Reformulating Integrals as ODE Problems

[JohanL]

How do you reformulate an integral as an ode-problem?

 

[Nylex]

Differentiate both sides of the equation? Or do you mean something else? :/

 

[JohanL]

But is that possible if you have an integral between two values. I know its possible if you have an indefinite?! integral...now i have

M(y)=int(a,b) m(x,y) dx

 

[gerben]

But is that possible if you have an integral between two values. I know its possible if you have an indefinite?! integral...now i have

M(y)=int(a,b) m(x,y) dx

m(x, y) is a function of x and y if you integrate this with respect to x you will get a function of y, so that will then be M(y):

$$\int_a^b m(x, y) dx = [f(x, y)]_a^b = f(b, y) - f(a, y) = M(y)$$

 

[JohanL]

But I want to reformulate this integral to an ode-problem

 

[gerben]

well I think the only way you can do that is like Nylex said, like this:

$$\frac{d(\int_a^b m(x, y) dx)}{dy} = \frac{dM(y)}{dy}$$

 

[JohanL]

But with this I still need to solve the integral, don't I?
And I want to rewrite it as an ode so i don't have to solve the integral.

 

[matt grime]

From your posts it isn't clear if y is a function of x or not, since it usually isn't that the integral has limits a and b.

An integral that can be turned into a differential (ODE) would be something like:


$$\int_0^xf(t)dt = y$$

which has associated differential equation dy/dx = f(x)


so why do you even think that the type of equation you wrote has an ODE equivalent?

Even if y were a function of x, then the integral you wrote would still only yield a number, and that isn't the function y.

 

[JohanL]

y isn't a function of x. a and b are reell numbers.
The question in the book is(numerical analysis)

M(y)=int(a,b) m(x,y) dx
a. Use quadl in MATLAB to determine M
b. An alternative way to determine M is to rewrite the integral as an ODE-problem. Do that and use ode45 in MATLAB to solve the ode-problem. Compare with the solution in a.

b is really strange...

 

[gerben]

what ode45(f, [a, b], y0) does is simply integrating f from a to b, using inital value y0...

so you can use that to get your answer (you will need to use the @ again ;-), like this ode45(@f, [a, b], 0)

 

[JohanL]

Do you mean that the initial value is f(a). Then
ode45(@f, [a, b], y0)
should yield the same value as
QUADL(@f,a,b);
but it dont.
I want them to yield the same value...what am i doing wrong...

 

[gerben]

I think the inital value should be f(0)