Introductory Differential Equations using Sage - William Stein

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38 CHAPTER 1. FIRST ORDER DIFFERENTIAL EQUATIONS y ′ (x) ∼ = y(x + h) − y(x) , h h > 0 is a given and small. This an the DE together give f(x,y(x)) ∼ = y(x+h)−y(x) h . Now solve for y(x + h): y(x + h) ∼ = y(x) + h · f(x,y(x)). If we call h · f(x,y(x)) the “correction term” (for lack of anything better), call y(x) the “old value of y”, and call y(x + h) the “new value of y”, then this approximation can be re-expressed y new = y old + h · f(x,y old ). Tabular idea: Let n > 0 be an integer, which we call the step size. This is related to the increment by h = b − a n . This can be expressed simplest using a table. x y hf(x,y) a c hf(a,c) a + h a + 2h c + hf(a,c) . . b ??? xxx The goal is to fill out all the blanks of the table but the xxx entry and find the ??? entry, which is the Euler’s method approximation for y(b). Example 1.6.1. Use Euler’s method with h = 1/2 to approximate y(1), where y ′ − y = 5x − 5, y(0) = 1. Putting the DE into the form (1.8), we see that here f(x,y) = 5x + y − 5, a = 0, c = 1. x y hf(x,y) = 1 2 (5x + y − 5) 0 1 −2 1/2 1 + (−2) = −1 −7/4 1 −1 + (−7/4) = −11/4 so y(1) ∼ = − 11 4 = −2.75. This is the final answer. Aside: For your information, y = e x − 5x solves the DE and y(1) = e − 5 = −2.28.... Here is one way to do this using Sage : .
1.6. NUMERICAL SOLUTIONS - EULER’S METHOD AND IMPROVED EULER’S METHOD39 Sage sage: x,y=PolynomialRing(QQ,2,"xy").gens() sage: eulers_method(5*x+y-5,1,1,1/3,2) x y h*f(x,y) 1 1 1/3 4/3 4/3 1 5/3 7/3 17/9 2 38/9 83/27 sage: eulers_method(5*x+y-5,0,1,1/2,1,method="none") [[0, 1], [1/2, -1], [1, -11/4], [3/2, -33/8]] sage: pts = eulers_method(5*x+y-5,0,1,1/2,1,method="none") sage: P = list_plot(pts) sage: show(P) sage: P = line(pts) sage: show(P) sage: P1 = list_plot(pts) sage: P2 = line(pts) sage: show(P1+P2) The plot is given in Figure 1.13. Figure 1.13: Euler’s method with h = 1/2 for x ′ + x = 1, x(0) = 2. 1.6.2 Improved Euler’s method Geometric idea: The basic idea can be easily expressed in geometric terms. As in Euler’s method, we know the solution must go through the point (a,c) and we know its slope there is m = f(a,c). If we went out one step using the tangent line approximation to the solution curve, the approximate slope to the tangent line at x = a + h,y = c + h · f(a,c) would be
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176 CHAPTER 4. INTRODUCTION TO PART
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202 BIBLIOGRAPHY [D-spr] Wikipedia
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204 BIBLIOGRAPHY [NS-intro] General
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206 BIBLIOGRAPHY
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208 CHAPTER 5. APPENDICES 5.1 Appen
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Introductory Differential Equations using Sage - William Stein

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