Question

Write a project on Numerical Solutions of First Order Linear Differential Equations with Applications Modified Euler's Method, Runge-Kutta Method Applications: a) Solving linear first-order equations numerically. (Students may write code to accompany their project.) [mechanical, Electromechanical, chemical, electronics] The purpose of your project is for you to explore a topic in mathematics where calculus has substantial use. You will also explore the application of the topic to an applied context related to the Electronics Engineering program.

Answer 1

Answer:

Let us consider a first-order differential equation y'a dy - f(y) with initial condition y (wo) - Yo da To solve equo numorically a sequence of approximation to the value of exact solution will be generated at the points У), , , , 21, X253, where, where , nith anth m = 0, 1,-- h = real number. Using Euleris method. at the points written as, the affroximate solution of ego (M=0,1,2,...) nh can be Ant -xot entl m = 0, 1, 2, .. y = y + hf (xm, Yom) mt! 2

0 Modified Euler Method: The modified Euler method is also called the Heun's method. Hores by using predictor-corrector method, modify Eulez's method of approximate solution we will by We will modify the approximate solution given by 29" @ applying Euler's method once more to eqh If we take the average of the slops from point (xm, Ym) to (sants you through the curre, Smits = yon + z' " ! f (xem. You) + f (this is MH s [<1,4(x1)] atx=21 modified Euler affrox. {x,y) Exael solution of Euler approx.at x=2, Cai, yok) (x0,90) P h/ >x 20 not h/ xi Figil. Modified Euler method for met So, from Figit. modified Euler approximations are coming from first stepping to the point P ( xesthes you + bf (x0.9))

So, alo ng the tangent line to the solution carve through cxm, yo). the modified Euler method for afproximate solution of eq" 0 be expressed as, your = Ym toth [F(xnYm) + f (Ponte en ;)] can at Enti = xo +mh (M=0,1,2,...) DUE where, , m=0,1,2,.. Imth f (xn, yon) Youth 3 0 Runge- Kulta Method : second onder Runge-Keita method 3 4. From og "@ Euler's method gived, y = yot h. f (xo, Yo) and from egy ☺ Euler's modified method says, % = %+h[ f(x0,9.)+ f(x,y,*) 7 If we replace y,* logged by using eq". 9,2 % +h[f (*0.7.)+ f (41, 4o +bf(x,y))] it hof (246, Yo) = ki and hof (21,90 + ki) = K2 L 5 b

then, eg" becomes, L 66 y = yot [kit kz] In general forama у In + 2 [kit ka k= 12 (Xm, 9m) K2 = hf (BCmpia Yotki) where m = 0, 1, 2,.. mul and 22 where n= 0, 1, 2, -- Third onder- Runge- Kutta method) : Thind order Runge – Kutta method in general form is givenby, manus = Yn tot [*1 +4K4+ ks] ki= h flens yn) kq = hf (cath/2, Youth bf [amth, Yn+hf (kmati 'm+ and ز yout ki ) 7 Kz Fourth order Runge Kutta method Do In ti = Yn + ó[ k+2K2+ 2kg + ka ) at K041526 + 20h where m= 0, 1, 2, and, Ki = hf (Xove Yom ) K2 = hf (unt 1/2 " Ynt kyz) K3 = hf (xn+b/2 , yat k2/2) K4 hf (kn+1 9 Intl Kg