Exercise · ZW-VARIATION-OF-PARAMETERS-010 · ★★★ Advanced
Find a particular solution of \(y''+y=\sec t\), \(0<t<\pi/2\), with \(y_1=\cos t\), \(y_2=\sin t\).
Choose one option.
Show answer
- D. \(\cos t\,\ln\cos t+t\sin t\)
Explanation
With \(W=1\): \(u_1'=-\sin t\sec t=-\tan t\) gives \(u_1=\ln\cos t\); \(u_2'=\cos t\sec t=1\) gives \(u_2=t\). So \(y_p=\cos t\ln\cos t+t\sin t\).
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
- ZW-VARIATION-OF-PARAMETERS-001 Compute W(y_1,y_2) for y_1=e^t, y_2=e^-t, solutions of y''-y=0.
- 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).