Math 2280 - Assignment 1 Dylan Zwick Spring 2014 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 Section 1.4-1, 3, 17, 19, 31, 35, 53, 68 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 = 3x 2 ; y = x 3 + 7 2
1.1.12 Verify by substitution that the given function is a solution of the given differential equation. x 2 y xy + 2y = 0; y 1 = x cos (ln x), y 2 = x sin (ln x). 3
1.1.15 Substitute y = e rx into the given differential equation to determine all values of the constant r for which y = e rx 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, y(0) = 10 5
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? Note - In order to solve this problem you need a result from problem 43 of this section. Namely, that a differential equation of this form has a solution of the form: P(t) = 1 (C kt). 6
More room for Problem 1.1.45, if you need it. 7
Section 1.2 - Integrals as General and Particular Solutions 1.2.1 Find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. dy = 2x + 1; y(0) = 3. dx 8
1.2.6 Find a function y = f(x) satisfying the given differential equation and the prescribed initial condition. dy dx = x x 2 + 9 y( 4) = 0. 9
1.2.11 Find the position function x(t) of a moving particle with the given acceleration a(t), initial position x 0 = x(0), and initial velocity v 0 = v(0). a(t) = 50, v 0 = 10, x 0 = 20. 10
1.2.15 Find the position function x(t) of a moving particle with the given acceleration a(t), initial position x 0 = x(0), and initial velocity v 0 = v(0). a(t) = 4(t + 3) 2, v 0 = 1, x 0 = 1. 11
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? 12
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. 13
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/s 2. 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 10 8 m/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? 14
thole x FIGURE 1.3.16. f the (14) of s._ clx ixis alue nstance, 7 / / I Il//I 1 / \ 3 0 x 2 I I 2 / I / / / / 1,1 / 1W /.7 / I / / / I I 2 3 j 2 ditional points marked in each slope field. /11 solution curves. Sketch likel solution curves through the ad the left solution -3 rigin ider a through \ISS5\\\\ irantees I I I \\\\ / I I I I \ / I I I 13) the indicated difft i-ential equation, together with one or inon clx I/Ill In Pivblenis I thmugh 10, ne hai e provided the slope field of Ill/Il n Fig. 3,I,, /1/ II / / 7 7 3 2 \ 7/I/I/I II I//I / / 1,/Ill /1 / 7/1/7/ clx I / / / / / / - clx 5. = y x + 1 / / / /1/1/1/ 7, /1/7/7 dy //1// / fl/i//i / I/Il / S \ I I/ I / / I / I / / / 51 / / / I / / III / / / / / / / Ill / / I ii I I 2 dx II / Il/Il / y many /1/1/ / II hus the / 3 7,I.IlIö\I IS.._ _- 7/ / / // /1/ \\//\\ SS,/ 7 3 dy I I/Ill Ill SI\l / Ill II. S\S\ / / / I/I//I,, 7/,, 7/, 1.3.1 and 1.3.6 See Below FIGURE 1.3.17. FIGURE 1.3.20. 2 I S 5. I I 3 2 l 3 3. =y sinx 2 1 III / / I / II \I\\. /IIlIlII\ I\..- 7/..I I / I I 1111//Ill S - 7///..-j- I/l\\\S /1/f i/i\\\ss...,i//1,::--- Ill.S.- ////IIII 2. - = x ± v FIGURE 1.3.15. 2 1 0 I 2 f/i I/Il/f /7/11 1/1/ 1/I,, 1 ki._._, / I / I / / / I / I / / /.- - --... \\\ \\\\\ III /1/1/ l / I I/ / / I,,,, / / \ \ \ \ \ I\ \ \ \ /1 1/ _/ I I, / / I. / /.3 S S.. I /1//Il/Il / 7e I I / / II I/S / / I I / / / 7 I I I \ S S / I / /5 I / I I S S S ( 3 /1 I, I I / / / / / / /7 S \\.