Solve the first order linear initial value problem
dy/dx +2y = 2 - 7x, y(0)=0

The given problem is a first-order linear differential equation with an initial condition:

\[
\frac{dy}{dx} + 2y = 2 - 7x, \quad y(0) = 0
\]

### Step 1: Identify the standard form
A first-order linear differential equation has the form:

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

In this case, \( P(x) = 2 \) and \( Q(x) = 2 - 7x \).

### Step 2: Find the integrating factor
The integrating factor is given by:

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

### Step 3: Multiply the differential equation by the integrating factor
Multiply the entire equation by \( e^{2x} \):

\[
e^{2x} \frac{dy}{dx} + 2e^{2x} y = e^{2x} (2 - 7x)
\]

The left-hand side simplifies to:

\[
\frac{d}{dx} \left( e^{2x} y \right) = e^{2x} (2 - 7x)
\]

### Step 4: Integrate both sides
Now integrate both sides with respect to \( x \):

\[
\int \frac{d}{dx} \left( e^{2x} y \right) dx = \int e^{2x} (2 - 7x) dx
\]

The left side simplifies to:

\[
e^{2x} y = \int e^{2x} (2 - 7x) \, dx
\]

### Step 5: Compute the integral
To compute \( \int e^{2x} (2 - 7x) \, dx \), break it into two parts:

1. \( \int 2e^{2x} \, dx = \frac{2}{2} e^{2x} = e^{2x} \)
2. \( \int -7x e^{2x} \, dx \)

Use integration by parts for the second integral. Let:

- \( u = x \), so \( du = dx \)
- \( dv = -7e^{2x} dx \), so \( v = \frac{-7}{2} e^{2x} \)

Using the formula \( \int u dv = uv - \int v du \), we get:

\[
\int -7x e^{2x} dx = \frac{-7x}{2} e^{2x} + \int \frac{7}{2} e^{2x} dx = \frac{-7x}{2} e^{2x} + \frac{7}{4} e^{2x}
\]

Thus, the integral becomes:

\[
e^{2x} y = e^{2x} - \frac{7x}{2} e^{2x} + \frac{7}{4} e^{2x}
\]

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

\[
y = 1 - \frac{7x}{2} + \frac{7}{4}
\]

Simplify:

\[
y(x) = \frac{11}{4} - \frac{7x}{2}
\]

### Step 7: Apply the initial condition
Use the initial condition \( y(0) = 0 \):

\[
0 = \frac{11}{4} - \frac{7 \cdot 0}{2} = \frac{11}{4}
\]

This leads to a contradiction with the initial condition, suggesting an issue in the setup. Let's revisit

Solve the first order linear initial value problem dy/dx + 2y = 2 - 7x, y(0) = 0

Learn how to solve the first-order linear differential equation dy/dx + 2y = 2 - 7x, with the initial condition y(0) = 0. This solution involves using the integrating factor method, making it ideal for students studying Calculus II or Differential Equations.

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