In my differential equations class, we are reading from "The Qualitative Theory for Differential Equations." My question is about one of the requirements for uniqueness of a solution to a differential equation.
An example from the book writes:
$y'=\alpha y$
and I can understand this from my computational diffyq class, $y(t)=ce^{\alpha t}$, but then the book rewrites this as: $f(t,y)=\alpha y$, as a function of the (t,y) plane.
This is where my understanding breaks down. Perhaps I am overthinking it or intimidated, but I cannot come to terms with "the t-y plane". I am used to thinking of it as the XY plane, as in elementary school, yet $y(t)=ce^{\alpha t}$ is always unequal to zero, so how could it be thought of as an axis to a plane? The book also mentions regions in (n+1)-dimensional euclidean space, where the variables are functions of t.
Anything that might help my intuition is welcome, thanks!
In some application of differential equation the independent variable is time. For example if you put your money in the bank and the interest rate is $5$ percent per year, the differential equation is $y'=0.05y$ where y=y(t) is the balance at time $t$.
If you want to graph your solution you will get a curve, which is the graph of $y=y(0)e^{0.05 t} $ and you graph it in the $ty$ plane instead of $xy$ plane.
Now if you have a system of differential equations you may have to consider the $tY$ space where$ Y=(y_1,y_2,..., y_n)$ is a vector whose components are functions of $t$
In summary, instead of $x$ you can call your independent variable $t$ or whatever else depending the model that you are working on.