Question

1. Solve the following system of equations (find the general solution). If the system has no lution, then say so: dar dt2 dy dt =1-t dy - + 2 dt = 4et dt

Answer 1

We have dt² I-t - dt da dt -x+ 2 dy = 40€ dt from ① and @ have. 2 da (het dt2 da dt + tx) = -t > 2 2 dn + dt da dt x = 2-atthet Auxiliary equation is. 2 m² tm-1=0 =) (2m-1) (m+1) =D -> m = 1/2 m =-1 2 Complementary function is Ke (t) = (^ (A Cl2t + Bet A Panticulan integral is Xp(t) (2-2+ +40+) 2 (20'+D -) whene Ded at

(26²+D-1) (2) - (at) (20'+D1 + (407 (2010-1 (2) + I 1- (20+ D) 1-(2070) J (20) 4 et + 2+1-1 [1:40 = 20794 and f() #0 {1+(20's) +60+)? +(20² + D) + (28?+ D) } 2 (2t) + aet - - 2 + (at + 2) + 20t. [:D = d and. D (2) = 0 + D) (2+) = 2 and all other highen powers I make at as o (2D o] Hence, xp (t) = attaet :. the genenal solution for x (t) is x €t) - Xc (t) + Xp (+) = (A CH2 + Bert taet +at) where A and B are anbitrany constants.

Using equation 2 we get. dy dť - (4et tx- da => day 140 2 dx) 4et + (A etla + Be + + 2 et + 2t) - (A No. Bet + 2 * 72}} 2 {4e + A et +27 -2] dy dt. ta Bert = dy - 2 et + zet + Apy Bet+t-1 dt. 4

Now, integrating both side with respect to 4 we get the_ Be t + + ² -t + c y (t)= a é + Ae X(t)- Act/2 Hence, the general solution of the given differential equation is + 2 et +at +zett where, A, B, C are arbitrany constants, t Bert y (t) = A 242 -Bet 2 ttc +t2 र

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