Exercise · ZW-VARIATION-OF-PARAMETERS-008 · ★★☆ Intermediate
In \(y''+y=t\) with \(y_1=\cos t\), \(y_2=\sin t\) (\(W=1\)), compute \(u_2=\int y_1g\,dt\) (no constant).
Choose one option.
Show answer
- C. \(t\sin t+\cos t\)
Explanation
By parts, \(\int t\cos t\,dt=t\sin t-\int\sin t\,dt=t\sin t+\cos t\).
Review the concept: Variation of parameters
A method that builds a particular solution \(y_p\) of \(y''+py'+qy=g(t)\) from two homogeneous solutions \(y_1,y_2\) and their Wronskian \(W\), by letting the constants vary.
\[y_p=-y_1\!\int\!\frac{y_2\,g}{W}\,dt+y_2\!\int\!\frac{y_1\,g}{W}\,dt\]
More practice problems: variation of parameters
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- ZW-VARIATION-OF-PARAMETERS-006 In y''+y=tan t with y_1=cos t, y_2=sin t (W=1), what is u_2'=(y_1g)/(W)?
- ZW-VARIATION-OF-PARAMETERS-007 With y_1=e^t, y_2=e^-t and W=-2, find y_p for y''-y=e^2t by variation of parameters.
- ZW-VARIATION-OF-PARAMETERS-009 With y_1=e^t, y_2=te^t (W=e^2t), find y_p for y''-2y'+y=(e^t)/(t), t>0, taking u_1 and u_2 with no…
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