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: Given the differential equation: \[ (x + 1)\frac{dy}{dx} - y = e^{3x}(x + 1)^2 \] This is a linear first-order differential equation. Rewriting it in the standard linear form: \[ \frac{dy}{dx} + P(x) y = Q(x) \] where \( P(x) = -\frac{1}{x+1} \) and \( Q(x) = e^{3x}(x + 1) \). 
Step 2: Use the integrating factor (IF): \[ IF = e^{\int P(x) dx} = e^{\int -\frac{1}{x+1} dx} = \frac{1}{x+1} \] 
Step 3: Multiply both sides of the equation by the integrating factor: \[ \frac{3y}{x+1} = e^{3x} + C \] Thus, the solution is: \[ \frac{3y}{x+1} = e^{3x} + C \]