Section 8.2 First Order ODEs: Notation and Theorems
¶FIXME Add explanation
Subsection 8.2.1 Notation for First Order ODEs
Standard Form:
\begin{equation}
\frac{dy}{dx}=f(x,y)\label{standard}\tag{8.2.1}
\end{equation}
Differential Form: Write
\begin{equation}
f(x,y)=-\frac{M(x,y)}{N(x,y)}\tag{8.2.2}
\end{equation}
(There are many ways to do this. Choose a way that is helpful for the problem at hand.) Then Eqn(8.2.1) becomes
\begin{equation}
M(x,y)\, dx + N(x,y)\, dy =0\tag{8.2.3}
\end{equation}
Subsection 8.2.2 First-Order ODEs: Uniqueness Theorem
If \(f\) and \(\frac{\partial f}{\partial y}\) are continuous in a rectangle \(\vert x-x_0\vert \le a\text{,}\) \(\vert y-y_0\vert\le b\text{,}\) then there exists an interval about \(x_0\) in which the initial value problem \(y^{\prime}=f(x,y)\text{,}\) \(y(x_0)=y_0\) has a unique solution.