Question

Problem 3. Consider the following second-order linear differential equation with the given initial conditions: I day = 6 x 10-6(x – 100) dx2 Initial Conditions at x = 0: y = 0 and dy dx = 0 Determine y at x =100, with a step size of 50 using: a) Euler's method, b) Heun's method with one correction.

Answer 1

Problém - 3 Consider the following following seconde linear difterential equation the given initial conditions. order with I= dzy = 6x 10-6(x-100) Initical conditions at x =0 and dy a Euler's Method.. yltay = 2-0-44 y(0)=1 with a stopsige use Euler's Method of h = onl to find appro. vanes Solution at of = 0.1,0.2, 0.3, 0.4 ando. 5 compare them to the exact Values

ام ريه This is a fairly Sirople linear ciften eretical cavacion leave it to you to chock the solution is y (4) In order to use Euler's Method we first need to rewrite the ditterential equation equation into the form guen in co. y = 2-e 4+ -zy From this we can see that flity 2-e-4t-ey to = 0 and yo=1 Also niste theat -261 = 1 fo=f(ai) = 2-e-460) y = yothto = 1+ Cadeiro = 0.9

PMKVY So, the appout to the solution at t = 0.1 is y, 6.900 next stop we have al fi fo.1,0.9) = 2-6410-13 -2000 --O470320046 y = yit hit, 20.94 Los (-0.4702 0.852967995 1 ... the approy to the solution at t = = 0.2 isy2 = 0.8529 f = -6.155264954, 9 = 0.83474415 fz0.023922788 - Yu = 0.839833 6.1184359245 .85=0.85180 ty

Here's we quick table given the app dox as well as the exact Value of the solution at given points - Timetn Approximation - Exact Error y = 0.4 #10) d2=0.2 t3-03 ty=0.4 ty=0.5 Yo=1 y (o)=1 0% y (0-1)=0.925 /2-79% 20-852967995 1 (0.2=0.88452-16% Yo 0.837441500 1% (0-3)=0.870 4.42% 50.839833779 ylo.4=0.8762 4.16% =0.85167737) Ys y (6.5=0.883 3.63% Percent Exact cepprox/xioo Excet