Question

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Example 2.4. Find a solution to the initial value problem tion of variables. = xy'), y(0) = 0 by separa- = 220/2 dx dy =xda yv f yolkdy - freda 1/2 y 2 Vy = 22 + c y colo 200) = 0 + c cao = 25 = 22 - ² - 4 Vy Show that the theorem on eristence and uniqueness of first-order initial value problems (See See ) does not guarantee that this solution is unique. (3) Verify that any piecewise function of the form (11 (2² - Q²)² rza is a solution to this initial value problem for any value of a > 0.

Answer 1

ut dy = f(x,y) and yota)=Yo - on comparing ② and ③ we get a tx,y)= xy12 and NOFO , Yo = 0 Existence theorem says that - 66 Suppose toay be a continuous function on some region D such that D-d brigje R?: 19-Yok<a,14-408555 9,60 fra, es cordin Louy on Di & tonig) is bounded on D. ie. J a positive constant M such that Honig &M then the TYP y(x) = faiy) 4(X)=Yo has atleast one a rolution on the interval ix-Yolsh where hamin (ab Uniqueness theorom says that - "$t in addition to the continuity of frorey)of als also continuous function on D then the colution of the IVP is unique ?" Эл ол салғ — fruyla sy which is a polynomial in x and y and hence continuous in the neighbourhood of 10,0) je bor yo) Bloco che non (my%a) = av ) = med y which is not continuous in the neighbourhood of 10,0). poy Lug I Scanned with CS can CamScanner

Since, toxyl is continuous around 10,0) therefore by existence theorem, o we only can asseure the existence of solution beet due to the failure of continuity of of around 10,0), the uniqueness of solution is not confirmed. ie. Uniqueness thorem doesn't guarantee that the CSllution its uniquel. CamScanner

The IVP is - = sylia y(0)=0 - Case I ! w dom X<a 1 Y = 0 => dy - 0 Putting it in t left hand side of 6 - LHS = dy = 0 → LHS = 0 RHS = xyra applying yo) = 0 = 0X0 = 0 + RHS = 0 Since LHS = RHS ₃ y=0 is a solution of arp for naa Care II : when x>0 , y = (x?q?j? se 26x2_a2j?"'* 2x 4 Therefore LHS. of ② will be - LAS • Folyx (x?a?) – @ 4 and RMS: xytrex Jeymany2 = 142_02) - © from and 6 - Lots. = RHS. - Ar y = / (x2-9272 ;xa is a solution of IVP. 4 CamScanne

- Thus the given of the IVP piece wise dy function is a solution sely 9 (0)=0. cs Scanned with CamScanner