(1)/(t)(dy)/(dt)=4sin^(2)(y), y(0)=(7\pi )/(4)

The given differential equation is:

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

with the initial condition:

\[
y(0) = \frac{7\pi}{4}
\]

### Step 1: Separate Variables
We aim to separate the variables \( y \) and \( t \) on different sides of the equation. First, rewrite the equation:

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

Now, separate the variables:

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

### Step 2: Integrate Both Sides
Integrate both sides to solve the differential equation.

#### Left-hand side:

\[
\int \frac{dy}{\sin^2(y)} = \int 4t dt
\]

The left-hand side can be simplified using the identity:

\[
\frac{1}{\sin^2(y)} = \csc^2(y)
\]

Thus, we have:

\[
\int \csc^2(y) dy = -\cot(y)
\]

#### Right-hand side:

\[
\int 4t dt = 2t^2 + C
\]

### Step 3: Combine the Results
Now combine the integrated results:

\[
-\cot(y) = 2t^2 + C
\]

### Step 4: Use the Initial Condition
Substitute the initial condition \( y(0) = \frac{7\pi}{4} \) to find the constant \( C \).

At \( t = 0 \):

\[
-\cot\left(\frac{7\pi}{4}\right) = 2(0)^2 + C
\]

Since \( \cot\left(\frac{7\pi}{4}\right) = -1 \), we have:

\[
1 = C
\]

Thus, the constant \( C = 1 \).

### Final Solution:
The solution to the differential equation is:

\[
-\cot(y) = 2t^2 + 1
\]

or equivalently:

\[
\cot(y) = -(2t^2 + 1)
\]

You can now solve for \( y \) in terms of \( t \), if needed.

Would you like more details on solving for \( y \), or any other clarification?

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Here are 5 questions based on this topic:

1. How can you solve for \( y \) in terms of \( t \) from the final equation?
2. What is the general approach to solving separable differential equations?
3. Can you explain the identity \( \csc^2(y) \) and how it's used in calculus?
4. How would the solution change if the initial condition were different?
5. What are some real-world applications of this type of differential equation?

**Tip:** Always verify your solution using the initial condition after solving the equation!

The query submitted by the user: (1)/(t)(dy)/(dt)=4sin^(2)(y), y(0)=(7\pi)/(4)

Explore how to solve the separable differential equation (1/t)(dy/dt) = 4sin^2(y) with the initial condition y(0) = 7π/4. This step-by-step solution covers differential equations, separation of variables, and the use of trigonometric identities. Ideal for university-level students studying calculus.

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