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Worksheet Key Terms & Concepts
β³οΈ Solutions & Initial Conditions.
- Satisfying a DE
A function satisfies a DE if substituting it into the dependent variable results in the equation simplifying to a true statement (e.g.,
\(0 = 0\)).
A function that satisfies the DE.
- General Solution
The common form (template) of all the solutions in the family. It contains constants that can take any value.
- Particular Solution
A single solution obtained by assigning specific values to the constants in the general solution.
- Family of Solutions
The collection of all possible particular solutions.
- Initial Conditions
Known values of the solution or its derivatives at a specific point, used to determine a particular solution from the general solution.
β³οΈ Direct Integration.
- Direct Integration
A method to solve differential equations of the form:
\begin{equation*}
\frac{d}{dx}\left[g(x,y)\right] = f(x),
\end{equation*}
by integrating both sides with respect to the independent variable \(x\text{.}\)
