solutions3

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 12/29/2010 for the course MATH 265 taught by Professor Leahkeshet during the Winter '10 term at UBC.

### Page1 / 5

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online