(2x-y+4)dy+(x-2y+5)dx=0

Jasonbradley

4 min read Jun 16, 2024

Solving the Differential Equation (2x-y+4)dy + (x-2y+5)dx = 0

This article will guide you through the process of solving the given differential equation:

(2x-y+4)dy + (x-2y+5)dx = 0

This equation is a first-order linear differential equation. We can solve it using the following steps:

1. Rearrange the equation

First, we rearrange the equation to make it easier to work with:

(2x-y+4)dy = -(x-2y+5)dx

(2x-y+4)dy / dx = -(x-2y+5)

2. Identify the integrating factor

To solve this differential equation, we need to find an integrating factor. This is a function that we multiply both sides of the equation by to make it easier to integrate.

In this case, the integrating factor is:

μ(x, y) = e^(∫(∂/∂y)(2x-y+4) - (∂/∂x)(x-2y+5) dx)

μ(x, y) = e^(∫(-1) - (1) dx)

μ(x, y) = e^(-2x)

3. Multiply both sides by the integrating factor

Multiply both sides of the rearranged equation by the integrating factor:

e^(-2x) (2x-y+4)dy / dx = -e^(-2x) (x-2y+5)

4. Simplify the equation

The left side of the equation can be rewritten as the derivative of a product:

d/dx (e^(-2x) (2x-y+4)y) = -e^(-2x) (x-2y+5)

5. Integrate both sides

Integrate both sides of the equation with respect to x:

∫d/dx (e^(-2x) (2x-y+4)y) dx = -∫e^(-2x) (x-2y+5) dx

e^(-2x) (2x-y+4)y = (1/2)e^(-2x) (x-2y+5) + C

Where C is the constant of integration.

6. Solve for y

Now we can solve for y:

y = (1/2) (x-2y+5) / (2x-y+4) + Ce^(2x) / (2x-y+4)

7. Simplify the solution

We can simplify the solution by combining terms:

y = (x + 5) / (4x - 2y + 8) + Ce^(2x) / (2x-y+4)

This is the general solution to the given differential equation.

Conclusion

We have successfully solved the differential equation (2x-y+4)dy + (x-2y+5)dx = 0 using the method of integrating factors. The general solution is: