224_pdfsam_math 54 differential equation solutions odd

# 224_pdfsam_math 54 differential equation solutions odd -...

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Chapter 4 where v 1 ( t )and v 2 ( t ) are determined by the system v 0 1 cos t + v 0 2 sin t =0 , v 0 1 sin t + v 0 2 cos t =3sec t. Multiplying the frst equation by cos t and the second equation by sin t and subtracting the results, we get v 0 1 = 3sec t sin t = 3tan t. Hence v 1 ( t )= 3 Z tan tdt =3ln | cos t | + C 1 . To fnd v 0 2 ( t ), we multiply the frst equation oF the above system by sin t , the second by cos t , and add the results to obtain v 0 2 =3sec t cos t =3 v 2 ( t )=3 t + C 2 . ThereFore, For this frst equation (with g 1 ( t )=3sec t ), by letting C 1 = C 2 =0 ,w ehav ea particular solution given by y p, 1 ( t )=3cos t ln | cos t | +3 t sin t. The nonhomogeneous term g 2 ( t )=
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Unformatted text preview: undetermined coeﬃcients. Thus, a particular solution to this nonhomogeneous equation will have the Form y p, 2 ( t ) = A 2 t 2 + A 1 t + A ⇒ y p, 2 ( t ) = 2 A 2 t + A 1 ⇒ y p, 2 ( t ) = 2 A 2 . Plugging these expressions into the equation y + y = − t 2 + 1 yields y p, 2 + y p, 2 = 2 A 2 + A 2 t 2 + A 1 t + A = A 2 t 2 + A 1 t + (2 A 2 + A ) = − t 2 + 1 . By equating coeﬃcients, we obtain A 2 = − 1 , A 1 = 0 , 2 A 2 + A = 1 ⇒ A = 3 . 220...
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