Exercise · ZW-VARIATION-OF-PARAMETERS-001 · ★☆☆ Basic
Compute \(W(y_1,y_2)\) for \(y_1=e^{t}\), \(y_2=e^{-t}\), solutions of \(y''-y=0\).
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- B. \(-2\)
Explanation
\(W=y_1y_2'-y_1'y_2=e^t(-e^{-t})-e^te^{-t}=-1-1=-2\).
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
- ZW-VARIATION-OF-PARAMETERS-002 Compute W(cos 2t,sin 2t), with y_1=cos 2t, y_2=sin 2t solutions of y''+4y=0.
- ZW-VARIATION-OF-PARAMETERS-003 To apply the formula to 2y''+2y=8t you must put it in the form y''+py'+qy=g(t). What is g(t)?
- ZW-VARIATION-OF-PARAMETERS-004 When computing u_1 and u_2 by integration, why can the constants of integration be omitted?
- ZW-VARIATION-OF-PARAMETERS-005 In which equation does variation of parameters work but undetermined coefficients do not?
- 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-008 In y''+y=t with y_1=cos t, y_2=sin t (W=1), compute u_2=∫ y_1g dt (no constant).
- 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…