Trigonometric functions using dsolve

This code works well. How can I use sin(theta2) instead of theta2. Its giving me error.

syms theta1(t) theta2(t) theta3(t)
ode1= diff(theta1)== 1+theta2-theta1;
ode2= diff(theta2)==1+ theta3-theta2;
ode3= diff(theta3)== theta2-theta3;
odes=[ode1;ode2; ode3];
S=dsolve(odes);
theta1Sol(t) = S.theta1;
theta2Sol(t)=S.theta2;
theta3Sol(t)=S.theta3;
cond1 = theta1(0) == 0.1;
cond2 = theta2(0) == 0.3;
cond3 = theta3(0) == 0.2;
conds = [cond1; cond2; cond3];
[theta1Sol(t),theta2Sol(t), theta3Sol] = dsolve(odes,conds)

theta1Sol(t) = 

theta2Sol(t) = 

theta3Sol = 

fplot(theta1Sol)
hold on
fplot(theta2Sol)
grid on
fplot(theta3Sol)
grid on
legend('theta1Sol','theta2Sol','theta2Sol','Location','best')

[Walter Roberson]

When you say to use sin(theta2) instead of theta2 are you referring to all three places where theta2 appears in

ode1= diff(theta1)== 1+theta2-theta1;
ode2= diff(theta2)==1+ theta3-theta2;
ode3= diff(theta3)== theta2-theta3;

??

And would

cond2 = theta2(0) == 0.3;

change to

cond2 = sin(theta2(0)) == 0.3;

??

[Urmish Hemant Haribhakti Parikh]

ode1= diff(theta1)== 1+sin(theta2)-theta1;

Like this and the conditions remaining same.

[Torsten]

Before, your system of ODEs was linear - easy to solve.

Now it's nonlinear and an analytical solution with dsolve out of reach.

[Urmish Hemant Haribhakti Parikh]

How do I calculate a numerical solution with this?

[Urmish Hemant Haribhakti Parikh]

If I use vpasolve getting the same error

[Urmish Hemant Haribhakti Parikh]

syms theta1(t) theta2(t) theta3(t)

ode1= diff(theta1)== 1+sin(theta2)-theta1;

ode2= diff(theta2)==1+ theta2-theta2;

ode3= diff(theta3)== theta2-theta3;

odes=[ode1;ode2; ode3];

S=vpasolve(odes);

theta1Sol(t) = S.theta1;

theta2Sol(t) = S.theta2;

theta3Sol(t) = S.theta3;

cond1 = theta1(0) == 0.1;

cond2 = theta2(0) == 0.3;

cond3 = theta3(0) == 0.2;

conds = [cond1; cond2; cond3];

[theta1Sol(t),theta2Sol(t), theta3Sol] = dsolve(odes,conds);

fplot(theta1Sol)

hold on

fplot(theta2Sol)

grid on

fplot(theta3Sol)

grid on

legend('theta1Sol','theta2Sol','theta2Sol','Location','best')

[Urmish Hemant Haribhakti Parikh]

How do I solve it using ode45 ? having a hard time

Answer 1

[Walter Roberson]

syms theta_1(t) sin_theta_2(t) theta_3(t)
ode1 = diff(theta_1) == 1 + sin_theta_2 - theta_1;
ode2 = diff(sin_theta_2) == 1 + theta_3 - sin_theta_2;
ode3 = diff(theta_3) == sin_theta_2 - theta_3;
odes = [ode1; ode2; ode3];
Snc = dsolve(odes)

Snc = struct with fields:
    sin_theta_2: C1 + t/2 - exp(-2*t)*(C2 - exp(2*t)/4)
        theta_1: C1 + t/2 + exp(-t)*(C3 + exp(t)) + exp(-2*t)*(C2 - exp(2*t)/4)
        theta_3: C1 + t/2 + exp(-2*t)*(C2 - exp(2*t)/4)

theta_1_Sol_nocond(t) = simplify(Snc.theta_1)

theta_1_Sol_nocond(t) = 

theta_2_Sol_nocond(t) = simplify(asin(Snc.sin_theta_2))

theta_2_Sol_nocond(t) = 

theta_3_Sol_nocond(t) = simplify(Snc.theta_3)

theta_3_Sol_nocond(t) = 

cond1 = theta_1(0) == 0.1;
cond2 = sin_theta_2(0) == 0.3;
cond3 = theta_3(0) == 0.2;
conds = [cond1; cond2; cond3];
Sc = dsolve(odes, conds);

theta_1_Sol_cond(t) = simplify(Sc.theta_1)

theta_1_Sol_cond(t) = 

theta_2_Sol_cond(t) = simplify(asin(Sc.sin_theta_2))

theta_2_Sol_cond(t) = 

theta_3_Sol_cond(t) = simplify(Sc.theta_3)

theta_3_Sol_cond(t) = 

fplot(theta_1_Sol_cond)
hold on
fplot(theta_2_Sol_cond)
grid on
fplot(theta_3_Sol_cond)
grid on
legend('theta1Sol','theta2Sol','theta2Sol','Location','best')

figure
lowerbound = solve(Sc.sin_theta_2 == -1)

lowerbound = 

upperbound = solve(Sc.sin_theta_2 == 1)

upperbound = 

bounds = double([lowerbound, upperbound])

bounds = 1×2

-0.8426 1.0491

fplot(theta_1_Sol_cond, bounds)
hold on
fplot(theta_2_Sol_cond, bounds)
grid on
fplot(theta_3_Sol_cond, bounds)
grid on
legend('theta1Sol','theta2Sol','theta2Sol','Location','best')

[Torsten]

But this is not the solution to

ode1= diff(theta1)== 1+sin(theta2)-theta1;
ode2= diff(theta2)==1+ theta3-theta2;
ode3= diff(theta3)== theta2-theta3;
cond1 = theta1(0) == 0.1;
cond2 = theta2(0) == 0.3;
cond3 = theta3(0) == 0.2;

[Walter Roberson]

At the time I was composing this, there had not been a reply indicating where the sin() was intended to go.

Answer 2

[Torsten]

fun = @(t,theta) [1+sin(theta(2))-theta(1);1+theta(3)-theta(2);theta(2)-theta(3)];
theta0 = [0.1, 0.3, 0.2];
tspan = [0,2*pi]
[t,theta] = ode45(fun,tspan,theta0)
plot(t,theta)