solutions3

solutions3 - 3.6 5. The functions y1 = cos t and y2 = sin t...

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3.6 5. The functions y 1 = cos t and y 2 = sin t form a fundamental set of solutions, with W ( y 1 ,y 2 ) = 1. The particular solution is given by y p = u ( t ) y 1 + v ( t ) y 2 in which (according to Thm. 3.6.1) u ( t ) = - Z sin t tan tdt = - Z 1 - cos 2 t cos t dt = - Z 1 cos t dt + Z cos tdt = - ln (sec t + tan t ) + sin t v ( t ) = Z cos t tan tdt = Z sin tdt = - cos t The particular solution is y p = - (cos t )[ln (sec t + tan t )] and the general solution is y ( t ) = c 1 cos t + c 2 sin t - (cos t )[ln (sec t + tan t )] 12. The functions y 1 = cos 2 t and y 2 = sin 2 t form a fundamental set of solutions, with W ( y 1 ,y 2 ) = 2. The particular solution is given by y p = u ( t ) y 1 + v ( t ) y 2 in which (according to Thm. 3.6.1) u ( t ) = - 1 2 Z t t 0 g ( s ) sin 2 sdt v ( t ) = 1 2 Z t t 0 g ( s ) cos 2 sdt The particular solution is y p = - 1 2 cos 2 t Z t t 0 g ( s ) sin 2 sds + 1 2 sin 2 t Z t t 0 g ( s ) cos 2 sds = - 1 2 Z t t 0 g ( s ) sin 2 s cos 2 tds + 1 2 Z t t 0 g ( s ) cos 2 s sin 2 tds Note that the second step we write the functions of t inside the integral because they are constant with respect to
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solutions3 - 3.6 5. The functions y1 = cos t and y2 = sin t...

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