-..S.. S S S S I I I I s S 5, / / / \, I / I II I I I / / / I / I, I I I / I III Il/Il SII\\\\\S I / / / / /.5 / / I//I/Il/I /1/ 111/11/1/ I, 7/7//Il / / 1_ Ill/Il I/I 7/I//I/Il ---. -1d I I/I, /!1Il5IIl IS 7//llIII I ( v sinx 3 2 I 2 -_,p,,,.,, 3 I 7/I / / / / / / I//I = x - v + I FIGURE 1.3.19. 3 2 1 0 U I/I/Il//I / SS / ///9// I 1111/Ill? I/I/I li/ill/il I//I 7 ill/il//i //J/// lssiiiili /1/fl/I FIGURE 1.3.18. 2 i U 1 2 1,/If/I /1/fl /1(1 7/7/f//I / ////I / I / I 1-3 5.-, C 1/ - / I / / / /1//I /1/ /1 1 5.-. 0 IIIss_. 7/Il//fl III\\I Sfl. 7/I/I \ 5\5\5..,, /11/1/ II Il/Il 7//I / / / / /1155 I/I / / / I/Il Il/IS / Ills S.. 7/I//I I 17/ /11 / / i//il / III / / / / / / / /,/ I / / / / / / / / I I! /,,.,j., I,.,.,.I,,. 11J ss\ I III /1/,/ II I S S s.. \ /.1.1 / / I I /7 / III \ \ / /,b /,, / 4. =x v
FIGURE 1.3.21. FIGURE 1.3.23. dv (li_. SS(( ((( /1) FIGURE 1.3.22. 555555555 E s. 3 2_.._V_i.VVV, 8. =.v- v V - ( i 13. - 21. v =rx+v, v(0)=0; H. (5/! _555VV._ (1/ ( I (/// 55 dx dv (((((5 \5555 ((((5 5V._V V.. 1.3.9 See Below /5/ -V I//_ 5 19._L. (/I_V_5I\\\ lution. 20. - In Problems 11 through 20, dete,,nine whether Theorem I does ma/ac problem. hf existence is guaranteed, determine whether Jimeoremn I does or does not guarantee uniqueness of that so 14. = (0) = 0 18. v = x 19. = In(1 + v2); y(o) = 0 dx dx dx dv dx vu) = 1 =xlnv: d or does not guarantee existence ofa solution oft/ic given initial FIGURE 1.3.24. /1/1,1,,, H ii/_ S/Il ((555 /Si /N..\( S (// S(I\ S -.. II\\5. / (f_.\((\\ \III\ _V_/ //I\\(1(/ 5 dx - I/Il/I/I / /lliiiiii I! Ill//I I//I/Il II I Il//f! I//I/Il I II (III I h//il/i I 5/I /1? / / /1 / -. --. / / / / / I., I / 1 S S S S S S S S S S S S S S S 5 S S S S., S 5 5 5 5 \ S S \ S S S S S S S S S S \ S S S 2 i U I 2 3 3 2 (/VVVS\\ 555 I S S / / / S S S S S S S 5 I I // sss S5,/I - II (S : VV ///(( I/h Il//I ( ///// I ( ( ( I / / / / -, I I ( I ( I I I I II II I -I 3 2 I) 1 2 I( I 5 9. -x v 2 3 2 l 2 I / \\ 5 I S S 55 I / / S S S III I / / / S I I / 5/ I / / II /1/, V, /5 5 / / I / I / /5 / / I I I 3 2 l 0 1 2 3 S _// / / II I desired ia/ac oft/ic solution v(x). tial condition. Finally, use this solution cane to estimate the Then sketch the solution cane corresponding to the given ini 12. 15. -=,jr-; y(2)=2 16. -=,/ ; v(2)=1 In Probleni.s 2 I and 22, first use the method of LUamnple 2 to construct a slope field for the given diffrrential equation. 22. v ==v x, y(4)=o; v( 4)=? v( 4)=? 17. v =x 1; v(0)=1 = v(o) -2-1 0 2 ( Is s \ ((((((55. SS\ 5(5. 5555 ((((((S S 555(5(5 ((((55 55 5\\\(\ (\\\5V. 555555-5 555 3 V.5 555(55 5 (I V (/.V 1 V 1; v(1) = 0 = x2 v2; (LU v(0) = 1 = 2.v 2v vu) = 1
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 dx = 2x2 y 2 y(1) = 1. 17
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 dx = x y y(2) = 2. 18
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; y( 4) =? 19
More room for Problem 1.3.21, if you need it. 20
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 satisfies the differential equation y = 3y 2 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 initial value problem y = 3y 3, 2 y(a) = b has either no solution or a unique solution that is defined for all x? Reconcile your answer with Theorem 1. 21
More room for Problem 1.3.29, if you need it. 22
Section 1.4 - Separable Equations and Applications 1.4.1 Find the general solution (implicit if necessary, explicit if convenient) to the differential equation dy dx + 2xy = 0 23
