94年期中考考题 1(20%) Writing the statement of the following theorem (1) Gronwall's inequality (2) Existence and Uniquencess Theorem for IVP [Picard's THM] 2(20%) Suppose that Φ1 and Φ2 are continuous function satisfying the IVP ┌ y'(X) = sin(y(X)^2) , for all x 属於 I =[a,∞) < └ y(a) = 3 , for a 属於 R is given Prove that Φ1=Φ2 on I 3(10%) Show that ┌ y' = sin(xy^3) - e^y^2 ．x IVP ＜ └ y(0) = 2 has a unique solution Φ on (-δ,δ) for some δ>0 4(20%) The IVP ┌ y'(t) = 2√y(t) 　　 < └ y(0) = y0 Show the following questions: (i) The IVP has a unique solution y(t) on some t-interval I containing 0 in its inyerior if y0 > 0 . (ii)The IVP has more than one solution if y0 = 0 . 5(30%) Find the solution of the following questions: (i) Find all solutions of the ODE yy' = (1-y^2)sinX . (ii) Find the explicit solution y=y(x) of the IVP ┌ y+1 | y' = ─── , and the largest < x+1 | └ y(1) = 1 . x-interval on which the solution is defined. (iii)Find the implicit general solution formula of the ODE ( 3y^2 + 2y +1 ) ．y' = xsin(x^2) . --

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