An effective way to deepen your understanding of solutions is through visualization. Even though a family of solutions includes infinitely many curves, plotting just a few helps reveal how the general solution, particular solutions, and initial conditions are related.
Think of a family of solutions like a map full of side-by-side paths. The map shows all possible routes a traveler could take, each one representing a particular solution. The general solution defines the layout of all these paths, and choosing an initial condition is like dropping a pin on the map: the curve that passes through that point is the specific path (or solution) you follow.
\begin{equation*}
y = c e^{x^2} + 3.
\end{equation*}
Since the constant \(c\) can take any value, there are many possible solutions as seen by the green curves in the figure below. Notice, each green curve can be identified by where it crosses the \(y\)-axis (e.g., \(y(0) = 2\)). This point is called the initial condition and it leads to both \(c\) and the particular solution in blue.
Incorrect. The value of \(c\) must make the solution pass through \((0, 5)\text{.}\) Hover over the curve in the figure that passes through \((0,5)\text{.}\)
\(\quad y = 0.5e^{x^2}+3\)
Incorrect. Remember that at \(x = 0\text{,}\) the exponential term \(e^{x^2}\) equals 1, so \(y(0) = c + 3\text{.}\) What value of \(c\) gives \(y(0) = 5\text{?}\) Hover over the curve in the figure that passes through \((0,5)\text{.}\)
\(\quad y = 2e^{x^2}+3\)
Correct! The value \(c = 2\) ensures that \(y(0) = 2 + 3 = 5\text{,}\) so this solution passes through \((0, 5)\text{.}\)
\(\quad y = 5e^{x^2}+3\)
Incorrect. The general solution would pass through \((0, 4)\) if \(c = 1\text{.}\) Hover over the curve in the figure that passes through \((0,5)\text{.}\)
