Question

Need help. Only intelligent people. Anyone good with Matlab??

Answer 1

what type of math

Answer 2

Consider the initial value problem
dy
dt = cos(t)y(t) + cos(t)
y(0) = 1
a.) Give the solution to the initial value problem.
b.) Find a time step h that would evenly divide the interval [0; 2] into N subintervals
and guarantee a local truncation error of at most 0.01 for the Euler method on the
same interval. Use this time step to apply the Euler method and the Improved
Euler method to this initial value problem over the interval [0; 2] and plot the
actual solution along with the two approximated solutions.

Answer 3

lolol

Answer 4

math like this is horrible and not really useful...

Answer 5

Intervals are [0. 2*pi]

Answer 6

is it the first ODE?

Answer 7

yes

Answer 8

Well...
\[\Large \frac{dy}{dt} = \cos(t)y(t) + \cos(t)\] right?

Answer 9

In that case, this equation is separable:
\[\Large \frac{\color{blue}{dy}}{\color{red}{dt}}=(y+1)\cos(t)\]

Answer 10

\[\Large \frac{\color{blue}{dy}}{y+1}= \cos(t) \color{red}{dt}\]and then integrate both sides:

Answer 11

the question looking for a numerical method

Answer 12

the forward euler scheme
for
\[\large \frac{dy}{dx}=f(x,y),\qquad y(0)=y_0\]is
\[
\large\hat y_{j+1}=\hat y_j+hf(x_j,\hat y_j)\]
where the step size is \( h\)

Answer 13

however the computer will prefer it if you set the initial value first
\[\hat y_1=y_0\]
then
```
for j=2:N
```

\[\large \large\hat y_{j}=\hat y_{j-1}+h f(x_{j-1},\hat y_{j-1})\]
```
end
```

Answer 14

the forward euler scheme
for
\[\large \frac{dy}{dx}=f(x,y),\qquad y(0)=y_0\]is
\[
\large\hat y_{j+1}=\hat y_j+hf(x_j,\hat y_j)\]
where the step size is \( h\)

Answer 15

```
function [] = DE (h)
%% Parameters %
t0 = 0; % min
tM = 2; % max
N ; % number of subintervals %% this should be a function of h,t,tMax %
t ; % domain %% this should be a function t0,tMax,N %
y0 = 1; % initial value
%% Exact Values %
y ; %% find the exact value using separation of variables, as terenzreignz started %
%% Approx Values %
yHat = zeros(1,N); % empty yHat
yHat(1) = y0; % yHat(0)
for j=2:N
yHat(j) ; %% = yHat +h.*f(x_j-1,yHat_j-1) %
end
%% Graph %
figure(2); clf; grid
hold on
plot(t, y, 'k')
plot(t, yHat,'b')
hold off
clear
```