Handwritten solving. 3 Euler's solver (#differentiation, #integration)Consider a differential equation of the typedydt(1)where f is a function that may depend on both t and y. We want to find the solutionto this differential equation, y(t), with initial conditions prescribed by y(to) = yo. Ourgoal in this problem is to derive a computational scheme known as Euler's method, andsee 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 equationthat describes the slope of the solution everywhere, Eq. 1. Use this information tofind 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 sufficientlysmall, the line tangent and the solution will be close, and y₁ will be a reasonableapproximation 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 thesame slope the solution would have at that point, as prescribed by the differentialequation.(d) We may once again use this line to estimate the value of the solution at a nearbypoint. Consider t₂ = t₁+h. Estimate the value of the solution at the point t2, whichwe will denote by y/2.(e) Generalize: come up with an algorithm to compute yn, the estimated value of thesolution at the point tn = to + nh.(f) We can model the population y of bunnies in a park by the differential equationdydt= ky(N - y)(2)where k is the rate of growth of the population, N is the carrying capacity, and t ismeasured 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 is7, use the algorithm you constructed in part (e) to estimate the number of bunniesthat will be living in the park after a year. Try at least three different values of thestep-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).

Note- In this question part (a) and part (c) alredy solve by experts and good explain each part we…

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: What's the answer for this

Q: solve #3, g and h

A: g):Consider the function h(x) = x5/3 -5x2/3.The first derivative of the function with respect to x…

Q: can i get the answer and the step to the question

A: As question have many subpart i have solved first three , please ask a separate question if you want…

Q: How to solve #53

Q: What does it means to know mathematics?

A: We will give answer by using the concept of uses of mathematics..

Q: How to solve?

Q: Trying to help my son and I am not sure how to solve

A: For two similar triangles, the corresponding angles are congruent and the corresponding sides are…

Q: what is the answer for the first question

Q: side x in simplest radical form 60 300 4

Q: Missing one of the solutions. fg?

A: The given functions are; f(x)=(x2+9x), g(x)=(1-x2).

Q: Solving

A: The given point is,

Q: Your missing the part d solution

A: First three parts are as solved before.

Q: how to solve this question

A: Step 1: Step 2: Step 3: Step 4:

Q: now could you solve part b of the question

Q: What are the answers to this ? #3

A: Denote p as the true proportion of people, who own cats.

Q: Part D?

A: As per your instructions i have to solve part(D)

Question

Handwritten solving.

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).

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1: part (a).

VIEW

Step 2: Linea approximation:

VIEW

Step 3: linear approximation:

VIEW

Step 4: part (b):

VIEW

Solution

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 5 steps with 33 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

What are the answers to this ? #2
how to solve this question
Part (iv) of the question please
Can you solve part D
Provide a coherent answer for items C and D.
What are the answers to this? #5
Solve all parts. It's appreciated
Calculation