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We are tasked with determining if the given differential equation is exact. If it is, we will solve it for the given initial conditions.

The differential equation provided is:

\[
\left(\frac{3y^2 - t^2}{y^5}\right) \frac{dy}{dt} + \frac{t}{2y^4} = 0
\]
with the initial condition \( y(1) = 1 \).

### Step 1: Rearrange the equation
We can rewrite the differential equation in the following form:
\[
M(t, y) \, dy + N(t, y) \, dt = 0
\]
where
\[
M(t, y) = \frac{3y^2 - t^2}{y^5}, \quad N(t, y) = \frac{t}{2y^4}
\]

### Step 2: Check if the equation is exact
For the equation to be exact, the following condition must hold:
\[
\frac{\partial M}{\partial t} = \frac{\partial N}{\partial y}
\]

Let's calculate the partial derivatives.

- \(\frac{\partial M}{\partial t}\):
\[
M(t, y) = \frac{3y^2 - t^2}{y^5}
\]
\[
\frac{\partial M}{\partial t} = \frac{\partial}{\partial t} \left( \frac{3y^2 - t^2}{y^5} \right) = \frac{-2t}{y^5}
\]

- \(\frac{\partial N}{\partial y}\):
\[
N(t, y) = \frac{t}{2y^4}
\]
\[
\frac{\partial N}{\partial y} = \frac{\partial}{\partial y} \left( \frac{t}{2y^4} \right) = \frac{-2t}{2y^5} = \frac{-t}{y^5}
\]

Since \(\frac{\partial M}{\partial t} \neq \frac{\partial N}{\partial y}\), the equation is **not exact**.

### Next Steps
Given that the differential equation is not exact, we need to find another method to solve it, such as an integrating factor. Would you like me to proceed by finding the integrating factor, or would you prefer further clarification on the process so far?

### 5 Related Questions:
1. How do you determine an integrating factor for a non-exact differential equation?
2. Can you explain the significance of exactness in differential equations?
3. What are some alternative methods for solving non-exact differential equations?
4. How does one approach solving linear versus non-linear differential equations?
5. What other conditions, besides exactness, ensure solvability of differential equations?

#### Tip:
When verifying if a differential equation is exact, always ensure you calculate the partial derivatives carefully as small mistakes can lead to incorrect conclusions.

Determine if the differential equation (3y^2 - t^2)/y^5 dy/dt + t/(2y^4) = 0 is exact. If it is, solve it for the initial condition y(1) = 1.

Determine whether the given differential equation is exact and solve it for the initial condition y(1) = 1. This problem involves checking exactness using partial derivatives and covers key topics in differential equations.

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