Question

1- Use the Reduction of Order method to find a second solution of the equation 4x2y" + y = 0 Given that yı = xì Inx 2- Solve the differential equation y" + 4y + 4y = 0 3- Solve the differential equation y" + 2y + 10y = 0 y” + 5y + 4y = cosx + 2e*

Answer 1

1. Auky" + y = 0 -> (1) First we put n=ez, then (1) reduces to a Cat - 1) y + y =0 → 4 2 1 - 4 dy fyty 20 (2) Now, given that y = x² Inn) = 8². z be a solution of (2). t= v(z)., be another solution Hence let 84 (2) · : ₂ = v.z. el = veht teu 1.22 ziet uz e 2/2 / 1 7/2 212 ₂ v 212 2.10 the o Ź 1 if -lee 2/2+bre te/2.2. v't o'z ezh a ve 2/2 & vet = 7/2./2.6"+(2+2) 6 + 4+20)

putting these values in (2) , we get 4 e32 (20'4 zol)zo. 26"+ 20' =0 (3) et v= W (day) . Hence (3) reduces to dwy + 2. w zo :: Integrating factor be S z de 2 z² . d (w.za) = 0 Integrating we get wizze, [ein integrating constant] o= a + where We take, o= + . i. Y2 = + 2 z en , [ by integrating ] is integrating constant X =te 2/2 =tetne In (29/2 [ Putting za Ina 7 = [e In (n)]² a Hence the second solution be

2. y +49 +49 20 (1) of (1) be The ausidiary equation mt amt 4 - 0 31 (m+2) =0 m =-2 ,-2 . Then the solution of y = (a + G 2) é 2x W) be constant mtegrating are ac a, where re 3. y" + 2y' +10y =0 The auxiliary equation be m" + 2m +10 20. -272 - 4.10 = -1 + 3 differential equation : The of the solution be yzék (a cos 3x + cq sim 3x) where all are integrating constant

cosu (1) y" + 5y't any = cosntzen The auscitiary equation be m+5m+4=0 > m= -5 + √5²-4.4 2 = -4,-1. Then the complementary solution of C) be Ye=Geth teen where eglę are integrating a constant (cosat 2.0 2.ok - D + 5D + 4 CS2 Now, the particulars solution be b +5048 (cos x + 2.24) - da sono cosx + 5074 22 > ... Cosx + 2 = 5.144 - Bez + $b+3 cosa Cosa cosa - tea 25 D² - 9 SD-3 25.(-12) -9 cosa sex + (34) (50-3) cosa the " - - (- 5 sima - 3 cosm) =

= 170 ( 3464 tiscosx 425 sina) Therefore the complete solution be be Tuare y = ye typ a ce ar t ex + + 20 (340+ 15 cosa +25 sinx)
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