University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2022
Views:

189

Pages:

18
Author:

Jadyn Chandler

Visualizing Solutions A differential equation is like a “road map”: y 0 = f (t, y ) That is, at each point (t, y ), we can compute the slope of the line tangent to the solution curve y (t). January 17, 2018 1/8 Visualizing Solutions A differential equation is like a “road map”: y 0 = f (t, y ) That is, at each point (t, y ), we can compute the slope of the line tangent to the solution curve y (t). If the function y is well behaved, the tangent line should be a good approximation to y . January 17, 2018 1/8 Visualizing Solutions A differential equation is like a “road map”: y 0 = f (t, y ) That is, at each point (t, y ), we can compute the slope of the line tangent to the solution curve y (t). If the function y is well behaved, the tangent line should be a good approximation to y . Definition: A direction field is a plot in the (t, y ) plane that give the local tangent lines to the solution to a first order ODE. January 17, 2018 1/8 Example: y 0 = t − y 2 January 17, 2018 2/8 Example: y 0 = t − y 2 In drawing a picture, we might consider curves of constant slope (a.k.a. Isoclines). For example, the points that correspond to zero slope: January 17, 2018 2/8 Example: y 0 = t − y 2 In drawing a picture, we might consider curves of constant slope (a.k.a. Isoclines). For example, the points that correspond to zero slope: 0 = t − y2 ⇒ y2 = t January 17, 2018 2/8 Example: y 0 = t − y 2 In drawing a picture, we might consider curves of constant slope (a.k.a. Isoclines). For example, the points that correspond to zero slope: 0 = t − y2 ⇒ y2 = t The points that correspond to a slope of 1: January 17, 2018 2/8 Example: y 0 = t − y 2 In drawing a picture, we might consider curves of constant slope (a.k.a. Isoclines). For example, the points that correspond to zero slope: 0 = t − y2 y2 = t ⇒ The points that correspond to a slope of 1: 1 = t − y2 ⇒ t = y2 + 1 January 17, 2018 2/8 Example: y 0 = t − y 2 In drawing a picture, we might consider curves of constant slope (a.k.a. Isoclines). For example, the points that correspond to zero slope: 0 = t − y2 y2 = t ⇒ The points that correspond to a slope of 1: 1 = t − y2 ⇒ t = y2 + 1 The points that correspond to a slope of −2: January 17, 2018 2/8 Example: y 0 = t − y 2 In drawing a picture, we might consider curves of constant slope (a.k.a. Isoclines). For example, the points that correspond to zero slope: 0 = t − y2 y2 = t ⇒ The points that correspond to a slope of 1: 1 = t − y2 ⇒ t = y2 + 1 The points that correspond to a slope of −2: −2 = t − y 2 ⇒ t = y2 − 2 and so on... January 17, 2018 2/8 Figure: Direction Field with Isoclines: y 0 = −2, y 0 = 0, y 0 = 1 January 17, 2018 3/8 Sample solution curves must be consistent with the direction field: January 17, 2018 4/8 Sample solution curves must be consistent with the direction field: January 17, 2018 4/8 Give an ODE of the form y 0 = ay + b whose direction field looks like: January 17, 2018 5/8 We see that the DE has y = 1 as an equilibrium, so we can try: y0 = y − 1 or y0 = 1 − y Which is it? January 17, 2018 6/8 We see that the DE has y = 1 as an equilibrium, so we can try: y0 = y − 1 or y0 = 1 − y Which is it? SOLUTION: y 0 = 1 − y January 17, 2018 6/8 Same question as before: January 17, 2018 7/8 Choose a DE 1 y0 = 3 − y 2 y 0 = y (y + 3) 3 y 0 = y (3 − y ) 4 y 0 = 2y − 1 January 17, 2018 8/8

Direction Fields Visualization

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