Exercise · ZW-CHARACTERISTIC-EQUATION-006 · ★★☆ Intermediate
Solve \(y''+2y'+5y=0\).
Choose one option.
Show answer
- C. \(y=e^{-t}(c_1\cos2t+c_2\sin2t)\)
Explanation
\(r=\frac{-2\pm\sqrt{4-20}}{2}=-1\pm2i\). The real part \(-1\) goes in the exponential and the imaginary part \(2\) in the sine and cosine.
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-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?
- ZW-CHARACTERISTIC-EQUATION-009 Solve y''-3y'+2y=0 with y(0)=1, y'(0)=0.
← ZW-CHARACTERISTIC-EQUATION-005 ZW-CHARACTERISTIC-EQUATION-007 →