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

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Avan Al-Saffar
Avan Al-Saffar 2014-11-13
评论： Roger Stafford 2014-11-13
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.
  1 个评论
Torsten
Torsten 2014-11-13
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.

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回答（1 个）

Roger Stafford
Roger Stafford 2014-11-13
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.
  3 个评论
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Roger Stafford
Roger Stafford 2014-11-13
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.

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