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The given differential equation is:

\[
(5y - x)dx + (3y - 7x)dy = 0
\]
with the condition that \( x = 2 \) and \( y = 5 \).

### Step 1: Check for Exactness
To solve the differential equation, we first check if it is exact. The general form of an exact equation is:

\[
M(x, y)dx + N(x, y)dy = 0
\]

Here, \( M(x, y) = 5y - x \) and \( N(x, y) = 3y - 7x \).

The condition for exactness is that:

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

Let's compute these partial derivatives.

\[
\frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (5y - x) = 5
\]
\[
\frac{\partial N}{\partial x} = \frac{\partial}{\partial x} (3y - 7x) = -7
\]

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

### Step 2: Find an Integrating Factor
Since the equation is not exact, we need to find an integrating factor. A possible integrating factor could depend on \( x \) or \( y \). Let's try finding an integrating factor that depends on \( x \).

If the integrating factor is a function of \( x \), say \( \mu(x) \), then we multiply the entire equation by \( \mu(x) \):

\[
\mu(x)(5y - x)dx + \mu(x)(3y - 7x)dy = 0
\]

For this to be exact, the condition becomes:

\[
\frac{\partial}{\partial y} \left( \mu(x)(5y - x) \right) = \frac{\partial}{\partial x} \left( \mu(x)(3y - 7x) \right)
\]

I will now solve for the integrating factor and proceed with solving the differential equation.The partial derivatives are:

\[
\frac{\partial}{\partial y} \left( \mu(x)(5y - x) \right) = 5 \mu(x)
\]
\[
\frac{\partial}{\partial x} \left( \mu(x)(3y - 7x) \right) = (-7x + 3y) \mu'(x) - 7 \mu(x)
\]

For the equation to be exact, we equate these two expressions. This leads to the equation:

\[
5 \mu(x) = (-7x + 3y) \mu'(x) - 7 \mu(x)
\]

This needs to be solved to find \( \mu(x) \), the integrating factor. Let me solve this differential equation next.The integrating factor is:

\[
\mu(x) = \frac{C_1}{(7x - 3y)^{12/7}}
\]

This shows that the integrating factor depends on both \(x\) and \(y\). However, solving this form directly for general cases is complex. Alternatively, we may consider solving this specific differential equation using known methods or try other forms of integrating factors, like those depending on \( y \), for easier computation.

Would you like me to explore this further, or attempt another method?

Here are some related questions:
1. How do we determine when a differential equation is exact?
2. What is the significance of integrating factors in solving differential equations?
3. How can integrating factors simplify non-exact equations?
4. What methods can be used when integrating factors aren't straightforward?
5. Can integrating factors depend on both \(x\) and \(y\) simultaneously?

**Tip:** When solving differential equations, exactness checks and integrating factors can be key tools to simplify the solution process.

(5y - x)dx + (3y - 7x)dy = 0, when x = 2, y = 5

Solve the differential equation (5y - x)dx + (3y - 7x)dy = 0 using integrating factors, with the given condition x = 2, y = 5. This problem covers exactness conditions and methods for solving non-exact equations.

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