94年期中考考題
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 屬於 I =[a,∞)
<
└ y(a) = 3 , for a 屬於 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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