Question

Consider the following equation: dy/dt = −20y+7e⁻⁰∙⁵ Given that the value of y=5 when t=0, a) Use the Euler's method to find y for t=0 to 0.1 using step size of Δt=0.01,0.001 and 0.0001 . Write a MATLAB code to do the calculations and show the results as a plot.

Answer 1

Sure! Here's a MATLAB code that implements Euler's method to solve the given differential equation and plot the results for different step sizes.

```matlab

% Parameters

t0 = 0; % initial time

y0 = 5; % initial value of y

h_vals = [0.01, 0.001, 0.0001]; % step sizes

% Solve the differential equation using Euler's method

for h = h_vals

t = t0:h:0.1; % time points

y = zeros(size(t)); % initialize y values

y(1) = y0; % initial condition

% Euler's method

for i = 1:length(t)-1

dydt = -20*y(i) + 7*exp(-0.5);

y(i+1) = y(i) + h*dydt;

end

% Plot the results

plot(t, y, 'DisplayName', sprintf('\\Delta t = %.4f', h)); hold on

end

% Add labels and legend

xlabel('t')

ylabel('y')

legend('Location', 'best')

```

This code will generate a plot showing the results for step sizes of 0.01, 0.001, and 0.0001.

Note: In the code, the term `exp(-0.5)` is used instead of `e^(-0.5)` to calculate the exponential function.