341h6 - M341 H6(S Zhang 3.5-6 1 EP 3.5 2 5 11 21 22 32...

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M341 H6 (S. Zhang) 3.5-6. 1. EP § 3.5: 2, 5, 11, 21, 22, 32, 34, (Nonhomogeneous equa- tions and undetermined coefficients) ans: 2. Find the general solution: y - y - 2 y = 3 x + 4 ans: 3.5:2 r 2 - r - 2 = 0 r = - 1 , 2 y H = c 1 e - t + c 2 e 2 t Since r = 0 is not a root, y p = ( A + Bx ) Plug it into the equaion, - 2 A - B = 4 - 2 B = 3 y p = - 5 4 - 3 2 x y = y H + y p = c 1 e - t + c 2 e 2 t - 5 4 - 3 2 x 3. Find the general solution: y + y + y = sin 2 x ans: 3.5:5 Rewrite y + y + y = 1 2 - 1 2 cos 2 x r 2 + r + 1 = 0 r = - 1 ± 3 i 2 y H = e - t/ 2 ( c 1 cos 3 t 2 + c 2 sin 3 t 2 ) Since r = 0 and r = 2 i are not roots, y p = ( A ) + ( B cos 2 x + C sin 2 x ) Plug it into the equaion, A = 1 2 - 3 B + 2 C = - 1 2 2 B - 3 C = 0 y p = 1 2 + 3 26 cos 2 x - 1 13 sin 2 x y = y H + y p = = e - t/ 2 ( c 1 cos 3 t 2 + c 2 sin 3 t 2 ) + 1 2 + 3 26 cos 2 x - 1 13 sin 2 x 4. Find the general solution: y + 4 y = 3 x - 1 ans: 3.5:11 For y H = y c (coplementary solution, or solution for the homogeneous equation), we find the roots of chracteristic eq r 3 + 4 r = 0 , r = 0 , ± 2 i y H = c 1 + c 2 cos 2 x + c 3 sin 2 x Since r = 0 is a root, we need an extra x in y p form, y p = x ( A + Bx ) Plug it into the equaion, 4 A = - 1 8 B = 3 y p = x ( - 1 4 + 8 3 x ) y = y H + y p = = c 1 + c 2 cos 2 x + c 3 sin 2 x + x ( - 1 4 + 8 3 x ) 5. Find a y P form, but do not find the coefficients. y - 2 y + 2 y = e x sin x ans: 3.5:21 For y H = y c (coplementary solution, or solution for the homogeneous equation), we find the roots of chracteristic eq r 2 - 2 r + 2 = 0 , r = 1 ± i y H = e t ( c 1 cos t + c 2 sin t ) Since r = 1 + i is a root, we need an extra x in y p form, y p = x ( A cos t + B sin t ) 1
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6. Find a y P form, but do not find the coefficients. y (5) - y (3) = e x + 2 x 2 - 5 ans: 3.5:22 For y H = y c (coplementary solution, or solution for the homogeneous equation), we find the roots of chracteristic eq r 5 - r 3 = 0 , r = 0 , 0 , 0 , 1 , - 1 Since r = 1 is a root, we need an extra x in y p form for the e x term, and because 0 is a triple root, we need an extra x 3 in y p form for the (2 x
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