Question

For the linear differential equation \[ \frac{dy}{dx} + \frac{2}{x}y = x^2 \] calculate the integrating factor (I.F.).

• A. \( x \)
• B. \( x^2 \)
• C. \( x^3 \)
• D. \( \ln(x) \)

Hint:

For equations of the form \(\frac{dy}{dx}+\frac{n}{x}y=Q(x)\), the integrating factor is usually \(x^n\).

Answer 1

The Correct Option is B

Concept: For a linear differential equation: \[ \frac{dy}{dx}+P(x)y=Q(x) \] the integrating factor is: \[ I.F.=e^{\int P(x)\,dx} \]

Step 1: Identify \(P(x)\).
Comparing with the standard form: \[ P(x)=\frac{2}{x} \]

Step 2: Calculate the integral.
\[ \int \frac{2}{x}dx = 2\ln|x| \]

Step 3: Find the integrating factor.
\[ I.F.=e^{2\ln|x|} \] Using logarithmic properties: \[ e^{2\ln|x|}=e^{\ln(x^2)}=x^2 \] Hence, the integrating factor is: \[ x^2 \]