Question

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

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

Hint:

Divide by x plus 1 to get the standard linear form and multiply by the integrating factor.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Write the equation in the form \(\dfrac{dy}{dx}+Py=Q\).

Step 2: Standard Form
Dividing by \(x+1\): \(\dfrac{dy}{dx}-\dfrac{y}{x+1}=e^{x}(x+1)\).

Step 3: Integrating Factor
\(\text{I.F.}=e^{-\int\frac{dx}{x+1}}=\dfrac{1}{x+1}\).

Step 4: Solve
\(\dfrac{y}{x+1}=\int e^{x}(x+1)\cdot\dfrac{1}{x+1}dx=\int e^{x}dx=e^{x}+c\).
So \(y=(x+1)(e^{x}+c)\).

Final Answer:
The general solution is y equal to (x+1) times (e to the x plus c). \[ \boxed{\text{(A) }y=(x+1)(e^{x}+c)} \]