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Math 2280 - Assignment 1 Dylan Zwick Spring 2013 Section 1.1 - 1, 12, 15, 20, 45 Section 1.2 - 1, 6, 11, 15, 27, 35, 43 Section 1.3 - 1, 6, 9, 11, 15, 21, 29 1 Section 1.1 - Differential Equations and Mathematical Models 1.1.1 Verify by substitution that the given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x. y ′ = 3x2 ; y = x3 + 7 2 1.1.12 Verify by substitution that the given function is a solution of the given differential equation. x2 y ′′ − xy ′ + 2y = 0; y1 = x cos (ln x), 3 y2 = x sin (ln x). 1.1.15 Substitute y = erx into the given differential equation to determine all values of the constant r for which y = erx is a solution of the equation y ′′ + y ′ − 2y = 0 4 1.1.20 First verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. y ′ = x − y; y(x) = Ce−x + x − 1, 5 y(0) = 10 1.1.45 Suppose a population P of rodents satisfies the differential equation dP/dt = kP 2 . Initially, there are P (0) = 2 rodents, and their number is increasing at the rate of dP/dt = 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? 6 Section 1.2 - Integrals as General and Particular Solutions 1.2.6 Find a function y = f (x) satisfying the given differential equation and the prescribed initial condition. dy = 2x + 1; dx 7 y(0) = 3. 1.2.6 Find a function y = f (x) satisfying the given differential equation and the prescribed initial condition. √ dy = x x2 + 9 dx 8 y(−4) = 0. 1.2.11 Find the position function x(t) of a moving particle with the given acceleration a(t), initial position x0 = x(0), and initial velocity v0 = v(0). a(t) = 50, v0 = 10, x0 = 20. 9 1.2.15 Find the position function x(t) of a moving particle with the given acceleration a(t), initial position x0 = x(0), and initial velocity v0 = v(0). a(t) = 4(t + 3)2 , v0 = −1, x0 = 1. 10 1.2.27 A ball is thrown straight downward from the top of a tall building. The initial speed of the ball is 10m/s. It strikes the ground with a speed of 60m/s. How tall is the building? 11 1.2.35 A stone is dropped from rest at an initial height h above the surface of the √ earth. Show that the speed with which it strikes the ground is v = 2gh. 12 1.2.43 Arthur Clark’s The Wind from the Sun (1963) describes Diana, a spacecraft propelled by the solar wind. Its aluminized sail provides it with a constant acceleration of 0.001g = 0.0098m/s2. Suppose this spacecraft starts from rest at time t = 0 and simultaneously fires a projectile (straight ahead in the same direction) that travels at one-tenth of the speed c = 3×108m/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? 13 1.3.1 and 1.3.6 See Below In Pivblenis I thmugh 10, ne hai’e provided the slope field of the indicated difft’i-ential equation, together with one or inon’ solution curves. Sketch likel’ solution curves through the ad ditional points marked each slope field. 13) dy 4. —=x—v clx \ \ /11 I I .1.1 / / /7 / III I/Ill’ I I I \\\\ / I I I I \ / I I I in ixis ( —v clx 2 sinx — s._ II 3 of alue 7 (14) 1 I Il//I ———‘//1// ki —2 ‘‘‘‘1/I,, /7/11 1/1/ f/I I/Il/f / I / /•/ / I I//I / II / / / I I I I/ l —2 f the —1 —2 / _/ FIGURE 1.3.18. 1,/Ill /1 I ,•/ / I. I 0 ‘I I’ I 2 5. = clx v 7 ,I,, /1/ / / /!1Il5IIl ’ ‘‘ I d 1 7/1/7/ /1//Il/Il 7/I/I/I 7//llIII I ////IIII I/I, .3 ---—‘‘‘.‘ / S S..———I — \ IS Ill ‘.S’.-— y = clx -3 x± 3 C 5.-, + 1 — x 1/ / / / I / / / lssIIIili ill/Il//I 1111/Ill? lI/Ill/Il I / / / /1//I /•///9//•I I/I/Il//I ‘ / - - /1/ /1 / / / / — /1/fl/I //J/// I/I/I 7——— I//I /1/ ,/ S\S\ —1 —7 ——.,I//1 i/I\\\SS.. I/l\\\S I I ‘‘ .‘.I I / 1111//Ill I I / / II I/S /IIlIlII\ •I I / / / 7 I I I III / / I / II / I / /5 I / —3 —2 —1 ..-j- — — / — U —1 7e S—-———7/// ‘—‘—————/ / —2 —3 I\.—.-—7/ II. III / / / Ill I/ I Ill’ / 51 / / I •—sS\ —2 /1/f SS / ,::---— ider a n Fig. 2 ‘‘‘I,,,,’ / / / / / / / FIGURE 1.3.15. - 1 U —i /1/1/1/ / 2. ////I / I/I//I,, 7/7/f//I / /1/fl /1(1 1,/If/I’ / fl/I//I / / ,b / ,•, / / I/Il I / I 1 —, / I / I / / / I —1 rigin 7/,, S s’.. — /1/7/7 7, 7/, —j —..