Exercise · ZW-CHARACTERISTIC-EQUATION-010 · ★★★ Advanced
In which equation do all solutions tend to \(0\) as \(t\to\infty\)?
Choose one option.
Show answer
- B. \(y''+2y'+5y=0\)
Explanation
Every root must have negative real part. \(y''+2y'+5y=0\) has \(-1\pm2i\); \(y''-y=0\) has \(\pm1\), \(y''+y=0\) has \(\pm i\) (no decay) and \(y''-2y'+5y=0\) has \(1\pm2i\) (grows).
Review the concept: Characteristic equation
The characteristic equation of \(ay''+by'+cy=0\), with constant coefficients \(a,b,c\), is the algebraic equation in the unknown \(r\) that results from trying \(y=e^{rt}\).
\[ar^2+br+c=0\]
More practice problems: characteristic equation
- ZW-CHARACTERISTIC-EQUATION-001 What is the characteristic equation of y''-5y'+6y=0?
- ZW-CHARACTERISTIC-EQUATION-002 Solve y''-y'-6y=0.
- ZW-CHARACTERISTIC-EQUATION-003 Solve y''+4y=0.
- ZW-CHARACTERISTIC-EQUATION-004 Solve y''+2y'+y=0.
- ZW-CHARACTERISTIC-EQUATION-005 What are the roots of the characteristic equation of y''-4y=0?
- ZW-CHARACTERISTIC-EQUATION-006 Solve y''+2y'+5y=0.
- ZW-CHARACTERISTIC-EQUATION-007 An equation y''+by'+cy=0 has characteristic roots 1 and -4. Which equation is it?
- ZW-CHARACTERISTIC-EQUATION-008 For which values of b does y''+by'+9y=0 have a double characteristic root?