2nd order ODE with time dependent parameters

I wanna solve a ODE question with matlab The question is

θ''(t) + 2θ'(t) + w* sin θ(t) = τ (t) , τ (t) = 5(1 − exp(−5t))

θ(0) = −π/6 and θ'(0) = 0

w = 10 when 0 ≤ t ≤ 5 or 8 otherwise

t = [0 10]

I wrote the code refer to examples in matlab ode45

ft = linspace(0,10,49); % w = f

for ft = 0:5

f = 10;

end

for ft = 5.2083:10 % w change of time range

f = 8;

end

gt = linspace(0,10,49);

g = 5.*(1-exp(-5.*gt)); % g = τ (t)

tspan = [0 10];

y0 = [-pi/6 0];

[t,y] = ode45(@(t,y) myode(t,y,ft,f,gt,g), tspan, y0);

plot(t,y)

function dydt = myode(t,y,ft,f,gt,g)

dydt = zeros(2,1);

dydt(1) = y(2);

dydt(2) = -2.*y(2)- f.*sin(y(1)) + g

f = interp1(ft,f,t); % Interpolate the data set (ft,f) at time t

g = interp1(gt,g,t); % Interpolate the data set (gt,g) at time t\

end

I got errors like this

numbers of left hand side and right hand side properties are different

오류 발생: untitled2>myode (33번 라인)

dydt(2) = -2.*y(2)- f.*sin(y(1)) + g

오류 발생: untitled2>@(t,y)myode(t,y,ft,f,gt,g) (20번 라인)

[t,y] = ode45(@(t,y) myode(t,y,ft,f,gt,g), tspan, y0);

오류 발생: odearguments (90번 라인)

f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.

오류 발생: ode45 (106번 라인)

odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

오류 발생: untitled2 (20번 라인)

[t,y] = ode45(@(t,y) myode(t,y,ft,f,gt,g), tspan, y0);

Answer 1

[Pratyush Roy]

Hi 민석 김,

While defining the myode function, one can find the interpolated function first and then compute the first and second order derivatives. The following code snippet might be helpful:

ft = linspace(0,10,49);   % w = f
f = zeros(1,49);

for idx = 1:49
    if ft(idx)<5.2083
        f(idx) = 10;
    else
        f(idx) = 8;
    end        
end

gt = linspace(0,10,49);
g = 5.*(1-exp(-5.*gt)); % g = τ (t)
tspan = [0 10];
y0 = [-pi/6 0];
[t,y] = ode45(@(t,y) myode(t,y,ft,f,gt,g), tspan, y0);
plot(t,y)
function dydt = myode(t,y,ft,f,gt,g)
dydt = zeros(2,1);
f = interp1(ft,f,t); % Interpolate the data set (ft,f) at time t
g = interp1(gt,g,t); % Interpolate the data set (gt,g) at time t
dydt(1) = y(2);
dydt(2) = -2.*y(2)- f.*sin(y(1)) + g;
end

Hope this helps!