Question

help with questions number 4 and 5 only

Section 4.5 - Method of undetermined coefficients, annihilator approach Solve the following using the method of undetermined coefficients, obtain the general solution y = yet Yp! 1. y" – 8y' – 48y = x2 + 6 2. y" – 6y' = sin (2x) 3. y' + 9y = xe6x

Section 4.5 - Method of undetermined coefficients, annihilator approach Solve the following using the method of undetermined coefficients, obtain the general solution y = yet Yp! 1. y" – 8y' – 48y = x2 + 6 2. y" – 6y' = sin (2x) 3. y' + 9y = xe6x

sorry I cropped it

Answer 1

(6 - 54_159"_74' +21 y = e 4x complementary function (36) Auniliury equation is 583_158% 18+21=0 (8-3) (55t+vi) (US 8-17) - 0 8=3, 8= -58, 8- 533 - the particular scretion (Yp) : wing me that of undetermined Gefficiants Assume - A e 4 - y = 44 e 48 9 - 16 A e 4 – 6 y = 64 A e 4* substitute , O , 4 in 6 5 (64A 4*) _15 (16A (4*) 7 (40.42) + 21 A 64 - 64x 73 A 64* - & 4 equate Coefficients of some terms on both sides

13 A=1 A=I 73 substitute A = 1 / 1 in a 4X ypten General solution is 4-4ct Up y - ches the then ③ 9"+ 3y - 287 = & 7+7+1 – 0 function (Vc): complementary s+38-28 20 duniliary equation is (8+7 (8-4)=0 8=-7,4 Y = Cield +6e4a.

particular station (up) : Crefficiente rusing methadt undetermined Ansune - Axé th Brut y = A * The 7") + B y = A (-70 Th 6 th 490e !) Y = A(-140 + 4800 Suchttute ②, ③, 4 in 0 - Al-146* +49*6*) +3 (A (et The ) ) +B) = 7+74+1 -28 (Axend +BA+C) AETH - 280 x + 3B-280 - = é 7 TAHI equate Coefficients at suure terms on hath sides - A _-288 = 1 3B-280 = 1 A = 1 - 31-12) - 2861 - 28602 Substitute A = 7 1 B = 4, C = 1 in * * Gerard solution is = 4+2 y24*+4e44-447* * 14. I