2
$\begingroup$

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!

$\endgroup$
2
  • 1
    $\begingroup$ It's just a special case... $\endgroup$ Apr 15, 2022 at 3:10
  • $\begingroup$ So it can be solved by Euler's method? Or is there a modification needed? $\endgroup$
    – sanjeev mk
    Apr 15, 2022 at 3:15

1 Answer 1

5
$\begingroup$

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).

$\endgroup$
1
  • $\begingroup$ Hi @Dr.Sundar, this is unrelated to your answer, hence I would be deleting this comment soonest. Please could you help out with this question math.stackexchange.com/q/4435979/585488 $\endgroup$
    – linker
    Apr 29, 2022 at 8:09

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .