Does the solution depend on initial conditions?

Question

I have started to learn numerical analysis of differential equations and after a bit information arose the question.

Let's assume we have the IVP for the first-order ODE. We solved it and got some solution which satisfies the problem and the theorem of existance and uniqueness. If i change initial conditions, some problems will lose the property to be satisfied to the theorem of existing and uniqueness (well-posedness would be broken). So, the question is: If i write in my essay the next sentence, "as we can see, the ODE should not depend on initial conditions", would it be correct? Any explanations are welcomed

Answer 1

You can speak of the (general) solution of the ODE. Another name for it is the flow function.

With an initial condition you get a solution of the corresponding IVP.

Take two IVP solutions with different parameters (initial conditions can also be seen as parameters). Under the usual smoothness assumptions, if the parameters are close, these solutions will also stay close for a time.

You might have been thinking of ODE like $y'(x)=y(x)^2$ where solutions with positive value at $x=0$ only exist for a finite time, and that time depends on the initial value. The divergence to infinity will of course make a large distance out of every small initial difference close to the first pole of the solutions.

Answer 2

You say the solution of ODE should not depend on the initial conditions"...that is correct, if by solution is meant the general solution.

When initial or boundary values are inserted into the general solution a particular integral/solution results.