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: \[ \frac{3y}{x+1} = e^{3x} + C \]

Answer 2

Step 1: Given Differential Equation
The given differential equation is: \[ (x + 1)\frac{dy}{dx} - y = e^{3x}(x + 1)^2 \] We need to solve this first-order linear differential equation.
Step 2: Rearrange the Equation
Rearrange the equation to isolate the derivative term on one side: \[ (x + 1)\frac{dy}{dx} = y + e^{3x}(x + 1)^2 \] This can be written in the standard form: \[ \frac{dy}{dx} - \frac{y}{x + 1} = e^{3x}(x + 1) \]
Step 3: Identify the Integrating Factor
The integrating factor \( \mu(x) \) is given by: \[ \mu(x) = e^{\int -\frac{1}{x + 1} \, dx} = e^{-\log_e (x + 1)} = \frac{1}{x + 1} \]
Step 4: Multiply the Equation by the Integrating Factor
Multiply both sides of the differential equation by \( \frac{1}{x + 1} \): \[ \frac{1}{x + 1} \frac{dy}{dx} - \frac{y}{(x + 1)^2} = e^{3x} \] This simplifies to: \[ \frac{d}{dx} \left( \frac{y}{x + 1} \right) = e^{3x} \]
Step 5: Integrate Both Sides
Now, integrate both sides with respect to \( x \): \[ \int \frac{d}{dx} \left( \frac{y}{x + 1} \right) dx = \int e^{3x} \, dx \] The left-hand side becomes: \[ \frac{y}{x + 1} \] The right-hand side is: \[ \frac{e^{3x}}{3} + C \]
Step 6: Solve for \( y \)
Now, multiply both sides by \( x + 1 \) to solve for \( y \): \[ y = (x + 1) \left( \frac{e^{3x}}{3} + C \right) \]
Step 7: Final Answer
To express the solution in a more simplified form, we multiply both sides by 3: \[ \frac{3y}{x + 1} = e^{3x} + 3C \] Let \( C' = 3C \), where \( C' \) is the constant of integration. Therefore, the solution is: \[ \frac{3y}{x + 1} = e^{3x} + C' \] Thus, the solution to the differential equation is: \[ \boxed{\frac{3y}{x + 1} = e^{3x} + C} \]