Exercise · ZW-INTEGRATING-FACTOR-007 · ★★☆ Intermediate
For \(x>0\), solve \(xy'+2y=x^2\).
Choose one option.
Show answer
- C. \(y=\dfrac{x^2}{4}+\dfrac{C}{x^2}\)
Explanation
First divide by \(x\): \(y'+\frac{2}{x}y=x\). Then \(\mu=x^2\) and \((x^2y)'=x^3\), so \(x^2y=\frac{x^4}{4}+C\).
Review the concept: Integrating factor
An integrating factor is a function \(\mu\) that, when multiplied by a first-order linear equation with coefficient \(p\), turns the left side into the derivative of a product.
\[\mu(x)=e^{\int p(x)\,dx}\]
More practice problems: integrating factor
- ZW-INTEGRATING-FACTOR-001 Find an integrating factor of y'+2y=x.
- ZW-INTEGRATING-FACTOR-002 For x>0, find an integrating factor of y'+(1)/(x) y=x^2.
- ZW-INTEGRATING-FACTOR-003 Find an integrating factor of y'+3x^2y=x^2.
- ZW-INTEGRATING-FACTOR-004 Write y'=y+e^2x in standard form and find an integrating factor.
- ZW-INTEGRATING-FACTOR-005 With μ=e^x, the equation y'+y=x becomes (e^xy)'=xe^x. What is its general solution?
- ZW-INTEGRATING-FACTOR-006 Solve y'+2y=e^-x.
- ZW-INTEGRATING-FACTOR-008 Solve y'+2y=4 with y(0)=3.
- ZW-INTEGRATING-FACTOR-009 On (-π/2,π/2), which is an integrating factor of y'+(tan x) y=1?