How to write a code for finding a definite integral for ( 1/x(t).^2 ) please?

My system is :

dx/dt = 2*x

1- I solved numerically using ode45.

2- Depending on the values that I got from ode45, I want to find this integral :

Integral ( 1/ x(t).^2 ) dt from 0 to 10

This is my code but I do not know if my result is correct or not,, how could I check that please?

function RunAvan

x0=0.1;

tspan=[0:0.1:10];

A=1;

[t,x]=ode45(@Avan,tspan,x0);

figure(1)

plot(t,x)

f = @(x) (A./x.^2)

Q1 = quad(f,1,20)

Q2 = quad(f,1,40)

Q3 = quad(f,1,60)

Q4 = quad(f,1,80)

Q5 = quad(f,1,101)

% I5=integral(f,1,101,'ArrayValued',true)

Q=[Q1./20 Q2./40 Q3./60 Q4./80 Q5./101]

B=[20 40 60 80 101]

figure(2)

plot(B,Q)

X = @(T) interp1(t,x,T);

f1 = @(T) (A./X(T).^2)

J1=integral(f1,0,2,'ArrayValued',true)

J2=integral(f1,0,4,'ArrayValued',true)

J3=integral(f1,0,6,'ArrayValued',true)

J4=integral(f1,0,8,'ArrayValued',true)

J5=integral(f1,0,10,'ArrayValued',true)

J=[J1./2 J2./4 J3./6 J4./8 J5./10]

R=[2 4 6 8 10]

figure(4)

plot(R,J)

1;

Note : The first time I integrated with respect to x and the second time with respect to t.

[Torsten]

Are the functions involved only test functions ?

Or why do you use ode45, quad and integral instead of the analytical solutions in all cases ?

Best wishes

Torsten.

Answer 1

[Roger Stafford]

You can check it by comparing its results with the exact mathematical solution to your differential equation. The solution to dx/dt = 2*x is

   x = K*exp(2*t)

where K can be determined by the initial condition. In your case that would be K = 0.1 . Taking the integral of 1/x^2 from 0 to u would give you

   int(1/(K*exp(2*t))^2,'t',0,u) =
   1/K^2*int(exp(-4*t),'t',0,u) =
   100*(-exp(-4*u)/4+exp(-4*0)/4) =
   25*(1-exp(-4*u))

where u is whatever upper limit you wish to use. Note that for the larger values of u your answer will be very close to 25.

[Avan Al-Saffar]

Many thanks. I see what do you mean but when I am trying to run this routine, unfortunately, it does not work and I am getting an error which is:

 Undefined function 'int' for input arguments of type 'double'.

The second thing I want to ask you about is my code, do you think I am using a wrong routine for integration??

Regards

[Zoltán Csáti]

Function int only works with symbolic objects.

[Roger Stafford]

My use of 'int' was meant only in place of the integral symbol and not to be used with matlab. For example

   int(1/(K*exp(2*t))^2,'t',0,u)

meant only the integral of (K*exp(2*t))^2 with respect to t from t = 0 to t = u.