# Is the following an Initial Value Problem or not?

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!

• It's just a special case... Apr 15, 2022 at 3:10
• So it can be solved by Euler's method? Or is there a modification needed? Apr 15, 2022 at 3:15

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 $$t_0$$).