Question

Sect. 4.2. Reduction of Order 1. In the following problems, the indicated function y(x) is a solution of the given differential equation. (a). Use the method of reduction of order, i.e., the formula 32(x) = x1(1) one-Plade de to find the second linearly independent solution 72(2). (b). After having determined yz(x), write down the general solution: y(x) = 4(x) + C292(2) The problems are given as follows: (1). 2y" – 7y' + 3y - 0, y = */2 (Answer: 92(x) = 2(x) = c+/2 + czer) (2). 3y" – 304 + 75y = 0, y = ez (Answer: y(x) - ret, (x) - CC + r ) (3). 2y" + 8y = 0, y = cos 2x (Answer: 32(x) = sin 21, 42(x) - cos2.r + y sin 2x) (4). xy" - (x + 1)y + y = 0, y = et (Answer: y(x)=x+1, 9(a) = c + (x + 1))

Answer 1

: رو ه = و3 + ارد -23 ه=وفي - او - الا compose ylt fylt dy=o لا الم | | dn . د دولے | در الماهر dn اگر مردی روم به لام . 2م في . . جع[G CS "Scanned with CamScanner

М.: ge І, 9 = 1, 192 (1) = (34 , soln then У 1) - Gч +(„Ч. 9 – 9, ал аlѕо soph 98) - я е 3х 39'' - 39 + 1Sч - D у" - Isy + 25y-o Р= -10 le lox an عا . y Semiole on - Yo son - 3 х 4 яе 5* ما 9 - 41, + (29, (98) = e5+ , xcs / Scanned with CamScanner

24"+8y = 0 Y"+"y=o p=0 Ya = y, l eroon du din (²2 n = y, s sechzx de == J Secten an yix tanza Y = (man хоч 2 и =sinan २ - (2n plankyo, L) 1 Y 2 (M) = Sikan Because - Y=64, es 1 = q (esant(₂514an) 1 Soly - then Sol 4 P = -1xH y=y, is also sol y los - el. e ner Yzzul nen an zyluer an eze . = one" - etxek = = (x+1) etxel Yx-(+)) e ° --(4+1) 192 = (x+1)) y=ay, tay, cs Scanned with y=9pM+ (z (n+1) Scanned with CamScanner