-—_._-—————--—. . / \ 0 1/ /1 / \ / / \ \ /1/1/ III I\ \ \ \ \ \ \ \\\‘\\\\\ S I — SI\l I •S \ I/Ill I / I / / 0 \ S S I \I\\.———— S S S — — — I I 3 7 ( FIGURE 1.3.19. FIGURE 1.3.16. 3. dy dx = —=y—sinx 3 2 the left hus the 3 ,/ / / / / / / / 7/I//I/Il /‘ / 1_ .7 / / I, /1 / / / / I I Ill/Il I/I / / 1,1 / 1 W 111/11/1/ I//I/Il/I I / / / / / .5 / / /7 7 I, 7/7//Il /1/ — 5.-. ‘ 0 x - v + I /1/ \\//\\ J 11 SS // / I/Il Il/IS 7/ ,I.IlIö\I’IS I/I / / / / /1155 7//I II Il/Il 7/I/I III\\I Sfl.’———— \•5\5\5..,____ /11/1/ —, 7/I//I I Ills S’.”. “.._——_- ,.,.,.I, ‘/1/1/ / y many S I I I I ‘s S 5, / / / II / / — SII\\\\\S through solution nstanCe, thole x —2 Ill/Il’’ III Il/Il’ I /•/ / \, / I I I / / I’ / II / I I / I, I I I S \\‘.-..S.. S S I S 5. I I \ISS5\\\\ -3 —3 —2 —l 3 I 2 —2 3 —3 —3 ————‘I//I -_,p,,,.,, 7/I / / / / / / 17/ /11 / / I —2 FIGURE 1.3.20. / I —I x FIGURE 1.3.17. II / / I’I! / / / / / / / / / / I / / III / / II / Il/Il / / / / / / Ill / / / •I / ,/ / i//Il / III / I ii I / ‘S iranteeS ,. 7/Il//fl IIIss_.___ I ,,.,j., 2 1.3.9 See Below 3 3 /1/1,1,,, 2_.._V_i.VVV__, -.—--—— -5 V —. H /1? / / /1 / / / / / / I., I / — - ———555 E’s. ——V.5 ((((5 S S S S S S S S S S S S S —1 5V._V__V.. “‘\ (((((5 \5555 ‘—‘‘ I U —i —2 5555 SS\ Is’s ( S S S S S S \ S 555(5(5 S 5(5’. 5 5 5 \ S S S S S S S S 5 S \ ‘5\\\(\ ((((((55’. —‘5 —., 55 ((((((S 555555555 S ‘•5 S S S “ S S ‘555(55 555555 (\\\5V. ((((55 2 0 -1 -2 3 2 FIGURE 1.3.24. FIGURE 1.3.21. In Problems 11 through 20, dete,,nine whether Theorem I does of a solution oft/ic given initial or does not guarantee is guaranteed, determine whether ma/ac problem. hf that so Jimeoremn I does or does not guarantee uniqueness existence of existence dv 8. ——=.v-—v lution. dx (I V 3 S S I 555”—— (/VVVS\\ I //‘—sSS (5/!’— S5—,/I I ._L. 5 9 1 • (S 13. ‘———‘-V 14. -, I) -I —2 3 vU) —1 — = 1 VV’ ///(( ‘‘Il//I ( I I//I/Il I//I/Il I h//Il/I I ( I (‘I I /llIIIIIi I I II II I( /5/ I/h’’ ( ///// I ( Ill//I I! Il//f! II I (III II /•/ / / I ( I/Il/I/I / I I I I ( i —‘5 d =xlnv: dx = II ——‘-———‘ ‘‘:‘ 12. 5/I ‘_555VV._ I//_ vU) (/.V (((‘—‘/1) S S S — 5 (lI_.—SS(( / / / — S S S S 2 v 2 2.v = (/I_V_5I\\\ 2 1 - = v(O) = (0) dx = 0 15. -=,jr-; y(2)=2 16. v(2)=1 FIGURE 1.3.22. dv 9. —-x—v—2 dx - 3 2 5 5 I /—\\ I ‘‘-.‘.‘‘ (f_.\((\\ ( iI/_ S 55—I / S II\\5.’/ —3 v(0)=1 £1 V 18. v— dx v(1) = x — 1; = 0 19. = In(1 + v ); 2 = 2 x y(O) = 0 \III\—_V_/ S ((555—/Si S/Il /N..\( —2 dv 17. v——=x—1; dx //I\\(1(/ (//‘—S(I\ —l -=,/‘‘; S 20. —- — ; 2 v v(0) = 1 (LU I / —— S S S (1/ III’’———— I / / / S I I / “ II /1/,— “ ‘ 5/ I / / —2 —l of’ H. (///—55 V———, / /5 / 5 I I / / II I /5 / / I I I / _// — 0 / 1 2 3 LUamnple 2 In Probleni.s 2 I and 22, first use the method diffrrential equation. given the for field slope a construct to Then sketch the solution cane corresponding to the given ini tial condition. Finally, use this solution cane to estimate the desired ia/ac oft/ic solution v(x). S 21. FIGURE 1.3.23. v’=rx+v, 22. v’==v—x, v(0)=0; y(4)=O; v(—4)=? v(—4)=? 1.3.11 Determine whether Theorem 1 does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem 1 does or does not guarantee uniqueness of that solution. dy = 2x2 y 2 dx 16 y(1) = −1. 1.3.15 Determine whether Theorem 1 does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem 1 does or does not guarantee uniqueness of that solution. dy √ = x−y dx 17 y(2) = 2. 1.3.21 First use the method of Example 2 from the textbook to construct a slope field for the given differential equation. Then sketch the solution 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(0) = 0; 18 y(−4) =? 1.3.29 Verify that if c is a constant, then the function defined piecewise by y(x) = 0 x ≤ c, (x − c)3 x > c 2 satisfies the differential equation y ′ = 3y 3 for all x. Can you also use the “left half” of the cubic y = (x − c)3 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 2 initial value problem y ′ = 3y 3 , y(a) = b has either no solution or a unique solution that is defined for all x? Reconcile your answer with Theorem 1. 19 More room for Problem 1.3.29 20