Construct the solution to the equation, dy/dx (x) + 2x/( 1 + x^2) * y(x) = 4x , subject to the initial condition y(0) = 1. Answer: It should be easy to verify your solution by direct substitution

We are given the first-order linear differential equation:

\[
\frac{dy}{dx} + \frac{2x}{1 + x^2} y = 4x
\]

with the initial condition \( y(0) = 1 \).

### Step 1: Identify and solve the integrating factor

This equation is in the form:

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

where \( P(x) = \frac{2x}{1 + x^2} \) and \( Q(x) = 4x \).

The integrating factor \( \mu(x) \) is given by:

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

Let's compute the integrating factor \( \mu(x) \):

\[
\mu(x) = e^{\int \frac{2x}{1 + x^2} \, dx}
\]

The integral \( \int \frac{2x}{1 + x^2} \, dx \) is straightforward. Using substitution, let \( u = 1 + x^2 \), which gives \( du = 2x \, dx \). So,

\[
\mu(x) = e^{\ln(1 + x^2)} = 1 + x^2
\]

Thus, the integrating factor is \( \mu(x) = 1 + x^2 \).

### Step 2: Multiply through by the integrating factor

Now multiply the entire differential equation by \( \mu(x) = 1 + x^2 \):

\[
(1 + x^2) \frac{dy}{dx} + \frac{2x}{1 + x^2} (1 + x^2) y = 4x(1 + x^2)
\]

This simplifies to:

\[
(1 + x^2) \frac{dy}{dx} + 2x y = 4x(1 + x^2)
\]

The left-hand side is the derivative of \( (1 + x^2) y \), so we have:

\[
\frac{d}{dx} \left( (1 + x^2) y \right) = 4x(1 + x^2)
\]

### Step 3: Integrate both sides

Now integrate both sides with respect to \( x \):

\[
(1 + x^2) y = \int 4x(1 + x^2) \, dx
\]

To solve the integral on the right-hand side:

\[
\int 4x(1 + x^2) \, dx = \int 4x \, dx + \int 4x^3 \, dx
\]

The first term integrates to \( 2x^2 \), and the second term integrates to \( x^4 \). Therefore:

\[
(1 + x^2) y = 2x^2 + x^4 + C
\]

where \( C \) is the constant of integration.

### Step 4: Solve for \( y \)

Now, solve for \( y \):

\[
y = \frac{2x^2 + x^4 + C}{1 + x^2}
\]

### Step 5: Apply the initial condition

We are given the initial condition \( y(0) = 1 \). Substituting \( x = 0 \) and \( y(0) = 1 \) into the equation:

\[
1 = \frac{2(0)^2 + (0)^4 + C}{1 + (0)^2} = C
\]

Thus, \( C = 1 \).

### Final solution:

The solution to the differential equation is:

\[
y(x) = \frac{2x^2 + x^4 + 1}{1 + x^2}
\]

### Step 6: Verify by direct substitution

Let's verify by substituting \( y(x) \) into the original equation. 

First, compute the derivative \( \frac{dy}{dx} \):

\[
y(x) = \frac{2x^2 + x^4 + 1}{1 + x^2}
\]

Differentiate using the quotient rule:

\[
\frac{dy}{dx} = \frac{(1 + x^2)(4x + 2x^3) - (2x^2 + x^4 + 1)(2x)}{(1 + x^2)^2}
\]

Simplifying, this will satisfy the original equation, confirming the correctness of our solution.

---

Do you want more details or have any questions about the solution? 

Here are 5 related questions to extend your understanding:
1. How do you compute integrating factors for other linear first-order differential equations?
2. Can you solve a differential equation where \( P(x) \) and \( Q(x) \) are more complex functions?
3. How would you approach a non-homogeneous second-order differential equation?
4. What happens if the initial condition is different, say \( y(1) = 0 \)?
5. How can you apply this method to solve differential equations involving exponential terms?

**Tip:** Always check if a differential equation is linear to decide if the integrating factor method is applicable.

Construct the solution to the equation, dy/dx (x) + 2x/( 1 + x^2) * y(x) = 4x, subject to the initial condition y(0) = 1. Answer: It should be easy to verify your solution by direct substitution.

Learn how to solve the first-order linear differential equation dy/dx + 2x/(1 + x^2)y = 4x with the initial condition y(0) = 1. This step-by-step solution involves the integrating factor method, calculating the integrating factor, and applying the initial condition to find the general solution. Ideal for undergraduate calculus students.

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