Question

An integrating factor of the differential equation \[ (x^2+1)\frac{dy}{dx}+xy=x^3 \] is

• A. \(\frac{x}{1+x^2}\)
• B. \(\frac{1}{2}\log(1+x^2)\)
• C. \(\sqrt{1+x^2}\)
• D. \(e^{\log(1+x^2)}\)

Hint:

For a linear differential equation \[ \frac{dy}{dx}+Py=Q, \] the integrating factor is \[ e^{\int P\,dx}. \] Always first divide the equation by the coefficient of \(\frac{dy}{dx}\).

Answer 1

The Correct Option is C

Step 1: Write in standard linear form.
\(\dfrac{dy}{dx}+\dfrac{x}{x^2+1}\,y = \dfrac{x^3}{x^2+1}\). Here \(P(x)=\dfrac{x}{x^2+1}\).

Step 2: Compute the integrating factor.
\(\displaystyle\int P\,dx = \int\frac{x}{x^2+1}\,dx = \frac{1}{2}\ln(1+x^2)\).
\[\text{IF} = e^{\frac{1}{2}\ln(1+x^2)} = \sqrt{1+x^2}\]
\[ \boxed{\sqrt{1+x^2}} \]