Question

solve. a & b

Let us der the Cwachy Problem (CP) where :D-R,D={(x) - Sex-al 6). Pano ways that a solution always exists in le-to Sa, where a mince and M arcos ift.) is (a) Giveawple to show that this is w h at it (b) Show, loy ch mple, the (CP) w ill be in and this solution may be unique, however, does not satisfy Lipsch condition. Justify yo

Answer 1

# Picard Existence & Uniqueness theorm of I-V.P Examples. let y = f(x,y) y (o)=0 whee f(x,y) = 14). claim f does not galsfies Lipschita condition since I f (x, y) = f(x, y) = 119,1-1811 7 19, Gal to yell. f (x,y) does not selles Lipschids endin ingverthe Next claim this I v. P has unique solutia y so. dy - 194 = 2 y, yo 1-y yo » Уtx) 7. du - A у іх) = qe1 fos genico, dy = -9 7 ody = y J (= bex. Now You = a ex Wo Iben rco since you is continous at o = 9(0) = lim year = lim yen) xotxone lim berk X0 → 70 -0 = this apx - min o=a=b Here 4(x) = zor = y 2 = 0 inique solution Conclusion. This is an examples I.v.p which does not sahisties lipschite condition, but still it has a inique soktion..

# consider dy ty = f(x) fwak 46000 Lo 10 x7d. D+ P+/ o ens 2171 C.F Y ) = a eti P.I in ozrel I ex = t.ex=1 in ay t.o.com o - particeles inthal y(n) = 21 oenal ) D 270 Hence year = aed & Sp (1). Y(0) 0 = oz a co ty, @) a = -9p ( = -1 so ser osxst, y(x) = – en + 1 = 1- e-n and for art , y(x) = a ex- - osus, Y 617 = t.ex e-x N71 an Hexe dy - If fun Take F(ny) = ye frag then F is not continens an et 1, but still er has a solution. conclusias In '= fun, of 98%) 7o, Is fmy is not continens still s.r. py - 8) Las a solution.