(1+x)dy/dx-xy=x+x^2

Jasonbradley

3 min read Jun 16, 2024

Solving the Differential Equation: (1+x)dy/dx - xy = x + x^2

This article will guide you through the process of solving the first-order linear differential equation:

(1 + x) dy/dx - xy = x + x^2

1. Identifying the Form

The given differential equation is in the form of a linear first-order differential equation:

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

where:

P(x) = -x / (1 + x)
Q(x) = (x + x^2) / (1 + x)

2. Finding the Integrating Factor

The integrating factor (IF) is calculated as:

IF = exp(∫P(x)dx)

Let's find the IF for our equation:

IF = exp(∫(-x / (1 + x))dx)

Using integration by substitution (u = 1 + x), we get:

IF = exp(-∫(u - 1) / u du)

IF = exp(-(u - ln|u|))

IF = exp(-(1 + x) + ln|1 + x|)

IF = (1 + x) * exp(-(1 + x))

3. Multiplying the Equation by the Integrating Factor

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

(1 + x) * exp(-(1 + x)) * (dy/dx) - x * (1 + x) * exp(-(1 + x)) * y = (x + x^2) * exp(-(1 + x))

4. Simplifying and Integrating

Notice that the left side of the equation is the derivative of a product:

d/dx [y * (1 + x) * exp(-(1 + x))] = (x + x^2) * exp(-(1 + x))

Integrate both sides with respect to x:

y * (1 + x) * exp(-(1 + x)) = ∫(x + x^2) * exp(-(1 + x)) dx + C

To solve the integral on the right side, you can use integration by parts twice. After solving the integral, you'll get:

y * (1 + x) * exp(-(1 + x)) = -x * exp(-(1 + x)) + C

5. Solving for y

Finally, solve for y to obtain the general solution:

y = (-x + C * exp(1 + x)) / (1 + x)

Conclusion

The solution to the differential equation (1 + x) dy/dx - xy = x + x^2 is: