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: Guessing and checking the pattern:
Notice \(\dfrac{d}{dx}(x^2y)=x^2\dfrac{dy}{dx}+2xy\); multiplying the given equation \(x\,dy/dx+2y=x^2\) by \(x\) gives exactly \(x^2\dfrac{dy}{dx}+2xy=x^3\), the left side of that derivative.

Step 2: Reading off the factor:
So multiplying by \(x\) turns the left side into \(\dfrac{d}{dx}(x^2y)\) — but by the standard formula this multiplier is \(x\cdot(\text{I.F. in standard form})\), confirming \(\text{I.F.}=x^2\) directly via \(e^{2\ln x}\).

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