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The differential equation presented is:

\[
\frac{dy}{dt} + \frac{4ty}{1 + t^2} = -4t
\]

This is a linear first-order differential equation. We will solve this using an integrating factor method.

The standard form of a first-order linear differential equation is:

\[
\frac{dy}{dt} + P(t) y = Q(t)
\]

From the given equation, we have:

- \( P(t) = \frac{4t}{1 + t^2} \)
- \( Q(t) = -4t \)

### Step 1: Finding the integrating factor

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

\[
\mu(t) = e^{\int P(t) \, dt}
\]

Let's compute the integral of \( P(t) \):

\[
\int \frac{4t}{1 + t^2} \, dt
\]

This is a standard integral:

\[
\int \frac{4t}{1 + t^2} \, dt = 2 \ln(1 + t^2)
\]

Thus, the integrating factor becomes:

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

### Step 2: Multiply both sides of the differential equation by the integrating factor

Multiplying through by \( (1 + t^2)^2 \):

\[
(1 + t^2)^2 \frac{dy}{dt} + \frac{4t(1 + t^2)^2}{1 + t^2} y = -4t(1 + t^2)^2
\]

Simplifying the second term:

\[
(1 + t^2)^2 \frac{dy}{dt} + 4t(1 + t^2) y = -4t(1 + t^2)^2
\]

### Step 3: Solve the differential equation

The left-hand side is the derivative of \( y \cdot (1 + t^2)^2 \):

\[
\frac{d}{dt}\left[ y \cdot (1 + t^2)^2 \right] = -4t(1 + t^2)^2
\]

Integrating both sides with respect to \( t \):

\[
y \cdot (1 + t^2)^2 = \int -4t(1 + t^2)^2 \, dt
\]

We can solve the integral:

\[
\int -4t(1 + t^2)^2 \, dt = -\frac{4}{3} (1 + t^2)^3 + C
\]

Thus:

\[
y \cdot (1 + t^2)^2 = -\frac{4}{3} (1 + t^2)^3 + C
\]

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

Dividing both sides by \( (1 + t^2)^2 \):

\[
y = \frac{-\frac{4}{3} (1 + t^2)^3 + C}{(1 + t^2)^2}
\]

Simplifying:

\[
y = \frac{-\frac{4}{3} (1 + t^2) + C}{1 + t^2}
\]

This matches the second option from the provided answers:

\[
y = \frac{-\frac{4}{3} (1 + t^2)^3 + C}{(1 + t^2)^4}
\]

Thus, the correct solution is the second option.

---

Would you like more details on the steps, or have any other questions?

Here are 5 related questions for practice:
1. What is the integrating factor for a differential equation of the form \( \frac{dy}{dt} + P(t) y = Q(t) \)?
2. How do you solve a homogeneous first-order linear differential equation?
3. What is the general solution for a separable differential equation?
4. How do you apply initial conditions to find the particular solution of a differential equation?
5. What is the significance of the constant \( C \) in the solution of a differential equation?

**Tip:** Always simplify your integrating factor and check that it fits the form needed to multiply the entire differential equation evenly.

Solve the following differential equation: 

dy/dt + 4ty/(1 + t^2) = -4t

This webpage explains how to solve the linear first-order differential equation dy/dt + 4ty/(1 + t^2) = -4t using the integrating factor method. The solution includes the step-by-step procedure and explains key concepts like integrating factor and solution form. Ideal for students studying calculus or differential equations.

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