Exercise · ZW-UNDETERMINED-COEFFICIENTS-009 · ★★★ Advanced
Find a particular solution of \(y''+2y'+y=e^{-t}\).
Choose one option.
Show answer
- A. \(\tfrac12 t^2e^{-t}\)
Explanation
The root \(-1\) is double (\(r^2+2r+1=(r+1)^2\)), so \(e^{-t}\) and \(te^{-t}\) solve the homogeneous equation. Guess \(y_p=At^2e^{-t}\): \(y_p''+2y_p'+y_p=2Ae^{-t}\), and \(A=\tfrac12\).
Review the concept: Undetermined coefficients
A method that finds a particular solution \(y_p\) of \(ay''+by'+cy=g(t)\) by guessing a function of the same type as \(g\), with coefficients \(A_i\) fixed by substitution.
\[y_p=t^{s}\,(A_0+A_1t+\cdots+A_nt^n)\,e^{kt}\]
More practice problems: undetermined coefficients
- ZW-UNDETERMINED-COEFFICIENTS-001 Find a particular solution of y''-y=6e^2t using the guess y_p=Ae^2t.
- ZW-UNDETERMINED-COEFFICIENTS-002 A particular solution of y''+4y=8t has the form y_p=At+B. Which one is it?
- ZW-UNDETERMINED-COEFFICIENTS-003 For y''-2y'-3y=e^3t, which guess for y_p works?
- ZW-UNDETERMINED-COEFFICIENTS-004 With y_p=Ate^3t for y''-2y'-3y=e^3t, what is A?
- ZW-UNDETERMINED-COEFFICIENTS-005 For which g(t) can undetermined coefficients be used in ay''+by'+cy=g(t)?
- ZW-UNDETERMINED-COEFFICIENTS-006 Find a particular solution of y''+3y'+2y=10sin t.
- ZW-UNDETERMINED-COEFFICIENTS-007 For y''+4y=cos 2t, which guess for y_p works?
- ZW-UNDETERMINED-COEFFICIENTS-008 Find the general solution of y''-y=4t.
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