What does $y^k(t_0) = y_k$ mean and how is it pronounced?

Question

I just started studying differential equations and came across two equations in the Initial Value Problems section and can't proceed until I understand what the above means. Also, what is the significance of the uppercase Y on the right side of the equation?

Context:

The interval of validity for an IVP with initial condition(s)

$y(t_0) = y_0$ and $y^k t(0) = y_k$

is the largest possible interval on which the solution is valid and contains $t_0$.

Answer 1

You didn't get the formatting right yet. I think you mean $$ y^k( t_0) = y_k $$ or, better, $$ y^{(k)}( t_0) = y_k . $$ That's read as

The $k$th derivative of $y$ at time/point $t_0$ is $y_k$.

The first statement tells you the value of $y$ at $t_0$ (which can sometimes be thought of as the zeroth derivative).

The overall meaning is that in this initial value problem you are told the value of $y$ and its first $k$ derivatives at a particular point.