I'm trying to solve an Initial Value Problem, but I'm not sure now if the problem I have in hand is even an Initial Value Problem. Notes from Paul Dawkins' Course states IVP has the definition below – those are the initial values we have:
$$
\frac{dy}{dt} = f(t,y), \quad y(t_0) = y_0
$$
The problem I'm trying to solve on the other hand, is as follows:
$$
\frac{dy}{dt} = f(y), \quad y(t_0) = y_0
$$
In my problem $\frac{dy}{dt}$ is only a function of $y$, not $t,y$.
Does my problem qualify as an Initial Value Problem and can it be solved by Euler's method? The confusion arises, because I have an initial value and an initial direction. But the rate of change does not depend on time $t$ .
Thanks!
Best Answer
To post a clear answer to your query, kindly note that $$ y' = f(y), \ \ y(t_0) = y_0 \tag{1} $$ is an Initial Value Problem (with an autonomous ODE and an initial condition at $t_0$) and this is a special case of $$ y' = f(t, y), \ \ y(t_0) = y_0 \tag{2} $$ (general case - when $f$ involves $t$ and $y$, the IVP (2) involves a non-autonomous ODE and an initial condition at $t_0$).
Euler's method or any finite-difference method used for finding estimates for the solution to the IVP (2) can be also applied for finding estimates for the solution to the IVP (1).