Exercise · ZW-INTEGRATING-FACTOR-006 · ★★☆ Intermediate
Solve \(y'+2y=e^{-x}\).
Choose one option.
Show answer
- A. \(y=e^{-x}+Ce^{-2x}\)
Explanation
With \(\mu=e^{2x}\): \((e^{2x}y)'=e^{x}\), so \(e^{2x}y=e^{x}+C\) and \(y=e^{-x}+Ce^{-2x}\).
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-007 For x>0, solve xy'+2y=x^2.
- 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?