How do I graph an ordinary differential equation with an initial condition?

I'm trying to solve a problem for a matlab problem set homework, and I was having issues getting the code below to work. The question in the book is "Plot the solution satisfying the initial condition y(2) = 1.5". Thank you!

syms y(t)
cond = y(2)== 1.5; %describes initial condition
ode = exp(y) + (( t*exp(y)-sin(y))* diff(y, t)) == 0; % the equation
ySol = dsolve(ode, cond)
figure
ezplot(ySol)

Answer 1

[Ameer Hamza]

dsolve() shows that the ODE might not have a symbolic solution. So you will need to resort to a numerical solution. Following shows how it can be done in MATLAB

syms y(t)
ode = exp(y) + (( t*exp(y)-sin(y))* diff(y, t)) == 0; % the equation
V = odeToVectorField(ode);
odefun = matlabFunction(V, 'vars', {'t', 'Y'});
IC = 1.5;
tspan = [2 10]; % interval in which solution is needed
[t, y] = ode45(odefun, tspan, IC);

f = figure();
ax = axes();
plot(t, y);

This graph follows the initial condition.

[Ramadan M]

Thank you! I understand this a lot more now!

Also, would you happen to know how to find y(some specified t) with code, and mark it on the graph? It's the follow up question to this one.

ex. y(1), mark this point on the graph.

I made a few attempts at it, but the textbook that I'm working out of seems to be fairly outdated.

[Ameer Hamza]

See the following code

syms y(t)
ode = exp(y) + (( t*exp(y)-sin(y))* diff(y, t)) == 0; % the equation
V = odeToVectorField(ode);
odefun = matlabFunction(V, 'vars', {'t', 'Y'});
IC = 1.5;
tspan = [2 10]; % interval in which solution is needed
sol = ode45(odefun, tspan, IC);

f = figure();
ax = axes();
hold(ax);
plot(sol.x, sol.y);
plot(5, deval(sol, 5), '*', 'MarkerSize', 8, 'LineWidth', 2);