Question

Solve the IVP using the method of Laplace transforms AND one other method of your choice. y" +5y' +6y= 2e ; y(0)=1, y'(0) = 3 TABLE 7.2 Properties of Laplace Transforms L{f'}(s) = s£{f}(s) - f(0) L{f"}(s) = s?L{f}(s) – sf(0) – f'(0) . TABLE 71 Brief Table of Laplace Transforms 50 F(x) = ${f}(s) s>0 S 1 => a S a p", n=1,2,... s>0 +1 sin bt s > 0 . s? + b2 s cos bt s> 0 8² +6²

Detailed answer using the Laplace Transforms method

Answer 1

2 - y" +54 +64 = zeit, Y(0)=1.50) = 3 | Abby Laplace toans form both sides [{yll}+5L841}+628y} = 2 $413) - 3460) – 4 (0) + Set 3 413) -54401 7 6413) = (37) (8 +58 +688-3-5 = 2 (8+1) (3+3)(8+2) 413) (8+8). (8+1) (8+3)(8+2) 415) 8² +98 +10 +ES+8) (8+ (St) 8?+98 +10 418) (3+1)(3+2) (3+3) Noce). we will do partial fraction, 8:+98 +10 न Star 3+1 6+2) (8+3) (S+1) (3+2) 8:+98 +10 A8+2) 843) + BISH)(83) + C(81) (3+2) Put 8=7, Put 8=-2 -4 put 8=-3, - 8 = 20 3 + + 2A B - 9 8P+98+10 3+1 3+2) 3+3) (5+1) 18+ 2) + (8+3) -+ q Y(8) (3+2) (3+3) (8+1) Apply inverse Laplace tranform, t qe + 96) = 47 28

y" + 54'+by zeit m² +5m +6=0 term Hence Yelt) = Get Gebut. 4 4 (t) = -2 ē Let ytoj = 1, y(0)3 corresponding Auxiliary egh is (m+3)(4+2) So m = -2,-3. # complementry solution is given by Now, we ceill find particular solution by method of under termined cofficient. for this RoHs of differen equation and its derivative contains the our guess for I '(t) = -Aēct Yp is also a solution given D.E. Aēt - sl-Aét) + 6 Aēt = zet 2 Aēt = zeit A = 1. the general solution is given by yt) = Gelt 8 y 10) = 1 G+Q+1=1 at C =0 Yplt) = A bit Yp' (t) = A ect Yp (t) = è it. èit + 3₂ e 3ct - et 9/10) = 3 3 = -2G - 3 Cg - 1 DO 29+36 = 4 - c 12 -2t -3t g) = 40 -4e te