Question

Differential Equations for Engineers II Page 2 of 6 2. Consider the nonhomogeneous ordinary differential equation XY" + 2(x – B)y' + (x – 2B)y = e-1, x > 0, (2) 5 marks where ß > 0 is a given constant. (a) A solution of the associated homogeneous equation is yı = e-*. Use the formula for the method of reduction of order, as described in the lecture notes / record- ings, to find a second solution, y2, of the associated homogeneous equation. You MUST express y2 in its simplest form. (b) Use the formulae for the method of variation of parameters, as described in the lecture notes / recordings, to find a particular solution, yp, of equation (2). (c) Hence state the general solution to (2). 5 marks 1 mark

Answer 1

2. @ 1 The giveesi non-homogeneous differential ecuation nyl +200-B]yl + (0-2B3y z e-x x>0, o) Pt's corresponding homogeneous equationis ay" +2 (aphyl. + (2B)y zo 2 Given that YUX) = eig es solution of arow, let us consider the you. Y(A) = roue-2 is the complete sowion of @ Enote that waitake ser recogy, omy] & Co a veny ex yl Cra) 2 erk Gre zu() 2 era che-201–20'+1911) ep 120"-2001 + r) putting these reame in xex (211-200 ) + +1) + 2021 m) e-do(201-19) +12-20) e-X20 =0 - 218489) +(-20){re ad or x rell ~ 280 281 20 C Reduce differente escucution) #201) {-29 +200-10).} re't face { x -210 Or, n2011 om 2011 201 2 2.3 a mo 2 28 inating [ By integrating jinci's constanta 28 cin om, 201 om du In Gex 2B 20+1 ori in Goa ta 210+1 Hence the complete or general solution of @ is ya) a reny, la a a a om, yen) " AYI & BY here, A za 2.2+le- +ge-c 20271 B = 9 20H are constants 2 = x ed, ( 'Another or second solution of nomiogeneous equarion) CS Scanned with CamScanner y, zen And 2 BHI

te-2x 2pel 2.-toen b est Yr the wronskian of y,; 4 =WCY, Y₂) (x) = Or, W = y, yly,'n = e-x x (121) x2 Pe-x 2 + 2x) - (-ex) (221871-x) (21011) e -28.4280 L 32+1 erox 7 e e-2.X = (20 + 1) x200 w By Naniatiation parameter! method the perticular integral Sp. 9, confiry + 7210) gesy where fix) 21 f I. (0) R de and gess Sy wR de Here Raena part for which the ODE will be mon nomogeneous] - 2PNe* xe- foa) 2 (289+1)%20e-2nda an nh 2 -S dre (2 P0+1) 2 (20+1) 2+ 4.8 'da 1-28 gins . J 9. 2 2 2011) (112) (1-28) Seranea "(2.8321)72e-2x n-ap (A - ABD 10+ sopottert ( era C yporn 1:10) arenk 27 am 24Aro) a²e + al-200 -ABV HARA27AB nean 2(1+28) (1-10) (2271)2 ne-a Cena 2(28+10 (1+20) (1-20) 2-AP E) The general soution is ; AY, I A2 + y play yon) Baaotle ny Aen + era + Scanned with CamScanner