Exercise · ZW-INTEGRATING-FACTOR-005 · ★☆☆ Basic
With \(\mu=e^{x}\), the equation \(y'+y=x\) becomes \((e^{x}y)'=xe^{x}\). What is its general solution?
Choose one option.
Show answer
- B. \(y=x-1+Ce^{-x}\)
Explanation
By parts, \(\int xe^{x}\,dx=(x-1)e^{x}+C\). Dividing by \(e^{x}\): \(y=x-1+Ce^{-x}\). The constant must be divided too.
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-006 Solve y'+2y=e^-x.
- 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?