ODE for frequency response

I would like to model above ODE and see the frequency response for w being frequency.

Can anyone please give me an advice? I am not sure this is even possible to solve..

Answer 1

[Star Strider]

This takes a while to run, so I used relatively coarse mesh intervals (for ‘tspan’ and ‘omega’) to run it here.

Experiment with it —

syms x(t) omega t x0 Y
sympref('AbbreviateOutput',false);
Dx = diff(x);
Eqn = 2*Dx*exp(1i*omega*t) + x*(1i*omega)^2 + 1000*x*exp(1i*omega*t) == exp(1i*10*t)

Eqn(t) = 

X = dsolve(Eqn, x(0)==x0)

X = 

X = subs(X,{x0},{0})

X = 

X(omega,t) = simplify(X,500)

X(omega, t) = 

[VF,Subs] = odeToVectorField(Eqn)

VF = 

Subs = 

Xfcn = matlabFunction(VF, 'Vars',{t,Y,omega})

Xfcn = function_handle with value:
    @(t,Y,omega)[(exp(omega.*t.*-1i).*(exp(t.*1.0e+1i)-exp(omega.*t.*1i).*Y(1).*1.0e+3+omega.^2.*Y(1)))./2.0]

tspan = linspace(0, 15, 11);
omegav = linspace(0, 600, 7);
for k = 1:numel(omegav)
    [t,x] = ode45(@(t,x)Xfcn(t,x,omegav(k)), tspan, 0);
    tv{k} = t;
    xv{k} = x;
end

tvm = cell2mat(tv);
xvm = cell2mat(xv);
omegavm = repmat(omegav,11,1);

figure
surf(tvm, omegavm, abs(xvm), 'FaceAlpha',0.5)
xlabel('Time')
ylabel('Frequency')
zlabel('|X(\omega,t)|')
colormap(turbo)
set(gca,'ZScale','log')

.