Math Assignment. Dylan Zwick. Spring Section 2.4-1, 5, 9, 26, 30 Section 3.1-1, 16, 18, 24, 39 Section 3.2-1, 10, 16, 24, 31

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1 Math Assignment 4 Dylan Zwick Spring 2013 Section 2.4-1, 5, 9, 26, 30 Section 3.1-1, 16, 18, 24, 39 Section 3.2-1, 10, 16, 24, 31 1

2 -z Section Method Numerical Approximation: Euler s Apply Euler s method twice to approximate the solution to the initial value problem below on the interval [0, ], first with step size h =.25, then with step size h = 0.1. Compare the three-decimal-place values of the two approximations at x = with the value y() of the actual solution, also given below. y = y, y(o) = 2; y(x) = 2e. A 5 Eci I \47 y+(-z)(-z) -; L - z N ) 1V - y3 /-L (/fl-/z) yj Jcj5 V1 I ±(t)(1 Y y ( 1J lz 1/3 (-JIzz ±(/)(/z) F h?5 o - 2

3 2.4.5 Apply Euler s method twice to approximate the solution to the initial value problem below on the interval [0, ], first with step size Ii =.25, then with step size h = 0.1. Compare the three-decimal-place values of the two approximations at x = with the value y() of the actual solution, also given below. = y x 1, y(o) = 1; y(x) = 2 + x ex. yq( iil57(o) yo XZ5 I - H (z)(i -z-i). _L ii I L c( ((zi) x-j va (L( ) - y (9) 2-fr- C - - c - _ f k I 3

4 2.4.9 Apply Euler s method twice to approximate the solution to the initial value problem below on the interval [0, ], first with step size h.25, then with step size h = 0.1. Compare the three-decimal-place values of the two approximations at x with the value y() of the actual solution, also given below. y = (1 +y 2), For y(o) = 1; 1 y(x) = tan ((x + u)). I XQ l(ii)) 1 (ii) I t (o5± -I(4(/Tuoy)y - I7 ( (i(i ( ( (I) ((i (1-/1 Vct( (; (-5) +c1( ( 5)) ) z J15 / h Z? lii7 /7 (- ()( (1 liz I7 77 /2ry /c!

5 Suppose the deer population P(t) in a small forest initially numbers 25 and satisfies the logistic equation dt = O.0225P P (with tin months.) Use Euler s method with a programmable calcu lator or computer to approximate the solution for 10 years, first with step size h = 1 and then with h.5, rounding off approximate P values to integrals numbers of deer. What percentage of the limiting population of 75 deer has been attained after 5 years? After 10 years? Al 11 mcii 1)7 I,ii4 7 1i 4Fh L L Z L5 J7 7 /OcC/6// 25 C, yei to -F 1Q P1ce4y t / 7 (2 h / A 5 I CG( 1

6 Md More room for problem

7 Apply Euler s method with successively smaller step sizes on the interval [0, 2] to verify empirically that the solution of the initial value problem =y2+x2, (0)0 dx has vertical asymptote near x = C) A Iz 5 CI 0 I 1 3, L if Cf x L y e - / 9 1) 7 T4 13 i Id

8 More room for problem

9 y Section Second-Order Linear Equations Verify that the functions yi and Y2 given below are solutions to the second-order ODE also given below. Then, find a particular solution of the form y = c1y1 +c2y2 that satisfies the given initial conditions. Primes denote derivatives with respect to x. yi=ex = 0; y2=e; y(o) = 0 y (O) = 5. yj D : /1 - C -y \/(_ y 0 V I -,{ -Y yl -y= e -e =6hz \/ (x) e y ( ) - L y(o)= C ±( =0 y lo) ( ) (D5 7-9

10 Verify that the functions y and Y2 given below are solutions to the second-order ODE also given below. Then, find a particular solution of the form y = ciyi +c2y2 that satisfies the given initial conditions. Primes denote derivatives with respect to x. xy + y + y = 0; Yi cos (in x), Y2 = sin (in x); y(l) = 2, y (l) 3. /i yi 4[Ii] () L (IilK) L I /7 = Y y 5i1 (/k1)- coi (Ii)- (/i) f(o (/) 1) o tl,i) üi(h y() ( (o()i) y(i)z y1 (i ) 1- s-1v1 (I/li) O - c-i y () - 5) x CL \/ (/) 10

11 cj Ic-ID (JD C)D II C C C CD Ci r (Th _p I_i 3! I 4 -x c7n II S -I- t.z -C:.

12 c) LID I. SI o I. 0 H -d 5 Ct a) a) C H CI] C.) C.) S. -d S I a) Cl,

13 3xy (a) Show that y = x and Y2 = are linearly independent solu tions on the real line of the equationx2y + 3y = 0. (b) Verify that W(yi, y2) is identically zero. Why do these facts not contradict Theorem 3 from the textbook? a) t /I( 1T y y ytx L($) 3L)tQ / i) 10 (a c c A c 1i e fe / (() cve \pe14 *A iei 1i Ihese ;iifi / I I 1 ii I (c I I i-he H h//= ( 1 o C o J I 3 I( / / Y+yO 4r (i3,] Cce,IE lih

14 Section General Solutions of Linear Equa tions Show directly that the given functions are linearly dependent on the real line. That is, find a non-trivial linear combination of the given functions that vanishes identically. f(x) = 2x, g(x) = 32 h(r) = 5x 8x2. () - c) () () f f(l) 5/ Iie to 14

15 Use the Wronskian to prove that the given functions are linearly independent. f(x) ex, g(x) = x2 li(x) x2 lnx; x > 0. e xl 6 x 5 Li4 /io - x - fl/) 2<6 7J x x 7O 5) i ec In 15

16 5y 2 = Find a particular solution to the third-order homogeneous linear equation given below, using the three linearly independent solutions given below. + 8y 0; y(o) = 1, y (O) = 4, y (O) = 0; Yi = e, Y2 = e2x, y = xe 2. y () C1 L ) C c x Ce ± I (ye Lic Y 1(e) Ct +L \i() (( / 3 C ±LI 1 1 ) C-/O C(

17 2y Find a solution satisfying the given initial conditions for the differ ential equation below. A complementary solution Yc, and a particular solution y are given. + 2y = 2x; y(o) 4 y (O) = 8; y( Ycdle0+c2e111x C( (05 K Xf YpX+l. y- ci C eco) CLe(0Y ±(5 / y (o) C H = 7 ctj yt(o) Ct C Cf 17

18 2y 5y This problem indicates why we can impose only n initial conditions on a solution of an nth-order linear differential equation. (a) Given the equation y +pg +qy 0, explain why the value of y (a) is determined by the values of y(a) and y (a). (b) Prove that the equation = 0 has a solution satisfying the conditions if and only if C = 5. g(o) = 1, y (O) = 0, y (O) C, y () -)y () q)yl), b) (o) Zo)5(i)% ms 18

19 More room for problem J 19

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