Diference between numerical and analitical sollution (Matlab)

[mat_ziv]

Hello All,

I am new on this forum.

I am trying to solve the following first order ODE:

dy/dt=(w*i-const_G)*y + 2*exp(w*t*i)

Analitacly, the result should have a form Y*exp(w*t*i). When I put it in to the equation I am getting:

Y*exp(w*t*i)*w*i=(w*i-const_G)*Y*exp(w*t*i) + 2*exp(w*t*i)

Now deviding with exp(w*t*i), I am getting:

Y*w*i=(w*i-const_G)*Y + 2

Now Y is:

Y=2/const_G

The solution is y(t)=2/const_G * exp(w*t*i). Which meens that the amplitude is reduced by const_G, the frequency is the same.

Using MATLAB to solve this equation and to get a graph, I have become confused.

This is the function in MATLAB:
------------------------------------------------------------------------------------
function result = mode_decay(inputs)
time=inputs(1);
frequency = inputs(2);
const_G=inputs(3);
y_re=inputs(4);
y_im=inputs(5);

y=y_re+y_im*1i;

input = 2*exp(2*pi*frequency*time*1i);
dydt=(2*pi*frequency*1i-const_G)*y + input;

result = [input; dydt];

end
------------------------------------------------------------------------------------
On the image you can see the simulation and resulting graph.



I am confised, how am I getting this amplitude rising graph. From where it comes.

Any advice will be very helpful for me.

Thanks in advance

 

[HallsofIvy]

First, the word is "analytically".

You have the differential equation dy/dt=(w*i-const_G)*y + 2*exp(w*t*i)
The associated homogenous equation is dy/dt= (wi- const_G)y so dy/y= (wi- const_G)dt. Integrating \(\displaystyle ln(y)= (wi- const_G)t+ C\). Taking the exponential of both sides, \(\displaystyle y(t)= C'exp(wi- const_G)t)\) where \(\displaystyle C'= e^C\) is still an undetermined constant.

For a solution to the entire problem, since the "non-homogeneous" part, 2exp(wit), is an exponential, we try y(t)= Aexp(wit). Then y'= Awi exp(wit). The equation becomes Awi exp(wit)= (wi- const_G)Aexp(wit)- 2exp(wit). We can divide through by exp(wit) leaving Awi= (wi- const_G)A- 2. const_G A= -2 so A= -2/const_G.

y(t)= C'exp(wi- const_G)t)- (2/const_G)exp(wit).

That does not appear to be what you got for your analytical solution.