Section3.3General, Particular, and Family of Solutions
You know that a solution to a differential equation is a function that satisfies the equation. But 🌌 Example 25 shows that the differential equation
\begin{equation*}
y' - 2y = 0
\end{equation*}
has multiple solutions, all of which differ by a constant factor:
\begin{equation*}
y = e^{2x},\quad y = 3e^{2x},\quad y = -5e^{2x},\quad y = \pi e^{2x}, \quad\text{ etc. }
\end{equation*}
Rather than listing all possible solutions, we can express them all at once using a formula that includes an arbitrary constant:
\begin{equation*}
y = ce^{2x},
\end{equation*}
where \(c\) can be any real number. This formula is called the general solution of the differential equation, and each choice of \(c\) gives a different particular solution.
A particular solution is a specific function that satisfies a differential equation. A general solution is the most general form (i.e., a formula) that represents all possible particular solutions. A family of solutions is the complete set of all particular solutions.
In this case, the general solution contains two arbitrary constants. To obtain a particular solution, we must assign specific values to both \(c_1\) and \(c_2\text{.}\)Table 28 shows how some particular solutions are generated from the general solution by selecting different values for the constants.
If we could somehow list every possible combination of \(c_1\) and \(c_2\text{,}\) we’d have the full family of solutions to the differential equation.
⚠️29.Not all solutions with constants are general solutions.
Keep in mind, solutions with arbitrary constants are not general solutions by default. For example, both of the functions
\begin{equation*}
y = \frac{1}{2}x^2 + c_1 x \quad \text{and} \quad y = \frac{1}{2}x^2 + c_1 x + c_2
\end{equation*}
are solutions to the equation \(y'' = 1\text{,}\) but only the second is the general solution. The first is a special case of the second, obtained when \(c_2 = 0\text{.}\)
Keep in mind, solutions with arbitrary constants are not general solutions by default. For example, both of the functions
\begin{equation*}
y = \frac{1}{2}x^2 + c_1 x \quad \text{and} \quad y = \frac{1}{2}x^2 + c_1 x + c_2
\end{equation*}
are solutions to the equation \(y'' = 1\text{,}\) but only the second is the general solution. The first is a special case of the second, obtained when \(c_2 = 0\text{.}\)
