Given equation: $\frac{dy}{dt} = 4t$

To solve this equation, we need to integrate both sides with respect to $t$.

Step 1: Integration of the left-hand side
$\int \frac{dy}{dt} dt = \int 4t dt$

Step 2: Simplify the integration
$y = \int 4t dt$

Step 3: Evaluate the integral
$y = 4 \int t dt$

Step 4: Apply the power rule of integration
$y = 4 \cdot \frac{t^2}{2} + C$

Here, $C$ is the constant of integration.

So, the general solution to the differential equation is $y = 2t^2 + C$.

Explanation:

The given equation $\frac{dy}{dt} = 4t$ represents a first-order linear ordinary differential equation. We are asked to find the solution for $y$ with respect to $t$.

To solve this equation, we need to integrate both sides of the equation. This is because the derivative of a function represents the rate of change of that function, and integrating the derivative gives us back the original function.

In Step 1, we integrate the left-hand side of the equation with respect to $t$. The integral of $\frac{dy}{dt}$ with respect to $t$ gives us $y$.

In Step 2, we simplify the integration on the left-hand side.

In Step 3, we evaluate the integral on the right-hand side of the equation. Here, we integrate $4t$ with respect to $t$, which gives us $4 \cdot \frac{t^2}{2}$.

Finally, in Step 4, we simplify the integral on the right-hand side and add the constant of integration $C$. The constant of integration accounts for the fact that there can be multiple solutions to a differential equation. The value of $C$ can be determined by any initial conditions or additional information given in the problem.

Therefore, the general solution to the given differential equation is $y = 2t^2 + C$, where $C$ is a constant.