1.4.3 Find the general solution (implicit if necessary, explicit if convenient) to the differential equation dy dx = y sin x 24
1.4.17 Find the general solution (implicit if necessary, explicit if convenient) to the differential equation y = 1 + x + y + xy. Primes denote the derivatives with respect to x. (Suggestion: Factor the right-hand side.) 25
1.4.19 Find the explicit particular solution to the initial value problem dy dx = yex, y(0) = 2e. 26
1.4.31 Discuss the difference between the differential equations (dy/dx) 2 = 4y and dy/dx = 2 y. Do they have the same solution curves? Why or why not? Determine the points (a, b) in the plane for which the initial value problem y = 2 y, y(a) = b has (a) no solution, (b) a unique solution, (c) infinitely many solutions. 27
More room for Problem 1.4.31, if you need it. 28
1.4.35 (Radiocarbon dating) Carbon extracted from an ancient skull contained only one-sixth as much 14 C as carbon extracted from presentday bone. How old is the skull? 29
1.4.53 Thousands of years ago ancestors of the Native Americans crossed the Bering Strait from Asia and entered the western hemisphere. Since then, they have fanned out across North and South America. The single language that the original Native Americans spoke has since split into many Indian language families. Assume that the number of these language families has been multiplied by 1.5 every 6000 years. There are now 150 Native American language families in the western hemisphere. About when did the ancestors of today s Native Americans arrive? 30
to 1.4.68 The figure below shows a bead sliding down a frictionless wire from point P to point Q. The brachistochrone problem asks what shape the wire should be in order to minimize the bead s time of descent from P to Q. In June of 1696, John Bernoulli proposed this problem as a public challenge, with a 6-month deadline (later extended to Easter 1697 at George Leibniz s request). Isaac Newton, then retired from academic life and serving as Warden of the Mint in London, re ceived Bernoulli s challenge on January 29, 1697. The very next day he communicated his own solution - the curve of minimal descent time is an arc of an inverted cycloid - the Royal Society of London. For a modern derivation of this result, suppose the bead starts from rest at the origin P and let y y(x) be the equation of the desired curve in a coordinate system with the y-axis pointing downward. Then a mechanical analogue of Snell s law in optics implies that sin ct V = constant where c denotes the angle of deflection (from the vertical) of the tan gent line to the curve - so cot c y (x) (why?) - and v /2jj is the bead s velocity when it has descended a distance y vertically (from KE = mv 2 = mgy = -PE). p 31
(a) First derive from sin α/v = constant the differential equation dy 2a y dx = y where a is an appropriate positive constant. 32
(b) Substitute y = 2a sin 2 t, dy = 4a sin t costdt in the above differential equation to derive the solution x = a(2t sin 2t), y = a(1 cos 2t) for which t = y = 0 when x = 0. Finally, the substitution of θ = 2a in the equations for x and y yields the standard parametric equations x = a(θ sin θ), y = a(1 cosθ) of the cycloid that is generated by a point on the rim of a circular wheel of radius a as it rolls along the x-axis. 33