Find the general solution of the differential equation $(1+x)\frac{dy}{dx}-xy=x+x^2$.

• A. $y=1+x+C(1+x)$
• B. $y=x+C(1+x)e^x$
• C. $y=1+x+C(1+x)e^x$
• D. $y=1+x+C(1+x)e^x$

Answer: (A)

Verified step by step guidance

1. Step 1: Rewrite the given differential equation in standard form. The given equation is (1 + x) * (dy/dx) - x * y = x + x^2. Divide through by (1 + x) (assuming 1 + x ≠ 0) to get dy/dx - (x / (1 + x)) * y = (x + x^2) / (1 + x).
2. Step 2: Recognize that this is a first-order linear differential equation of the form dy/dx + P(x) * y = Q(x), where P(x) = -x / (1 + x) and Q(x) = (x + x^2) / (1 + x).
3. Step 3: Find the integrating factor (IF). The integrating factor is given by e^(∫P(x) dx). Here, P(x) = -x / (1 + x). Compute the integral ∫(-x / (1 + x)) dx, which simplifies to -∫(x / (1 + x)) dx = -ln|1 + x|. Thus, the integrating factor is e^(-ln|1 + x|) = 1 / (1 + x).
4. Step 4: Multiply through the differential equation by the integrating factor (1 / (1 + x)) to make the left-hand side an exact derivative. This gives (1 / (1 + x)) * dy/dx - (x / (1 + x)^2) * y = (x + x^2) / (1 + x)^2. The left-hand side simplifies to d/dx [y / (1 + x)].
5. Step 5: Integrate both sides with respect to x. The left-hand side integrates to y / (1 + x), and the right-hand side requires integrating (x + x^2) / (1 + x)^2. Solve the integral on the right-hand side, and then multiply through by (1 + x) to isolate y. Add the constant of integration C to find the general solution.