Question

4. Solve y" - y - 12y = 144.+ 12.5 with the conditions y(0) = 5 and y'(0) = -0.5..

Answer 1

Sol" % - Given, Second order ordinery differential equation, 7"-'-12y = 144.23 +12.5 - and conditions, Yo = 5 YOO = -0.5 Now, The complete solution for this diferential equation would be To find for taking auxiliary equation by taking So, equation li (D2 - D - 12.) y = 14 4x3 + 12.5 For y CF Tanza (02-0-12) y = 0 is 02-0-120 =) D- 40+30-120 → (D-4) (0+3) =0 . . D = 4,- 3 3 Read and You = C, por + Cabr distinct a, b = rosto CF = 4x + C2 e 3 x

Now, To find Yps. of 12.5 i. Jas = (D2-D-12) 14412 . (D2-D-107 (D2-9-12) X3 (12.5) eo* we know, OPP. 292 so fofcan provided f(a) to Jos = 12 [1764] *2 + 12,5 (1-x)T' = 1+x +x2 + x34---- As. ' (0-0-12) si opis = -12 [1+ (0*70 ) + ( 0) (09:0). Je to Jeg =-12/+ (070) + o'487289 +(89+0°+ 30*40: + ----] x3 +- 12.5 9 Tong = -12[1+(8270)+(D'+0°42031+ +D# 304 30 1728 12 As we know. D for] = Differentiation of foxo 1 0 2 ² = 3x e and to fins a Integration of fox) (03136 s o n simplifying by differentiating corresponding Spis = -12[23+4 (624 3X2)+L (©+6x +12) + L_ **** times, I 1728 12

- Jos = -12[23+4+* +34 +32 + tex] - 12:6 = -12 x3 – X-37-38-120,5 12 Now. A So The Gmplete solution. - y = dox + y p. I 0 = c + c - (2 + 3v, 1, According to given condition, Yiol = 0 C;+ €11-10+ 0+0+2,5) = 0 di Gi+C2 = + 25 l Also, yol s0 y' = 46, 2424-362€> 36x2+6x7 134 12 46,-369 -13 = 0 .: 49-362 =@ 13 -nij Solving in and thin so Cartoon , C2 = 16 19 = 7+(12 x3 + 3x? +43x+2