Question

3. Modeling with Differential Equations a. Provide slope fields for the following differential equations: DE#1: y'-y-cos x; DE#3: y'-y-cos y. (4pts) DE#2: y-x-cos y, b. For each slope field, draw the solution curve for the initial condition y(0) 1. (4pts) Attach separate pages c. Use Euler's method to estimate y(2), using steps of h 0.5 and h0.1 '-y cosx,y(0)-1 You can use technology. Write your results accurate to 4 decimal places Create a separate table for h -0.1. (8pts) Vn. d. Of the three DE's above (in part a.), which one is autonomous? Describe what this means. (4pts)

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Answer 1

d. Out of the 3 differential equations only the 3rd one is autonomous as it does not contain x. Only the dependent variable is there.

The first two contains x term and hence they are not autonomous.

3.0 1. dy/dx- y" cos(x) Variables: dy/dx 2.0 (o, 1 with 20 segments with 15- segments 0.0 v with h 0.1 switching Euler To specify initial values for solution curves, either: enter (x.y)0 1submit The red curve is the solution curve passing through the point (0 1), that is y(0)-1. 3.0 .3 -2

3.0 2 dy/dxx*cos(y) Variables: dy/dx 1.0 with 20 segments with 15 segments 0.0 , with h = 0.1 Euler switching To specify initial values for solution curves, either enter (xy) 1submit The red curve is the solution curve passing through the point (0 1), that is y(0)-1. 2.0 ILIR 3.0 -2 -1 4

3.0 3. Variables: dy/dx 1.0 with 20 segments 4 SXS4 with 15- segments 0.0 with h 0.switching Euler To specify initial values for solution curves, either 1.0 . enter (xy)-( | | | ) | submit | 0 1 The red curve is the solution curve passing through the point (0 1), that is y(0)-1. -3.0 This is an autonomous equation as it does not contain x. -4 -2 -1 2 4

Euler's methog Graph of the solution by Euler method yy cos x -f(x y) y (i+1)-y i+h*f(xi, yi) 3.5 0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0 2.5 0.5 1.5 1.316373843 1.16607337 0.193906487 1.181098318 1.800672575 1.33380838 0.637153156 2.158187 1.5 2.741224 2.838177 2.5 2.247628 1.347291 3.5 0.680387 4 2.5 3.5

Euler's method 0.1 0.3 0.1 0.1 0.1 0.1 1.3 0.1 0.1 0.1 0.1 0.1 0.1 0.1 2.3 0.1 0.1 0.1