consider the initial value problem 𝑦’ = 𝑡 2 + 𝑐𝑜𝑠 𝑦, 𝑦(0) = 0. use matlabto plot the solution for 0 ≤ t ≤ 1 and find the approximate value of y(1). in matlab

To solve the initial value problem 𝑦’ = 𝑡^2 + 𝑐𝑜𝑠(𝑦), 𝑦(0) = 0 and plot the solution for 0 ≤ 𝑡 ≤ 1 in MATLAB, we can use numerical methods.

Here's an example of how you can do it using the built-in "ode45" function in MATLAB:

main.m
% Define the differential equation
dydt = @(t, y) t^2 + cos(y);

% Define the initial condition
initialCondition = 0;

% Define the time span
tSpan = [0, 1];

% Solve the differential equation
[t, y] = ode45(dydt, tSpan, initialCondition);

% Plot the solution
plot(t, y);
xlabel('t');
ylabel('y');
title('Solution of the Initial Value Problem');
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This code defines the differential equation using an anonymous function, specifies the initial condition and time span, uses the "ode45" solver to obtain the numerical solution, and then plots the result using the "plot" function.

To find the approximate value of y(1), you can simply evaluate the solution at t = 1:

main.m
approximateValue = y(end);
disp(approximateValue);
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This will display the approximate value of y(1).

Keep in mind that the numerical solution obtained by using "ode45" is an approximation, so the approximate value of y(1) will also be an approximation.

Make sure to replace "𝑐𝑜𝑠" with "cos" in the code, as MATLAB uses "cos" for the cosine function.

Note that the accuracy of the solution will depend on the properties of the differential equation, such as the smoothness of the function and the size of the time steps used in the numerical solver.