Question

Consider the initial value problem y' +y=e-, with y(0) = 0. PROJECT 1.) Find the exact solution to this equation, say 0(x). 2.) Use MATLAB to plot 6(x) in the interval [0.0, 4.0] . Use sufficient points to obtain a smooth curve. 3.) Now create a MATLAB program that uses Euler's Method to approximate the values of $(2) at N = 10 equally spaced points in (0,4). Plot these points on the same plot that was generated in part 2. Label the axes and title your plot with the number of points used. (See hints below). Plot the points and a dashed line connecting them. Your plot should like like Figure 2.7.3 in the textbook. 4.) Generate 2 more similar plots with N = 20 and N = 40. Use the same window for all three plots. 5.) Print out your 3 plots and the code that you executed to generate the Euler approximations. 6.) For the case N = 10 (and only the case N = 10) print out a table of values comparing Euler's approximate values to the values of the exact solution. This table should be set up to look like table 2.7.2 in the textbook (first 3 columns only)

Answer 1

y'+y=e-- +y=e-1

comparing with standard equation

4% + p(x)y= Q(x)

here p(x)=1

Q(x)=$e^{-x}$

now integrating factor I=

eſ p(x)dr p(x) dx = 2 I = p

Now we are multiplying $e^{x}$ both side of equation

dy te e * +e" *y = 1

dle *y) dr

integrating both side

*y = r +C y=I* -* +C* e-

so

f(x)=

c*e +cie

Matlab Code:

for i=1:40
x(i)=(1/10)*(i-1);
y(i)=exp(-x(i))*(x(i)+1);
end

plot(x,y)

0 0.5 1 1 .5 2 2.5 3 3.5

3. f=@(x) x*exp(-x)*1*exp(-x);
lv=0; // lowe value
uv=4; // upper value
invalue=1; // initial value
n=10; // value of n
YY=euler(f,lv,uv,invalue,n)
plot(YY(:,1),YY(:,2))

function E=euler(f,llimit,hlimit,ya,M)
h=(hlimit-llimit)/M;
Y=zeros(1,M+1);
T=llimit:h:hlimit;
Y(1)=ya;
for j=1:M
Y(j+1)=Y(j)+h*f(T(j));
end
E=[T' Y'];
end

4 and 5. ns=[10,20,40];
for i=1:3
f=@(x) x*exp(-x)*1*exp(-x);
lv=0;
uv=4;
invalue=1;
n=ns(i);
YY=euler(f,lv,uv,invalue,n)
plot(YY(:,1),YY(:,2))
hold on
  
end

6. f=@(x) x*exp(-x)*1*exp(-x);
lv=0;
uv=4;
invalue=1;
n=10;
YY=euler(f,lv,uv,invalue,n)
plot(YY(:,1),YY(:,2))
disp(YY)