Question

Exercise 2.5.152: Apply the method of undetermined coefficients to find the general solution to the following DEs. Determine the form and coefficients of yp

Exercise 2.5.152: Apply the method of undetermined coefficients to find the general solution to the following DEs. Determine the form and coefficients of yp a) y" – 2y' = 8x + 5e3x b) y'"' + y" – 2y' = 2x + e2x c) y'' + 6y' + 13y = cos x d) y'" + y" – 6y' = x2 + 2x + 4e3x

Answer 1

I given yu-ay' = 8x + 563* » (0? 20 y = 8x +5€34.-0 I th Auxiliary eans in m² am = 0 2=0, 2. Icf = Yc = (t cell .- Since the constant & occur in RHS of and it acho occuls as constant c, in the SF. Corresponding to the root of multiplicity one. tre can choox trial soletion - * - & [ Aot A, &] + Be3x. by* = A + 20, x + 3 Best py= 0 + 2A + 9Be p’yt - 2 Dyk 8x +5034 9 (24, +98 e32 )- 2/4 + 24,2 +3Rexh) 32 (24,-970 ) - 4 ASC + 3Be39, 8x+5034 - 4 A, - 8 ; 3B=5

24,- 2A = 0 24, -210 = 0 2(-2) -2000 - 240 = 4 JA=n2) K 1 y* = And +4,2? + BeB* Yo-y* = -2x - 4x + 5e3a general solution y = Yc+Up. 4-(4+cze 946-22 - 2X E Icon

5 Given +D+-20%= ga teba -0 I its Auxiliary ean in m²m - am = 0 mimit m - 2) =0. m (mi +9m -m-3)=0 m ( m ( m42 )-1( m42))-0 m (m-1)(m+2) = 0 m= 0, 1, -2, Cof=% = eithek + lze 2x . Trial soln y* = x (A + A,*) + 8 Dy* - Ao + 34,8 + 982 22 byt - 0 + 3A + 4 Be?? Dy* = 0+0 + 8 Be ? 0?y* +p?y* -204* - 92 teze (Bezl) + (SA, +4822) -2(40 +24,212re) [ 40 + 472 )+ Be22. = 227827 Y (122*48e24]+(241-2 m.) – 44,22 42.2X 2& 4e24

98822 + ( 2Ap-210) 4-4AX = Atel | 3g = 'PA, - 2 Ao - 0, -1 = 2 Anthony W 2(-) - 24.20 -|- 240 .0 -240 = 1 22 4-y*. Let the rest The general soln n a y = Ye typ Y = c + cez tez e 22 2 .

c) y" + 6y'+13y = cosa | 9 (61 60 +13) : (* Its Auxiliary equation is m'tom +13=0. mos -6 + 36 - 4(13) 2. M = -6 + 36-52. m -6 + 7-16 m2 = -6 + 41 m ; -3+27 Cof: Y = 332 [C, cosz2 +cSinzx] Trial solution y* = a colz +Blink byt - - A Sinx +Bcosx Dy* - - A Code – B Sinx. (6760713 )yy - Cook (- Acox – 85198 +66-A sinx +3.603x) +13 ( . À cox + D sinx) = cook SSVESNESESNYA

- (-A Cosx-B sinx) +6 (-Aşinx + B Cose) +13 ( Acoux +8 & 3) - Cox , Cose [- A+63 +13A] +sine-B-6A +18) Cosa 12A +613=1 #2B-6A = O mar 12 A HGB II -12 A 24 B=0 B= 30 no B=t/ 30

12A = 10613 12A = 1-6 Av 30 94 . 12 A = 30 50. Kan Ants :- Yp= y* = - cose + 1/ sinx The general solution y = y + Up y= 34 (8,60) 22 +¢ sın2x) + cox tez sina

? 2 Mmm = ta d) Given [103 +5²60)y = x + 2x + 4 682 Auxiliary ean 's mtm-6m = 0 m( m+M-6)=0 m ( n + 3 = 9m -4) 6. m (m+3) 6M-2) = 0 m=0,2,-3. Y = G + C el tege34 etter of Ar Esx+c) +zfox tt) + Fe32 Trial solut at & ty * = x (AX fis& te) Dyit = 32++232 +C D'yi = 6AX +2B. Byt - GA =) (6A) + (64242B)-6(342" +262 +C) ==*

-19Ax' + 6A – 123 )2 + 6A +2B-60 = x2 6A-12 B O -18A = 1 . - 12 B = B B= -1/36 . - 36. $ 61 t) + 21-* )-60=0. en té -6650 -60= the - 6 C 151 18 x 6 Trial solution y + = a (AX +B) Dy te 2A2+B. po s 30

Byt +04.-604ex 0+ 2A - 6(2A2+B) = 21 QA -6B)ě – 12A2 = 2x - 12A = 0. 1213)-63=0 야공 pertular scution Up at ex = 2 x 2 - 1 The trial o solution of 4 e 3x es ру, - за ее by t = 9A 63% By * = 27 Ae3t. 97A134 + 94232-6131459 ) = 428? (36 – 18 ] A e37 438 » y

18A-4, - ①. - 4 이 1 ( 37] - 27 + ) + 3 - 2 (-) + ( ILL | log The general solut Y=yet Up 며qte 4gt 3y