Please show all work! y' (t) = f(t,y(t))in Lecture 1, together with the initial condition y(a)=ß, we noted that iff = f(t) depends only on the independent variable t, you get the simplest typeof ODE that can be solved via Calc II: FTC impliesy(t) = ß + f* f(s) dsand you might be able to find an explicit formula for the integral on the righthand side using techniques from Calc II.We also noted that in the more interesting case f = f(t, y), FTC is stillapplicable and givesy(t) = ß + [* f(s, y(s)) dswhich doesn't immediately look useful, since the integrand of the integral onthe right hand side entails the unknown function y that you are trying to find.However: This formula can be used to iteratively approximate the solutionof the general ODE. Consider starting with a very simple initial guess for whatthe unknown function is; let's take it to be the constant function yo(t) = ß thatat least satisfies the initial condition. Plugging it into the right hand side ofthe integral formula, we get a new (hopefully improved) approximation; call ityı (t):3₁(t) = ß + [ * ƒ(s, yo(s)) ds = ß+ [ f(s, ß) ds.Because the integrand is now an explicit function of s, you can attempt toevaluate the integral via Calc II methods.Consider applying this idea iteratively: At the n-th stage, having obtainedthe n-th approximation y(t), you plug it into the right hand side to generatea new (hopefully improved) approximation yn+1(t):Yn+1(t)=B+ -[f(s, yn (s)) ds. Again, because you will have already determined a formula for yn (t) at thisstage, the integrand will be an explicit function of s that you can attempt tointegrate via Calc II methods.Apply this iterative procedure to the following ODE:y = yy(0) = 1.Starting the procedure with yo(t) = 1, find explicit formulas for y₁ (t), y2(t),..., y4 (t).Then, observing the pattern, determine the formula for yn (t). Plot y₁ (t),..., y4 (t)along with the exact solution of the initial value problem (find it) on the sameaxis for t€ [0, 1]. Do you recognize yn (t) as something you have seen before,perhaps in Calc I or II? Based on this, explain why you think the approximatesolutions yn (t) are converging, or not, to the true solution of the ODE.

Answered: Image /qna-images/answer/a3ee0318-2066-4305-bed9-aa66a75afc0e.jpg

y'(t) = f(t, y(t)) in Lecture 1, together with the initial condition y(a) = f = f(t) depends only on the independent variable t, you get the simplest type of ODE that can be solved via Calc II: FTC implies = ß, we noted that if

y'(t) = f(t, y(t)) in Lecture 1, together with the initial condition y(a) = f = f(t) depends only on the independent variable t, you get the simplest type of ODE that can be solved via Calc II: FTC implies = ß, we noted that if

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

Similar questions

the second picture is part of the first picture, i could not fit it all in one screenshot.
could you write this on paper so its easier to understand?
Name the image of overline DE
i need worked out solution
Could you also look at the second page ?