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Chapter 6: Differential Equations
- §1. Definitions and Notation
- §2. First Order: Notation and Theorems
- §3. First Order: Separable
- §4. First Order: Exact
- §5. The Word "Linear"
- §6. Homogeneous (Const Coeff)
- §7. Linear Independence
- §8. Inhomogeneous (Const Coeff)
- §9. Linear, Series Solutions: Theorems
- §10. Linear, Series Solutions: Method
First Order ODEs: Definitions and Theorems
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Notation for First Order ODEs
Standard Form: \begin{equation} \frac{dy}{dx}=f(x,y) \label{standard} \end{equation}
Differential Form: Write $$f(x,y)=-\frac{M(x,y)}{N(x,y)}$$ (There are many ways to do this. Choose a way that is helpful for the problem at hand.) Then Eqn(\ref{standard}) becomes $$M(x,y)\, dx + N(x,y)\, dy =0$$
First-Order ODEs: Uniqueness Theorem
If $f$ and $\frac{\partial f}{\partial y}$ are continuous in a rectangle $\vert x-x_0\vert \le a$, $\vert y-y_0\vert\le b$, then there exists an interval about $x_0$ in which the initial value problem $y^{\prime}=f(x,y)$, $y(x_0)=y_0$ has a unique solution.