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Consider a differential equation of the type dy dt f(t,y), (1) where ƒ is a function that may depend on both t and y. We want to find the solution to this differential equation, y(t), with initial conditions prescribed by y(to) = yo. Our goal in this problem is to derive a computational scheme known as Euler's method, and see how it can be useful to approximate solutions to first-order differential equations. (a) From above, we are given an initial point in the solution (y(to) = y0), and an equation that describes the slope of the solution everywhere, Eq. 1. Use this information to find the linear approximation to the solution at the initial point (to, 30). (b) Use the linear approximation to estimate the value of the solution at a nearby point, t₁ = to+h. We will denote your approximation by y₁. Provided that his sufficiently small, the line tangent and the solution will be close, and y₁ will be a reasonable approximation of the actual value of the solution, ults).

Consider a differential equation of the type dy dt f(t,y), (1) where ƒ is a function that may depend on both t and y. We want to find the solution to this differential equation, y(t), with initial conditions prescribed by y(to) = yo. Our goal in this problem is to derive a computational scheme known as Euler's method, and see how it can be useful to approximate solutions to first-order differential equations. (a) From above, we are given an initial point in the solution (y(to) = y0), and an equation that describes the slope of the solution everywhere, Eq. 1. Use this information to find the linear approximation to the solution at the initial point (to, 30). (b) Use the linear approximation to estimate the value of the solution at a nearby point, t₁ = to+h. We will denote your approximation by y₁. Provided that his sufficiently small, the line tangent and the solution will be close, and y₁ will be a reasonable approximation of the actual value of the solution, ults).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3 Euler's solver (#differentiation, #integration) Consider a differential equation of the type dy dt (1) where f is a function that may depend on both t and y. We want to find the solution to this differential equation, y(t), with initial conditions prescribed by y(to) = yo. Our goal in this problem is to derive a computational scheme known as Euler's method, and see how it can be useful to approximate solutions to first-order differential equations. = f(t, y), (a) From above, we are given an initial point in the solution (y(to) = yo), and an equation that describes the slope of the solution everywhere, Eq. 1. Use this information to find the linear approximation to the solution at the initial point (to, yo). (b) Use the linear approximation to estimate the value of the solution at a nearby point, t₁ = to +h. We will denote your approximation by y₁. Provided that h is sufficiently small, the line tangent and the solution will be close, and y₁ will be a reasonable approximation of the actual value of the solution, y(t₁). (c) Now, find the equation of the line that goes through the point (t₁, y₁) and has the same slope the solution would have at that point, as prescribed by the differential equation. (d) We may once again use this line to estimate the value of the solution at a nearby point. Consider t₂ = t₁+h. Estimate the value of the solution at the point t2, which we will denote by y/2. (e) Generalize: come up with an algorithm to compute yn, the estimated value of the solution at the point tn = to + nh. (f) We can model the population y of bunnies in a park by the differential equation dy dt = ky(N - y) (2) where k is the rate of growth of the population, N is the carrying capacity, and t is measured in months. Consider a population that grows at a rate k = 0.5 per month, and a carrying capacity N = 100 bunnies. If the initial population of bunnies is 7, use the algorithm you constructed in part (e) to estimate the number of bunnies that will be living in the park after a year. Try at least three different values of the step-size h. Which one provides the most accurate estimate? (g) Find the exact solution for the bunny population as a function of time, y(t).
Transcribed Image Text:3 Euler's solver (#differentiation, #integration) Consider a differential equation of the type dy dt (1) where f is a function that may depend on both t and y. We want to find the solution to this differential equation, y(t), with initial conditions prescribed by y(to) = yo. Our goal in this problem is to derive a computational scheme known as Euler's method, and see how it can be useful to approximate solutions to first-order differential equations. = f(t, y), (a) From above, we are given an initial point in the solution (y(to) = yo), and an equation that describes the slope of the solution everywhere, Eq. 1. Use this information to find the linear approximation to the solution at the initial point (to, yo). (b) Use the linear approximation to estimate the value of the solution at a nearby point, t₁ = to +h. We will denote your approximation by y₁. Provided that h is sufficiently small, the line tangent and the solution will be close, and y₁ will be a reasonable approximation of the actual value of the solution, y(t₁). (c) Now, find the equation of the line that goes through the point (t₁, y₁) and has the same slope the solution would have at that point, as prescribed by the differential equation. (d) We may once again use this line to estimate the value of the solution at a nearby point. Consider t₂ = t₁+h. Estimate the value of the solution at the point t2, which we will denote by y/2. (e) Generalize: come up with an algorithm to compute yn, the estimated value of the solution at the point tn = to + nh. (f) We can model the population y of bunnies in a park by the differential equation dy dt = ky(N - y) (2) where k is the rate of growth of the population, N is the carrying capacity, and t is measured in months. Consider a population that grows at a rate k = 0.5 per month, and a carrying capacity N = 100 bunnies. If the initial population of bunnies is 7, use the algorithm you constructed in part (e) to estimate the number of bunnies that will be living in the park after a year. Try at least three different values of the step-size h. Which one provides the most accurate estimate? (g) Find the exact solution for the bunny population as a function of time, y(t).
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