Question

= 2. We wish to 7. Consider the differential equation y' + y = 2.. with the initial conditions y(0) approximate y(1). (a) Set up Euler's method by hand and use it to approximate y(1) with a step size of h = 25. (You may use only a calculator for this question) (b) Use Mathematica to do Euler's method with the step sizes of h= .1,.01, .001, and .0001.

Answer 1

ANSSWER :

Consider the differential equation

$y'+y=\frac{1}{2-x}$ with the inital conditions y(0) =2 we wish to approximate y(1).

Didfferntial equation

$\frac{dy}{dx}+y=\frac{1}{2-x}$

1 dy dr = -Y + - 2-T

y(0) = 2

y(1) = 2, L = 0.25,20 = 0

Apply Euler's Method

f(x,y) = -y + (2-0)

$y_{1}=y_{0}+h(x_{0},y_{0})$

$\Rightarrow y_{1}=2+(0.25)\left ( -2+\frac{1}{2-0} \right )$

$\Rightarrow y_{1}=2+(0.25)\left ( \frac{-4+1}{2} \right )$

$\Rightarrow y_{1}=1.625$

$y_{2}=y_{1}+h(x_{1},y_{1})$

$\Rightarrow y_{2}=1.625+(0.25)\left ( -1.625+\frac{1}{2-0.25} \right )$

$\Rightarrow y_{2}=1.625+(0.25)\left ( -1.625+\frac{1}{1.75} \right )$

y2 = 1.361607

$y_{3}=y(0.75)+h(x_{2},y_{2})$

$\Rightarrow y_{3}(0.75)=1.361607+(0.25)\left ( -1.361602+\frac{1}{2-0.5} \right )$

$=1.1878719$

$\Rightarrow y_{4} (1)=y_{3}+h(x_{3},y_{3})$

$=1.1878719+(0.25)\left ( -1.1878719+\frac{1}{2-0.75} \right )$

$y_{1}=1.090903925$

b)

$h=0.1,0.01,0.001,0.0001$

$\Rightarrow y_1=y_0+h(x_0,y_0)$

$\Rightarrow y_1=2+(0.1)\left(-2+\frac{1}{2-0}\right)$

$\Rightarrow y_1=2+\frac{(0.1)(-4+1)}{2}=1.85$

$h=0.01$

$\Rightarrow y_1=y_0+h(x_0,y_0)$

$\Rightarrow y_1=2+(0.01)\left(-2+\frac{1}{2-0}\right)$

$\Rightarrow y_1=2+(0.01)(-1.5)$

$\Rightarrow y_1=2-0.015$

$\Rightarrow y_1=1.985$

$h=0.001$

$\Rightarrow y_1=y_0+h(x_0,y_0)$

$\Rightarrow y_1=2+(0.001)\left(-2+\frac{1}{2-0}\right)$

$\Rightarrow y_1=1.9985$

$h=0.0001$

$\Rightarrow y_1=y_0+h(x_0,y_0)$

$\Rightarrow y_1=2+(0.0001) \left(-2+\frac{1}{2-0}\right)$

$\Rightarrow y_1=1.99985$