This is a conceptual confusion I am facing rather than a mathematical example. So let's say we have a second-order linear differential equation $F(y, \frac{dy}{dt}, \frac{d^{2}y}{dt^{2}})$, our professor said that we need two initial conditions in order to uniquely determine a solution to the equations.
I am a little confused about this. Do these two initial conditions have to always give two essential pieces of information about y, and about the derivative of y? Can we also get information about two different initial conditions of y, if that is even possible? Why/Why not?
Best Answer
Notice that if you are given any two of $y(x)$, $y'(x)$ and $y''(x)$ at a particular time, you can use the equation to determine the third. If you are given combinations that allow you to solve for two of them, you again can get the third from the equation. (Easy examples of such combinations are invertible linear combinations, for example $y(0) + 2 y'(0) = 1$ and $2y(0) - y'(0) = 7$. However, any combinations, including nonlinear combinations, that can be used with the equation to find all three is sufficient. An example of what I mean when I say "used with the equation" is to combine the above example with the equation $y'' - y' + 2y = 0$, which immediately gives $y''(0) + 7 = 0$, even without solving for $y'(0)$ or $y(0)$ explicitly.)
It is also possible to specify "boundary conditions" to produce a boundary value problem. So for instance, you could specify $y(0)$ and $y(1)$. Similar techniques are used for solving initial value problems and boundary value problems. Different techniques are used for numerically approximating solutions. One need not use the same derivative for both boundary conditions, e.g. $y'(0) = 2$ and $y(1) = -1$.
There are few restrictions on sets of initial conditions that will lead to the existence of solutions for second order linear equations. However, there can be many constraints on sets of boundary values that permit solutions. (For more, see, for instance, https://www.sciencedirect.com/science/article/abs/pii/0362546X94900361 .)