Question

Find the integrating factor of the differential equation \(x\dfrac{dy}{dx}+2y=x^2\ (x\neq0)\).

Hint:

Convert to standard linear form \(y'+Py=Q\), then compute \(e^{\int P\,dx}\).

Answer 1

Step 1: Understanding the Concept:
Write the equation in the standard linear form \(\dfrac{dy}{dx}+Py=Q\) so \(P\) can be read off.

Step 2: Standard form:
Dividing by \(x\): \(\dfrac{dy}{dx}+\dfrac{2}{x}y=x\), so \(P=\dfrac{2}{x}\).

Step 3: Computing the integrating factor:
\(\text{I.F.}=e^{\int P\,dx}=e^{\int \frac{2}{x}dx}=e^{2\ln x}=x^2\).

Final Answer:
Integrating factor \(=\boxed{x^2}\).