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

Let's analyze the differential expression 

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

to determine if it is an exact differential.

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

First, expand the given expression:

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

Distribute and simplify:

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

Combine like terms:

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

Here, we identify:

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

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

A differential expression is exact if:

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

Compute the partial derivatives:

\[
\frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (4xy + 5xy^3) = 4x + 15xy^2
\]

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

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

Now, compare the two partial derivatives:

\[
\frac{\partial M}{\partial y} = 4x + 15xy^2, \quad \frac{\partial N}{\partial x} = 4x + 3y^4
\]

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

### Conclusion

The given differential expression \(x (4y \, dx + 2x \, dy) + y^3 (3yx \, dy + 5x \, dx)\) 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 do you have any questions?

### Related Questions:

1. What techniques can be used to solve non-exact differential equations?
2. How do you find an integrating factor for a non-exact differential equation?
3. Are there cases where the integrating factor depends on both \(x\) and \(y\)?
4. How do exact differential equations relate to potential functions?
5. What are the physical interpretations of exact differential equations?

**Tip:** If a differential equation is not exact, look for an integrating factor to transform it into an exact equation.

Learn about exact differentials and how to analyze the differential expression x (4ydx + 2xdy) + y^3(3yx + 5xdx). Explore the conditions for exactness, compute partial derivatives, and determine if the differential is exact. Suitable for advanced undergraduate students and enthusiasts of differential equations.

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