y(2x - y + 1) * dx + x(3x - 4y + 3) * dy = 0

The given differential equation is:

\[
y(2x - y + 1) \, dx + x(3x - 4y + 3) \, dy = 0
\]

To solve this, we can check if the equation is exact. A differential equation of the form \( M(x, y) \, dx + N(x, y) \, dy = 0 \) is exact if:

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

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

We have:
- \( M(x, y) = y(2x - y + 1) = 2xy - y^2 + y \)
- \( N(x, y) = x(3x - 4y + 3) = 3x^2 - 4xy + 3x \)

### Step 2: Compute partial derivatives

1. \(\frac{\partial M}{\partial y}\):
   \[
   \frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (2xy - y^2 + y) = 2x - 2y + 1
   \]

2. \(\frac{\partial N}{\partial x}\):
   \[
   \frac{\partial N}{\partial x} = \frac{\partial}{\partial x} (3x^2 - 4xy + 3x) = 6x - 4y + 3
   \]

### Step 3: Check for exactness

For exactness, we need:
\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
\]
Comparing:
\[
2x - 2y + 1 \quad \text{and} \quad 6x - 4y + 3
\]
Since these are not equal, the equation is **not exact**.

### Step 4: Try to find an integrating factor

An integrating factor might depend on \(x\), \(y\), or a combination of both. A common approach is to try an integrating factor based on \(x\) or \(y\). If you'd like, I can continue by testing for possible integrating factors.

Do you want to explore integrating factors, or should I attempt a different method?

---

**Relative Questions:**

1. Would you like to continue solving using an integrating factor?
2. Should we explore whether an integrating factor depends only on \(x\) or \(y\)?
3. Do you want me to explain more about the conditions for exactness?
4. Are you interested in alternative solution methods for non-exact equations?
5. Should we attempt a substitution to simplify the equation?

**Tip:** For non-exact differential equations, integrating factors can often turn them into exact equations, making them easier to solve.

Explore the step-by-step solution of a non-exact differential equation: y(2x - y + 1)dx + x(3x - 4y + 3)dy = 0. Learn how to check for exactness, compute partial derivatives, and apply integrating factors to simplify and solve the equation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation