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216_pdfsam_math 54 differential equation solutions odd

216_pdfsam_math 54 differential equation solutions odd - |...

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Chapter 4 The system (9) on page 195 in the text becomes v 1 ( t ) cos 2 t + v 2 ( t ) sin 2 t = 0 2 v 1 ( t ) sin 2 t + 2 v 2 ( t ) cos 2 t = tan 2 t. (4.5) Multiplying the first equation in (4.5) by sin 2 t , the second equation by (1 / 2) cos 2 t , and adding the resulting equations together, we get v 2 ( t ) = 1 2 sin 2 t v 2 = 1 2 sin 2 t dt = 1 4 cos 2 t + c 3 . From the first equation in (4.5) we also obtain v 1 ( t ) = v 2 ( t ) tan 2 t = 1 2 sin 2 2 t cos 2 t = 1 2 1 cos 2 2 t cos 2 t = 1 2 (cos 2 t sec 2 t ) v 1 ( t ) = 1 2 (cos 2 t sec 2 t ) dt = 1 4 (sin 2 t ln | sec 2 t + tan 2 t | ) + c 4 . We take c 3 = c 4 = 0 since we need just one particular solution. Thus y p ( t ) = 1 4 (sin 2 t ln | sec 2 t + tan 2 t | ) cos 2 t 1 4 cos 2 t sin 2 t = 1 4 cos 2 t ln | sec 2 t + tan 2 t | and a general solution to the given equation is y ( t ) = y h ( t ) + y p ( t ) = c 1 cos 2 t + c 2 sin 2 t
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Unformatted text preview: | sec 2 t + tan 2 t | . 2. From Example 1 on page 196 in the text, we know that ±unctions y 1 ( t ) = cos t and y 2 ( t ) = sin t are two linearly independent solutions to the corresponding homogeneous equation, and so its general solution is given by y h ( t ) = c 1 cos t + c 2 sin t. Now we apply the method o± variation o± parameters to fnd a particular solution to the original equation. By the ±ormula (3) on page 194 in the text, y p ( t ) has the ±orm y p ( t ) = v 1 ( t ) y 1 ( t ) + v 2 ( t ) y 2 ( t ) . 212...
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