solution ODE using IFFT

Hello,

I want to solve one ordinary differential equation using inverse Fourier transform.

But when I compare with ode45 solver results are very bad,

would you please help me?

my equation is:

my code is:

clc
clear all
t=[0:1e-3:30];
ff=sin(t);
Fs=1/t(2);
Y = fft(ff);
NFFT=length(Y);
f1 = Fs/2*linspace(0,1,NFFT);
YY=Y;
for ii=1:length(f1);
    f=f1(ii);
    w=2*pi*f;
    x(ii)=(-w^2+i*w+1)^-1*YY(ii);
end
figure(2)
plot(t,real(ifft(x)))

[Yu Jiang]

In this line

x(ii)=(-w^2+i*w+1)^-1*YY(ii);

why do you have i*w, not w?