Exercise · ZW-INTEGRATING-FACTOR-004 · ★☆☆ Basic
Write \(y'=y+e^{2x}\) in standard form and find an integrating factor.
Choose one option.
Show answer
- D. \(\mu=e^{-x}\)
Explanation
Standard form is \(y'-y=e^{2x}\), with \(p=-1\). Hence \(\mu=e^{-x}\). The sign of \(p\) changes when \(y\) moves to the left side.
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-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-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?