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

3 min read Jun 16, 2024

Solving the Differential Equation (x + 2y) dy/dx = 2x - y

This article will guide you through the process of solving the first-order differential equation (x + 2y) dy/dx = 2x - y. We'll use the method of integrating factors to achieve this.

1. Rearranging the Equation

First, let's rearrange the given equation to put it in a standard form:

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

This can be rewritten as:

dy/dx + (y/(x + 2y)) = (2x)/(x + 2y)

2. Identifying the Integrating Factor

Now, we can identify the integrating factor (IF). The equation is in the form:

dy/dx + P(x)y = Q(x)

Where:

P(x) = 1/(x + 2y)
Q(x) = 2x/(x + 2y)

The integrating factor is calculated as:

IF = exp(∫P(x) dx)

IF = exp(∫(1/(x + 2y)) dx)

IF = exp(ln|x + 2y|)

IF = x + 2y

3. Multiplying by the Integrating Factor

Multiply both sides of the rearranged equation by the integrating factor (x + 2y):

(x + 2y)(dy/dx) + (x + 2y)(y/(x + 2y)) = (x + 2y)(2x/(x + 2y))

Simplifying, we get:

(x + 2y)dy/dx + y = 2x

4. Integrating Both Sides

Now, we can integrate both sides with respect to x:

∫[(x + 2y)dy/dx + y] dx = ∫2x dx

The left side can be simplified using the product rule for differentiation:

d/dx[(x + 2y)y] = 2x

Integrating both sides:

(x + 2y)y = x² + C

Where C is the constant of integration.

5. Solving for y

Finally, we can solve for y:

y(x + 2y) = x² + C

2y² + xy - x² - C = 0

This equation represents the general solution to the given differential equation.

Conclusion

We have successfully solved the differential equation (x + 2y) dy/dx = 2x - y using the method of integrating factors. The solution is expressed as an implicit equation involving y and x.