Exercise · ZW-UNDETERMINED-COEFFICIENTS-010 · ★★★ Advanced
Find a particular solution of \(y''+y=t+e^{t}\).
Choose one option.
Show answer
- B. \(t+\tfrac12e^{t}\)
Explanation
By superposition solve each term: \(y''+y=t\) gives \(t\) and \(y''+y=e^t\) gives \(\tfrac12e^t\) (since \(A(1+1)=1\)). The sum is \(t+\tfrac12e^t\).
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.