}{dx} + 2y = 3e^{-x}
\]

### Step 2: Find the integrating factor
The standard form of a linear differential equation is:

\[
\frac{dy}{dx} + P(x)y = Q(x)
\]

Here, \( P(x) = 2 \) and \( Q(x) = 3e^{-x} \). The integrating factor is given by:

\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int 2 \, dx} = e^{2x}
\]

### Step 3: Multiply through by the integrating factor
We multiply both sides of the differential equation by \( e^{2x} \):

\[
e^{2x} \frac{dy}{dx} + 2e^{2x}y = 3e^{2x}e^{-x}
\]

Simplifying the right-hand side:

\[
e^{2x} \frac{dy}{dx} + 2e^{2x}y = 3e^{x}
\]

### Step 4: Solve the left-hand side as a derivative
The left-hand side is the derivative of \( y \cdot e^{2x} \):

\[
\frac{d}{dx} \left( y \cdot e^{2x} \right) = 3e^{x}
\]

### Step 5: Integrate both sides
Now we integrate both sides with respect to \( x \):

\[
\int \frac{d}{dx} \left( y \cdot e^{2x} \right) \, dx = \int 3e^{x} \, dx
\]

On the left-hand side, the integral of the derivative is just \( y \cdot e^{2x} \), and on the right-hand side, the integral of \( 3e^{x} \) is \( 3e^{x} \). Thus, we have:

\[
y \cdot e^{2x} = 3e^{x} + C
\]

### Step 6: Solve for \( y \)
To solve for \( y \), divide both sides by \( e^{2x} \):

\[
y = \frac{3e^{x} + C}{e^{2x}} = 3e^{-x} + Ce^{-2x}
\]

### Step 7: Apply the initial condition
We now use the initial condition \( y(0) = 6 \) to find \( C \). Substitute \( x = 0 \) and \( y = 6 \) into the equation:

\[
6 = 3e^{0} + C e^{0} = 3 + C
\]

This gives:

\[
C = 6 - 3 = 3
\]

### Final Solution:
Thus, the solution to the initial value problem is:

\[
y(x) = 3e^{-x} + 3e^{-2x}
\]

Would you like to go over any details or have further questions? Here are some related problems you might find interesting:

1. Solve \( \frac{dy}{dx} + y = 2e^{-x} \) with \( y(0) = 0 \).
2. Solve \( \frac{dy}{dx} + 3y = 5e^{-x} \) with \( y(0) = 1 \).
3. Solve \( \frac{dy}{dx} + 4y = 10e^{x} \) with \( y(0) = 2 \).
4. Solve \( \frac{dy}{dx} + 2y = e^{2x} \) with \( y(0) = 1 \).
5. Solve \( \frac{dy}{dx} + y = \sin(x) \) with \( y(0) = 0 \).

**Tip:** When solving linear differential equations, always check if multiplying by an integrating factor simplifies the equation into a form that can be easily integrated.

Solve the initial value problem dy/dx + 2y - 3e^(-x) = 0 where y(0) = 6

Learn how to solve the initial value problem dy/dx + 2y = 3e^(-x) with the condition y(0) = 6. This step-by-step guide covers linear first-order differential equations using the integrating factor method. Suitable for advanced high school or undergraduate students.

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