Question

Determine the values of a, if any, for which all solutions tend to zero as t > oo. Also determine the values of a, if any, for which all (nonzero) solutions become unbounded as t -> oo: y" (2a 1) a(a - 1)y 0

Answer 1

We know that, a if the root of the auxiliary equation is real and distinct my mo, then the general solution is y = ce me the more b) it the roots are real and exceal, m, = mg=m the the general solution is Wor e o it the rooks are complex conjugare XIIB then the general solution is g exx (a cospe + 6 sinfer). & equation ar tbatcSo. Also, the roots of a quadratic artbe+C =0 is given by a = -65 J63-400 2a Ivan 2a - Gwen problem is t ! ) 2"-(2x-1) 4' + & (2-1) YZ0 0 Let y=AmE (170) be the trial solution of a

then the corresponding auxiliary egration is nt - 22-12 mt 2 (0-1)=0. ng is a quadratic equation of m. the roove are given by me (24-1) + | 24-152 – 42 (0-1) 2 2x -1) + -4 1 – - = (x-1) + X, . So, the root of the auxiliary Egnation are ac 1). Then the general solution is 0 vet verdt Now, from & we have if 200 then 970 ap 170. Since line 10 also it 2=0 then go & ar to

| Now, if 270, then y o ap 20 since time at the Euro 20270) thus if sa co] then 220 of 0. it ayo then you as to Hence, 1 0 then solution tend to o t200 lif as the solution becomes embounded as Fed.