x (4ydx + 2xdy) + y (3y x + 5 d x ) using exact differential method

To determine if the differential expression \(x (4y \, dx + 2x \, dy) + y (3y \, dx + 5 \, dy)\) is an exact differential, we first need to rewrite it in the form of:

\[
M(x, y) \, dx + N(x, y) \, dy
\]

where \(M(x, y)\) and \(N(x, y)\) are functions of \(x\) and \(y\). 

### Step 1: Expand and identify \(M(x, y)\) and \(N(x, y)\)

First, let's expand the given expression:

\[
x (4y \, dx + 2x \, dy) + y (3y \, dx + 5 \, dy) 
\]

Distribute and simplify:

\[
= (4xy \, dx + 2x^2 \, dy) + (3y^2 \, dx + 5y \, dy)
\]

Now, we can combine like terms:

\[
= (4xy + 3y^2) \, dx + (2x^2 + 5y) \, dy
\]

Here, we have:

\[
M(x, y) = 4xy + 3y^2, \quad N(x, y) = 2x^2 + 5y
\]

### Step 2: Check if the differential is exact

A differential expression is exact if:

\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
\]

Let's compute these partial derivatives:

\[
\frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (4xy + 3y^2) = 4x + 6y
\]

\[
\frac{\partial N}{\partial x} = \frac{\partial}{\partial x} (2x^2 + 5y) = 4x
\]

### Step 3: Determine if the expression is exact

Now, compare the two partial derivatives:

\[
\frac{\partial M}{\partial y} = 4x + 6y, \quad \frac{\partial N}{\partial x} = 4x
\]

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

### Conclusion

The given differential expression \(x (4y \, dx + 2x \, dy) + y (3y \, dx + 5 \, dy)\) is not an exact differential because the mixed partial derivatives are not equal.

Would you like to learn about methods to solve non-exact differentials, or have any questions? Here are some related questions:

1. How do you solve non-exact differential equations?
2. What is an integrating factor and how does it make a differential exact?
3. Can you provide an example where the integrating factor is a function of \(x\) only?
4. How do you determine if a differential equation is linear or nonlinear?
5. What are some applications of exact differential equations in physics?

**Tip:** To make a non-exact differential equation exact, try finding an integrating factor.

Explore the analysis of the differential expression \(x (4y \, dx + 2x \, dy) + y (3y \, dx + 5 \, dy)\) using the exact differential method. Understand the criteria for determining exact differentials and the step-by-step solution process. Suitable for advanced undergraduate students studying differential equations and mathematical analysis.

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