Question

7. Find the solution to the following IVP in terms of . Find all possible long term behaviors of the solution at 1- 0. If this behavior depends on the value of a give this dependence. 2y -6y= 6-76' y (0)=4-a

Answer 1

The given ODE in 24'-6y= 6-7 et the characteristic equation of the homogenious part of the given ODE in 2 m - 6 = 0 i, m=3 Hence, the solution of the homogenians part of this ODE ine In = A et I A in an arbitrary constant] Now, to find a particular solution of this ODE: (2D-6) y = 6-7 et ie (0-3) 4 = 3-3.5. et ie, Yp = 1 (3-3.5 t) = 10) (3) - (35) 0 2 cet) =65) 0-() - (6.5) * is met | - -* ( 1 + + t ) (3) + 3x = (-1) + zet = 7 et - 4 Heme, the general solution is given by: y (t) = Yu+Y ie, y (t) = A et + 76 - 4 sunce, y (0) = 4 - d2 A + 25 = 4 - 22 is, A - 13 – x2 D-3 4 4

in y(t) = (1 - x) et + ret - = (20 -4&2) et 1 = (5-2) et-1 & Now, if 5-2 co, ie, ^ > 5, ie, 15 or a <-rs) Then, lim y(t) = – o ta * I rs. 9f 5 - x = 0 ie, a = then, lim y(t) = -1 ta * If 5-2 0 ii, - isla <rs Then, lim y(t) = to. ta