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HW  5,  MAT  303

Try your math skills on problems 9,24a,27a in section 2.3, problems 1-4 in section 2.4. (These problems from the text book will not be graded.)

Also complete the following four problems, which will be graded.

(1) (3 points) Consider initial value problem dydx  = f (x), y(a) = y0, where a denotes a ixed real number, and where f (x) denotes a real valued function deined and continuous on the real number line.

(a) Verify that for each number b > a we have y(b) = y0 +la(b) f (x)dx.

(b) Choose a = x0  < x1   < x2  < .... < xi  < xi+1  < ... < xn  = b so that △x = xi+1 − xi  = b—an  for all 0 ≤ i < n; so the xi  devide the interval  [a, b] into n intervals  [xi, xi+1] of equal lengths.  Now use Euler’s method (see page 106, and/or lecture 5b) to construct the values y1, y2 , ...., yn . Show that

n — 1

yn = y0 +工 f (xi)△x.

i=0

(c) Now use what you learned in Calculus II (about deinite integrals),

together with parts (a),(b) above, to verify that

limitn→∞ yn = y(b).

(Hint:  Think about“Riemann sums”approximating deinite in- tegrals.)

(2) (3 points) Consider the initial value problem dydx  = 13y2, y(0) = 1. (a) Solve this initial value problem; then compute y(1).

(b) Use Euler’s method with step size equal .25 to approximate the value y(1).  (In otherwords let xi  = .25i for i=0,1,2,3,4; then use Euler’s method to compute y1, y2, y3, y4, and use y4  as the approximation for y(1)).

(c) Is the diference y(1) − y4  less than 1100?

(3)   (2 points) Do problem 10 in section 2.3.

(4) (2 points) Do problem 26 in section 2.3.