Question

1.3 Show that the IVP r=rt, (0) = 0 has infinitely many solutions. Explain why Theorem 1.3 does not apply to give you uniqueness.

Answer 1

solution: giver Ivpis, x= x3 & 10)20. de that is da = ab - dt. (which is of separable form) forato) integrating, Sdt, ato 323_ ttc , &$o using given intial condition alo)=0 2 2 (4 +C) = 2 + + 2 6 a *?") buth side square root on both side. taking x= 3 (²+ + 2 Gole using initial condition x 10) = 0 - o= (3 (0) + 2 632 - 0 (3 g) squaring, 50 - 8015 GEDI Therefore the solution is x=(389220512 for #+0.

the solution of givent initfel Therefore we get value problem, That 1-00 >=(2, 2)! and X=-180) *20 * X= 252 2124*3 - 252 42 is 353 The uniquenes theorem is not apply here. because at to the function fro)= x 3 has stope infinite slope at x=0. since at x=0, f'(a)= x 35 atx=0 f'(x) does not exists. - slope of fro) ata o is infinite. There done fit) is not lipschitz continuous. which violating the hypothesis of the theorem.