Question

The solution of the differential equation \( (x + 1)\frac{dy}{dx} - y = e^{3x}(x + 1)^2 \) is:

• A. \( y = (x + 1)e^{3x} + C \)
• B. \( 3y = (x + 1) + e^{3x} + C \)
• C. \( \frac{3y}{x+1} = e^{3x} + C \)
• D. \( ye^{-3x} = 3(x + 1) + C \)

Hint:

For linear first-order differential equations, always start by finding the integrating factor and multiplying through to solve.

Answer 1

The Correct Option is C

Step 1: The provided differential equation is \( (x + 1)\frac{dy}{dx} - y = e^{3x}(x + 1)^2 \). This is a first-order linear differential equation. To express it in the standard form \( \frac{dy}{dx} + P(x) y = Q(x) \), we rearrange it as: \( P(x) = -\frac{1}{x+1} \) and \( Q(x) = e^{3x}(x + 1) \).
Step 2: The integrating factor (IF) is calculated as: \( IF = e^{\int P(x) dx} = e^{\int -\frac{1}{x+1} dx} = \frac{1}{x+1} \). 
Step 3: Multiplying the standard form equation by the integrating factor yields: \( \frac{3y}{x+1} = e^{3x} + C \). The derived solution is therefore: \( \frac{3y}{x+1} = e^{3x} + C \).