Math Assignment. Dylan Zwick. Spring Section 1.2-1, 6, 11, 15, 27, 35, 43 Section Section 1.1-1, 12, 15, 20, 45

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1 Math Assignment 1 Dylan Zwick Spring 201 Section 1.1-1, 12, 15, 20, 45 Section 1.2-1, 6, 11, 15, 27, 5, 4 Section , 6, 9, 11, 15, 21, 29 1

2 Section Differential Equations and Mathe matical Models Verify by substitution that the given function is a solution of the given differential equation. Throughout these problems, primes de note derivatives with respect to x. 2; yrrx y =x +7 (hec5 O 2

3 C C.) 0 II I 1) - 0 IcI t (J Ii - (I Q I ) -r ç-j C C I. 0 C U I. I. I.. I

4 bubstitute y = en into the given differential equation to determine áll values of the constant r for which y = enx is a solution of the equation I! I y +y 2y=O I! L( rry e r /1/ 10 M Uf h o 1r- z (*+L)(r-)) So! 4

5 1, First verify that y(r) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. yl = y(x) = Ce + x y(o) = 10 I K id y-( x 1) -Ce-t / y I N/I (o) C e ( / L - c1 5) Jñl}{/ vq/ 5

6 Suppose a population P of rodents satisfies the differential equation dp/dt = kp2. Initially, there are P(0) = 2 rodents, and their number is increasing at the rate of dp/cit = 1 rodent per month when there are P = 10 rodents. How long will it take for this population to grow to a hundred rodents? To a thousand? What s happening here? ii2 khc =2 /) C k+ - j L /-?kf IZ( - ICC) lcd / (io) 10 P(q /00 I-)(10W I *0 6 / /-4 -Mq

7 Section Integrals as General and Particular Solutions,1-2Find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. =2x+1; y(o)=. y()= J(ti)( yty-c 7

8 1.2.6 Find a function y = f(x) satisfying and the prescribed initial condition. the given differential equation dx y( 4) = 0. y( Jxx L 2+ v ci 1,x q [/ \/ () / LJ)) y(x)= i9 8

9 Find the position function x(t) of a moving particle with the given acceleration a(t), initial position x0 = x(0), and initial velocity vo = v(0). a(t) = 50, = 10, v( ) $o+ V0 = 20. O (o) 50 / O (f) 5 t * Ic ± (CL) 1O() + 9

10 Find the position function x(t) of a moving particle with the given acceleration a(t), initial position = x(o), and initial velocity vo = V (0). (p5)) a(t) = 4(t + )2, V0 = 1, = 1. v()4 (o) +( V(o) v() 57 -i x() + 5) -J?(o)+C

11 A ball is thrown straight downward from the top of a tall building. The initial speed of the ball is lom/s. It strikes the ground with a speed of 60m/s. How tall is the building? Vj (I7L). 1c/ 55 xc L) () ± o llvn t 5f 11

12 1.2.5 A stone is dropped from rest at an initial height Ii above the surface of the earth. Show that the speed with which it strikes the ground is v() V ( F) 12

13 1.2.4 Arthur Clark s The Windfrom the Sun (196) describes Diana, a space craft propelled by the solar wind. Its aluminized sail provides it with a constant acceleration of O.OOlg = O.0098m/s 2. Suppose this space craft starts from rest at time t = 0 and simultaneously fires a projec tile (straight ahead in the same direction) that travels at one-tenth of the speed c = x 108m/s of light. How long will it take the spacecraft to catch up with the projectile, and how far will it have traveled by then? L (7 v+ v. (A k4- q c ye/j) 1 (A CL( /1- y )

14 FIGURE ider a through ihole x nstance, dx dv (lv 2 i + I -v (14) (15) rigin solution irantees y many 7 II\/\\\ \: -.- f/// / / /5= (16) (17) hus the the left 2 n Fig. m I//I/Ill I ll/ill, I,,, I - III // II, dx I i/il I/I I/Il : z:. ///./!I / dy I III Il//S. ue of f the S. I,,, Ill LXIS 1) d dy I Ill II I ) 1 2 /1/Ill/Il Il/Il/li II?, /1//ill/i I/I,, 1..1 and 1..6 See Below F1GUE FIGURE , I / i f/i / Il/I I f/i) / I / / I 2. =. +y FIGURE f, / I I I I / / I 7 / I I I S. S. I S. I S. / I I I S I 2 (1 I S. I S. S. S. 7 I S. 5 I S. 1/ S. / \ I I - I I / I: I S., I S. 2 l 0 I 2 S.\ Il I I I I IS.S.\ /1-5.::::::. j Il/uI/I. / Il/I,,!,, //f,,,,/ I////I I// c/f//f//f F11,,TT / / /. =v snx II Ill IS. II II Il/// I ll - JllJJ Il//I III I /1/, // =, ditional points marked in each slope field. solution curves. Sketch likely solution cune through the ad the indicated differential equation, together with one oi I11OFL In Problems I through 10, we have provided the slope field of - y+ I FIGURE I 0 I 2 S. S. S. I I / I = y FIGURE i, I//\I\s.. f//s II/I/II% S. S..5 2 C -- f/i I III II \\S.. I I I I I I I/%I\/ S. I I. _. I,.. i / Pt, ;i -/ / / I / / I I I I I I / P1 / I / / Il / l I I I / / S. / I I _\ S. I I S. I I / I I I / I / /1 I ( / i?///)i f-s I/I / 1/ / I /1 I 1/ I I I I 2 II /1/f II f/i/f =x v

