Question

Step (1) "Rules for Guessing": 1. If y(x) = ek *. guess that yp(I) = Acts then find A 2. If g(x) = sin(kx), guess that y(x) = A cos(kt) + B sin(kt) then find A and B 3. If g(x) = cos(kx), guess that yp(x) = A cos(kt) + B sin(kt) then find A and B 4. If g(x) is a polyonomial of degree n, guess that ypa) Ao + AX + A₂x² + ... + A," then find Ao..... An. 5. If g(x) = 91(x) + 92(x) is a sum of two functions, guess that ypx) = guess for 91(x) + guess for 92(x) 6. If g(x) = g(x)ga(x) is a product of two functions, guess that yp(x) = (guess for g(x)) (guess for 92(x)) 7. If you make a proper guess but it doesn't work out, then set new guess Cold guess) 1. Find the general solution for each of the following differential equations (10 points each): (a) y" - 2y' - 3y = 3er (b) - - 2y = -2x + 4x2 (c) y"+y' - 6y = 12e3x + 12e-21 (d) y" - 2y - 3y = 3xe* y" +2y + y = 2e- (Hint: you'll use Rule 7. at least once) 2. Find the solution to the following differential equation with initial value (20 points) y" + 2y' + 5y = 4e* cos(2x) with y(0) = 0 and y/(0) = 1
solve number 2 for me pls.

Answer 1

2. The given differential equation is +2y'+54 cos (2x) y" Te-x To find C.F. Let =ema - then the auxiliary y. #o be a troial solution, equation is m? +2.m + 5 =0 =) m - -2 I√4-20 2 -2 I 4i 2 1 - 1+2i CE is e-{c, cos(22) + Csin (2x)}

To find p. Yp: 4 e-x cosaa D. +2D +5 -4e-> cos 2x (0-1)2+2(0-1) +5 1 = 40-e D2-20+1+20-2+5 cos2x = te-x D2 + 4 Cos 2x 4e-2 le zix eau D²+4 te-zise 2 = 2e-x e mix -zix D²+4 + D²+4 gentia 2ise Now Dita D2 + 4 ezize 1 - (D+21)2 +4 = e rixe .240 D²+4 Di

- ezize xo Adil 1 + 41 Felix 1 4 Di seo (1 + )" 110 (1-0/41).** = e zix - ezize I · 1 4iD re - zixe e 4i Similarly, e-zise +4 22 e-zie 1. Yp: 2e- a ezix 41 A e-zise 2ix] e-zise = ze-x sinza Therefore, the gene er al solution (c, cos 2x +G sinax) ya e-x txe-x sinza

Now y(0)=0 gives co x Y'a-e- (cz sin 2x + x sin 2x) + 262 Cos 2x + Sin 2 x) e-se + 2xe-x Cosaa Now y'COD=1 gives 1 = 262 2 => C2 = 1 Therefore, the required general solution is y-be- Sin2x + He-2 sinza s ex sin2x (2x+1)