Question

Given the initial-value problem

y'=2-2tyt2+1,          0 ≤t≤1,        y0=1

With exact solution

yt= 2t+1t2+1

Using MATLAB use Euler’s method with h = 0.1 to approximate the solution of y

Answer 1

i use 'x' in replace of 't'

clear all

clc

f=@(x,y)(2-2.*x.*y)./(x.^2+1); %Write your f(x,y) function, where dy/dx=f(x,y), x(x0)=y0.

x0=input('\n Enter initial value of x i.e. x0: '); %example x0=0

y0=input('\n Enter initial value of y i.e. y0: '); %example y0=0.5

xn=input('\n Enter the final value of x: ');% where we need to find the value of y

%example x=2

h=input('\n Enter the step length h: '); %example h=0.2

%Formula: y1=y0+h/2*[f(x0,y0)+f(x1,y1*)] where y1*=y0+h*f(x0,y0);

fprintf('\n x y ');

while x0<=xn

fprintf('\n%4.3f %4.3f ',x0,y0);%values of x and y

k=y0+h*f(x0,y0);

x1=x0+h;

y1=y0+h/2*(f(x0,y0)+f(x1,k));

x0=x1;

y0=y1;

end

1) 86% (4) Sat 2:25 PM Q E G do 5 C O Q Search Documentation Log In 3 D e Run Section 6 RUN Workspace Name Í MATLAB Window Help MATLAB R2017b HOME PLOTS APPS EDITOR PUBLISH VIEW + G H Find Files Insert E fx 1 - Compare , Go To - Comment % %%% New Open Save Breakpoints Run Run and KL Advance Run and Print Find - Indent - - Advance Time FILE NAVIGATE EDIT BREAKPOINTS a / Users shubhamkamboj Documents MATLAB Editor - /Users/shubhamkamboj/Documents/MATLAB/HomeworkLib123.m 1 eular.m x HomeworkLib123. m x + | clear all clc f=@(x,y) (2-2.**.*y)./(x.^2+1); %Write your f(x,y) function, where dy/dx=f(x,y), x(x)=yo. x0=input('\n Enter initial value of x i.e. XO: '); %example x0=0 yo=input('\n Enter initial value of y i.e. yo: '); $example yo=0.5 xn=input('\n Enter the final value of x: ');% where we need to find the value of y example x=2 h=input('in Enter the step length h: '); %sexample h=0.2 *Formula: yl=y0+h/2*[f(x0,70)+f(x1, yl*)] where yl*=y0+h*f(x,y); fprintf('\n x y '); while x <=xn fprintf('\n%4.3f $4.3f ',x0,70); values of x and y k=y+h*f(x0, y0); x1=x0+h; y1=y0+h/2*(f(x0, y ) +f(x1,k)); XO=x1; yo=y1; end BBBBBB Value @(x,y)(2-2.*x.*y).... 0.1000 1.4486 1.1000 1.1000 1.4467 1.4467

-sexdlice X-2 Command Window Enter initial value of x i.e. x0: 0 Enter initial value of y i.e. yo: 1 Enter the final value of x: 1 Enter the step length h: 0.1 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 fx 1.000 1.000 1.187 1.344 1.466 1.549 1.597 1.615 1.608 1.583 1.545 1.498 >>