# Math for Moms and Dads: A dictionary of terms and by Kaplan

By Kaplan

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Additional resources for Math for Moms and Dads: A dictionary of terms and concepts...just for parents

Sample text

1) satisfying the initial condition ,,(to) -- o, ,,'(to) = o, is identically zero. 5. 11) holds, and the function ho : Ca, b) x R+ ~ R+ is nondecreasing on the second argument and lira if. b ho(t,x)a~ (h21,hn )(t) dt = O. 2i) has at least two solutions with exactly I zeros in the interval (a,b). 4, we easily verify that there exists a number z > 0 for which (g;1,g;2,g;i,g;2) 6 Vik((a,b)), whenever 9 gj~(t)= { g]~(t) for a < t ~< b - ~, min{gi~(t),hil(t) } forb-r (j=1,2), g~ { max{gi,(t),hi2(t)} for a < t < a + e , 2= g~'2(t) fora+e~

2i) is solvable. 31. Let f be nondecreasing on the second argument and for some Zo E (0, 1) let rues {t E (a,b) : f(t, zo) < 0} > 0. 21) to have a unique solution it is necessary and su~cient that f b(t-a)(b-t)lf(t,x)ldt < +co for z > 0. 26) PROOF: We first prove sufficiency. Since f is nonpositive and nondecreasing on the second argument, we have, for any b E (0,1), f;(t) = lf(t,6)l. 26). 8) and the function h ( t ) = 6of(t, zo) satisfies the condition / b(t - a)(b - t)lh(t)l dt < b - a. 24) hold.

8) satisfies these initial conditions. ,,,] [1(2 ,, ,,') (P'-u2t) 1/2+v ~/-~:u-~) = ~F (b-t)2(1-b'~'/'lu(t) ~-, j p2 ~>6u(t)>0 for0 0. From this it follows immediately that where 6 = ~,'(t) <0 anduCb-)=O. 9). 9) is solvable in the case p = 2 by taking as lower and upper functions sl (t) = 2 - t and s2 (t) = 2 - 2 -6 t 7. REMARK. 9) is a decreasing function. 3. 4,), (i =1,2). 42) under the assumption that f E Kloc((0,+oo) x R+ xR_) and f(t,O,O) = 0 , f(t,x,O)>~O x>O.