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.