Initial conditions evaluated at different $t_0$'s?

[nadia.chigmaroff]

Hello,
I have noticed that in the textbook, initial value problems are given as $y(t_0) = y_0$ and $y'(t_0) = y'_0$. That is, both initial conditions are defined at $t_0$ and moreover, the related Wronskian is also always evaluated at $t_0$. What happens if the initial conditions are defined at different $t_0, t_1$? Moreover, what if $t_0$ and $t_1$ are in different rectangles (meaning the areas where the functions $p, q, g$ are continuous)?
Thanks 

[Xinqiao Li]

From what I understand, when you are solving the second order initial value problems, you first get a general solution of that equation with two arbitrary constant of the form y(t)=C₁e^rt+C₂e^rt. Then you find the derivative y' of this general solution, and plug in the initial conditions to solve what the two constants are. So you might not necessary need both initial conditions to be defined at t₀.

[david]

Initial value problems, by definition, are problems where we have a differential equation and specified values of the solution (and its derivatives) at the same point. If we have conditions such as $y(t_0) = y_0$ and $y(t_1) = y_1$ (called boundary conditions), then this problem is called a boundary value problem. To solve these problems, the same process can be used to get the general solution, but you use the boundary conditions instead to find a particular solution. However, unlike initial value problems, where we only needed some continuity conditions for there to be a unique solution, boundary value problems may have infinite, one, or no solutions.

[Victor Ivrii]

As david pointed out, an initial value problem is 1-point problem. Otherwise it is not an initial value problem  and usually, even for higher order equations, there are only two points, which are the ends of the interval; then it is called a boundary value problem which is covered in the end of the textbook (if its title is ... with boundary value problems) but not in our class. On the other hand, BVPs for ODE are important in PDE class (like APM 346) and covered there.

Important is that BVP are not always solvable uniquely.

[nadia.chigmaroff]

Thanks! That makes sense.