Question

1. Consider the problem x' (t) = (1+t)x(t), x(0) = 5 (a) Find the Picard iterates xi(t) and x2(t). (b) Find the first two approximate solutions x, and x2 in the Euler scheme. (c) Find the exact solution. 2. For which initial conditions will the solution of x"(t) + 4x'(t) - 20x(t) = 0 tend to zero t → 00?

Answer 1

x') (1+t) x) 2 f(12)- (tt)x C flt,n-1) dt +9x4 t + oK 1x tp x(71) and 1 1tt), dt t)Sot 1. 2 StSt+5 2 J 1(t, ,) dt S + (1+t)1 ot c54 5t41)(t+ 11) dt 2.

t ) S+S 3 S+s 22 E + 2 Sty4 St3나 St니 st+ S 지 2. t141)at) -(1+t) e o Iutegratin Fastor R) -172 Sohuthon i )ZIP) dt + Yf4) (Q1-0) 4)et-t S. olt t C cet+t C1= 5 (0)2S tt

8-20x= o AvxilTlay euatien i-8u-20=0 u11-10w t24-2020 A(-10) + 2 (-10) =0 4-10) (u1) 20 lo,-1 - 2t e ఉంo t) bounded 1e C2=0 when co Hence )e2t,elot shen 1-co