Exercise · ZW-CHARACTERISTIC-EQUATION-009 · ★★★ Advanced

Solve \(y''-3y'+2y=0\) with \(y(0)=1\), \(y'(0)=0\).

Choose one option.

Show answer

C. \(y=2e^{t}-e^{2t}\)

Explanation

Roots \(1\) and \(2\): \(y=c_1e^t+c_2e^{2t}\). With \(c_1+c_2=1\) and \(c_1+2c_2=0\) we get \(c_2=-1\), \(c_1=2\).

Study Differential equations in ZWMath →

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?