Yes, integration of ODE involves extrapolation whereas integration of functions involves interpolation. Take for example the Euler's method, we are projecting the slope at the initial point and assume that it applies throughout the time step (which is obviously not the case in general). Worse, the errors accumulate through the time steps. That is, the starting value for the 2nd time step is inaccurate to start with.

Integration of functions involves interpolation of a certain polynomial to fit the functional values and evaluating the area under that polynomial.

In fact, many researchers have tried to apply functional integration methods to integration of ODEs and not the other way round! We know the Gaussian quadrature is one of the most efficient and accurate numerical integration techniques around. Several ODE solvers are based on the Gauss method - e.g. the fully-implicit Gauss-Legendre method developed by Hollingsworth (1955) and later generalized by Butcher (1964) for arbitrary orders as the Gauss methods (implicit Runge-Kutta processes).

Still later, such implicit Runge-Kutta processes are shown to be identical to the collocation methods (Wright, 1970).

Let me pose you a problem: Given an ODE y' = f(x) where the slope function is a polynomial (solely a function of x). Are you able to integrate this ODE exactly in one step using existing ODE solvers without transforming this to a functional integration problem?

Yes, integration of ODE involves extrapolation whereas integration of functions involves interpolation. Take for example the Euler's method, we are projecting the slope at the initial point and assume that it applies throughout the time step (which is obviously not the case in general). Worse, the errors accumulate through the time steps. That is, the starting value for the 2nd time step is inaccurate to start with.

Integration of functions involves interpolation of a certain polynomial to fit the functional values and evaluating the area under that polynomial.

In fact, many researchers have tried to apply functional integration methods to integration of ODEs and not the other way round! We know the Gaussian quadrature is one of the most efficient and accurate numerical integration techniques around. Several ODE solvers are based on the Gauss method - e.g. the fully-implicit Gauss-Legendre method developed by Hollingsworth (1955) and later generalized by Butcher (1964) for arbitrary orders as the Gauss methods (implicit Runge-Kutta processes).

Still later, such implicit Runge-Kutta processes are shown to be identical to the collocation methods (Wright, 1970).

Yes, integration of ODE involves extrapolation whereas integration of functions involves interpolation. Take for example the Euler's method, we are projecting the slope at the initial point and assume that it applies throughout the time step (which is obviously not the case in general). Worse, the errors accumulate through the time steps. That is, the starting value for the 2nd time step is inaccurate to start with.

Integration of functions involves interpolation of a certain polynomial to fit the functional values and evaluating the area under that polynomial.

In fact, many researchers have tried to apply functional integration methods to integration of ODEs and not the other way round! We know the Gaussian quadrature is one of the most efficient and accurate numerical integration techniques around. Several ODE solvers are based on the Gauss method - e.g. the fully-implicit Gauss-Legendre method developed by Hollingsworth (1955) and later generalized by Butcher (1964) for arbitrary orders as the Gauss methods (implicit Runge-Kutta processes).

Still later, such implicit Runge-Kutta processes are shown to be identical to the collocation methods (Wright, 1970).

Let me pose you a problem: Given an ODE y' = f(x) where the slope function is a polynomial (solely a function of x). Are you able to integrate this ODE exactly using existing ODE solvers without transforming this to a functional integration problem?

Yes, integration of ODE involves extrapolation whereas integration of functions involves interpolation. Take for example the Euler's method, we are projecting the slope at the initial point and assume that it applies throughout the time step (which is obviously not the case in general). Worse, the errors accumulate through the time steps. That is, the starting value for the 2nd time step is inaccurate to start with.

Integration of functions involves interpolation of a certain polynomial to fit the functional values and evaluating the area under that polynomial.

In fact, many researchers have tried to apply functional integration methods to integration of ODEs and not the other way round! We know the Gaussian quadrature is one of the most efficient and accurate numerical integration techniques around. Several ODE solvers are based on the Gauss method - e.g. the fully-implicit Gauss-Legendre method developed by Hollingsworth (1955) and later generalized by Butcher (1964) for arbitrary orders as the Gauss methods (implicit Runge-Kutta processes).

Still later, such implicit Runge-Kutta processes are shown to be identical to the collocation methods (Wright, 1970).

Let me pose you a problem: Given an ODE y' = f(x) where the slope function is a polynomial (solely a function of x). Are you able to integrate this ODE exactly in one step using existing ODE solvers without transforming this to a functional integration problem?