15 lution. 21. v = x + y. v(o) = 0; y( 4) =? 22. i =v x, y( FIGURE L di: 1../V; y(o) = I 15. = / ; (2) = 2 5 i ahie proble,n. If cvi.sie,ic e is guaranteed, determine whether 77meoremn I does or does not guarantee uniqueness of that so 12. =1nv; v(1)= =.1V; v(o)=0 /l,\l\ 5 ii. - =2i or does not guarantee e.ii,steiice ofa solution oj the given initial, 5/// 5.,,.55/ SkIS Il/Si/S S 55/51/S,,,j 1/ 2y2; v(i) = 1 In Problcmiii: II ihivugli 20, determine whether Theorem I does 17. =.i 1; v(0) = 18. =,i I; v(i) = 0 Il/I,, hi/i /1/1/ S/It/f Ill//f f/i/i, I/Ill? I 1..9 See Below I/Il I/_ SS S- / / II II!/ S..5 5\.S Ii -2-IO 2 FIGURE dl f/i ft / / fi,,,il,,_ 2 I 0 I 2 dv =.i- y 8. d,i. 7 I 2 ( _L. (5/// / I I I / /.1 I I S S S S S S III I I/It, I I II I t ( / I I I.--- -i i, r I//Sf- - S III I, P I s I/SIll i S I, I 2 l 0 I 2 9. di. TT re r; I 2 /5 I 1 / I,S J ith FIGURE / / / / I I S f//il / I III I II 11. SS 5 I I S. Tt,i,,,,, 4)=O; v( 4)=? desired tahie of the solution v(x). tial condition. Fincillv use this solution curie to estimate the Then sketch the solution curve corresponding to the given mi in Problems 21 amid 22, fin use the method of Eiamnple =x SI/k/S il/i Ilk /555 S.\kk /1 is.,, SS.i I S \- - S/S I S 1Sk\s.5/S S /k, - I 2 I 0 I _=,/Z; (2)=z1 19. = In( 1 -,,2. (0)= 1 to construct a slope field for the given differential equation..2). i (O) = 0 FIGURE

16 1..11 Determine whether Theorem 1 does or does not guarantee existence of a solution of the given initial value problem. If existence is guar anteed, determine whether Theorem 1 does or does not guarantee uniqueness of that solution. 2xy y(l) = 1. (\ cre iio5 raai/ n.y i (, Ift( (i, -JL 16

17 1..15 Determine whether Theorem 1 does or does not guarantee existence of a solution of the given initial value problem. If existence is guar anteed, determine whether Theorem 1 does or does not guarantee uniqueness of that solution. y - I To / Cz/l) 1- y 5, K)5 n( ) I 17

18 .Q zl First use the method of Example 2 from the textbook to construct a slope field for the given differential equation. Then sketch the solu tion curve corresponding to the given initial condition. Finally, use this solution curve to estimate the desired value of the solution y(x). y =x+y, y(o)=o; X\f_z - ( i 7 1 J Q 1 zl zl 0 r --5_ ( / / / 1/ / / / ) y(([); \\\\ \ \\z 18

19 1..29 Verify that if c is a constant, then the function defined piecewise by 1 0 x<c, (x c) >c yl- satisfies the differential equation y y for all x. Can you also use the left half of the cubic y = (x c) in piecing together a solution curve of the differential equation? Sketch a variety of such solution curves. Is there a point (a, b) of the xy-plane such that the initial value problem y = y, y(a) = b has either no solution or a unique solution that is defined for all x? Reconcile your answer with Theorem 1..O (x-c) y O oi 1 Cc 6c. y cc a A I i G 19

20 cd H - c- J c_i 0 D - U - s c-) c_i ri c_i cj -1:-, -j H-- (1) QI ]$-!..-- C