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Worksheet Key Terms & Concepts
β³οΈ Differential Equations and their Components.
Differential Equation (DE)
An equation that involves one or more derivatives of an unknown function.
Variables
The dependent variable represents the unknown function that you are solving for and always has derivatives applied to it. The dependent variable is a function of the independent variable . In
\begin{gather*}
12y'' + (x+5)y' - \ln(x) y = 5 - \cos x\text{,}
\end{gather*}
dependent \(\leftarrow y\quad\) & \(\quad\) independent \(\leftarrow x\text{.}\)
Terms & Free Terms
Parts of an equation separated by
\(+\text{,}\) \(-\text{,}\) or
\(=\) and each containing a different form of the dependent variable. The collection of all terms without a dependent variable is referred to as a
free term .
\begin{gather*}
\us{\large y'' \text{ term}}{\boxed{12y''}} + \us{\large y' \text{ term}}{\boxed{(x+5)y'}} - \us{\large y \text{ term}}{\boxed{\ln(x) y}} = \us{\large\text{free term}}{\boxed{5 - \cos x}}
\end{gather*}
Coefficients
The part of a term multiplied by the dependent variable or its derivatives.
\begin{gather*}
\boxed{12}\us{\Large y'',\ y',\ y \text{ coefficients}}{\us{\nwarrow}{\ y}'' + \us{\uparrow}{\boxed{(x+5)}}\ y' \us{\nearrow}{-\ }} \boxed{\ln (x)}\ y = 5 - \cos x
\end{gather*}
β³οΈ Order & Linearity.
Order
The highest order derivative present in a DE.
Linear Term
\begin{equation*}
a(t)\ y,\ a(t)\ y',\ a(t)\ y'',\ a(t)\ y''',\ \ldots\text{,}
\end{equation*}
where
\(y\) is the dependent variable, and
\(a(t)\) is a coefficient that depends only on the independent variable
\(t\text{.}\)
Linear DE
A DE composed entirely of linear terms.
Nonlinear DE
A DE that contains at least one nonlinear term.
