The given equation is:

\[
(2x + y - 1) \, dy = (x - 2y + 5) \, dx
\]

This is a first-order differential equation. To solve it, we aim to separate the variables or recognize any special type of equation (e.g., exact, homogeneous, or linear).

First, let's rewrite the equation as:

\[
\frac{dy}{dx} = \frac{x - 2y + 5}{2x + y - 1}
\]

Now we can check if it's exact or separable.

### Step 1: Check if the equation is exact
For an exact equation, we would have the form \( M(x, y) \, dx + N(x, y) \, dy = 0 \), where:

\[
M(x, y) = x - 2y + 5 \quad \text{and} \quad N(x, y) = -(2x + y - 1)
\]

To check if it's exact, we find the partial derivatives:

\[
\frac{\partial M}{\partial y} = -2 \quad \text{and} \quad \frac{\partial N}{\partial x} = -2
\]

Since \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\), the equation is **exact**.

### Step 2: Solve the exact equation
Since the equation is exact, there exists a function \( \psi(x, y) \) such that:

\[
\frac{\partial \psi}{\partial x} = M(x, y) = x - 2y + 5 \quad \text{and} \quad \frac{\partial \psi}{\partial y} = N(x, y) = -(2x + y - 1)
\]

#### Integrating \( M(x, y) \) with respect to \(x\):
\[
\psi(x, y) = \int (x - 2y + 5) \, dx = \frac{x^2}{2} - 2xy + 5x + h(y)
\]
where \( h(y) \) is a function of \( y \) that appears because of the partial integration.

#### Differentiate \( \psi(x, y) \) with respect to \(y\):
\[
\frac{\partial \psi}{\partial y} = -2x + h'(y)
\]

Now, set this equal to \(N(x, y)\):
\[
-2x + h'(y) = -(2x + y - 1)
\]

Simplifying:
\[
-2x + h'(y) = -2x - y + 1
\]

Thus:
\[
h'(y) = -y + 1
\]

#### Integrating \(h'(y)\) with respect to \(y\):
\[
h(y) = -\frac{y^2}{2} + y + C
\]

### Step 3: Write the solution
The general solution is given by:

\[
\psi(x, y) = \frac{x^2}{2} - 2xy + 5x - \frac{y^2}{2} + y + C = 0
\]

This is the implicit solution to the differential equation.

Would you like more details or have any questions?

Here are five related questions to explore:
1. How do we determine if a differential equation is separable?
2. What other methods can be used to solve non-exact differential equations?
3. What are the conditions for a differential equation to be homogeneous?
4. Can you derive a differential equation from real-life applications like population models?
5. How can we verify that a solution satisfies a differential equation?

**Tip:** When solving exact equations, always check that the mixed partial derivatives of the functions are equal.

Learn how to solve the exact first-order differential equation (2x + y - 1)dy = (x - 2y + 5)dx using the method of exact equations. This detailed solution covers checking for exactness, finding partial derivatives, and integrating to find the general solution. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation