- \(\ds y = -2e^{x^2}+3\)
- Incorrect. The value of \(c\) must make the solution pass through \((0, 5)\text{.}\) Hover over the curve in the figure that passes through \((0,5)\text{.}\)
- \(\ds y = 0.5e^{x^2}+3\)
- 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 \((0,5)\text{.}\)
- \(\ds y = 2e^{x^2}+3\)
- Correct! The value \(c = 2\) ensures that \(y(0) = 2 + 3 = 5\text{,}\) so this solution passes through \((0, 5)\text{.}\)
- \(\ds y = 5e^{x^2}+3\)
- Incorrect. The general solution would pass through \((0, 4)\) if \(c = 1\text{.}\) Hover over the curve in the figure that passes through \((0,5)\text{.}\)
Section 2.4 Visualizing Solutions
A powerful way to understand solutions to differential equations is by visualizing them. Consider the differential equation
\begin{equation*}
\frac{dy}{dx} = 2xy - 6x \text{,}
\end{equation*}
which has the general solution, \(y = ce^{x^2} + 3\) (solution verification details).
Since each member of the family of solutions comes from a different value of \(c\text{,}\) we can plot a few of them to see how they differ (see Figure 25).
Hover over the green curves in the figure to see each particular solution and corresponding value of \(c\text{.}\)
In terms of the types of solutions,
- Each green curve in the figure represents a particular solution, with its own specific value of \(c\text{.}\)
- The general solution, \(y = ce^{x^2} + 3\text{,}\) provides the framework for all these individual solutions.
Although the figure only shows a few members of the infinite family of solutions, notice that the green curves never overlap. This means that each curve represents a unique solution. If you were to randomly select a point on the plot, you would, in effect, be selecting a single curve—only one solution can pass through that specific point. This idea is at the heart of where particular solution come from and it will be the focus of the next section.
Reading Questions Check your Understanding
Answer the following questions using the figure and the fact that \(\ds y = ce^{x^2} + 3\) is the general solution to the differential equation
\begin{equation}
\ds \frac{dy}{dx} = 2xy - 6x. \tag{4}
\end{equation}
1. Select the particular solution in Figure 25 that passes through the point \((0,5)\).
2. Select the \(c\)-value for the solution in Figure 25 that passes through the point \((1,1)\).
- \(c = -2\)
- Incorrect. Hover over the curve in the figure that passes through \((1,1)\) and look at the coefficient on \(e^{x^2}\text{.}\)
- \(c = 0.5\)
- Incorrect. Hover over the curve in the figure that passes through \((1,1)\) and look at the coefficient on \(e^{x^2}\text{.}\)
- \(c = 2\)
- Incorrect. Hover over the curve in the figure that passes through \((1,1)\) and look at the coefficient on \(e^{x^2}\text{.}\)
- \(c = -1\)
- Correct! Hovering over the curve passing through \((1,1)\) shows the particular solution \(\ds y = -e^{x^2}+3\text{,}\) so \(c=1\text{.}\)
3. Two different particular solutions of (4) could have the same \(c\)-value.
- True
- Incorrect. The value of \(c\) is what makes each particular solution different.
- False
- Correct! Different particular solutions have different values of \(c\text{.}\)
4. There are infinitely-many particular solutions in a family.
- True
- Correct! Each value of \(c\) corresponds to a different particular solution.
- False
- Incorrect. There are infinitely many particular solutions in a family.
5. Which of the following is the best visual description of a family of solutions?
- Collection of all the possible solution curves.
- Correct! A family of solutions includes all possible solutions so visually it would be the collection of all solution curves.
- A single solution curve.
- Incorrect. A family of solutions includes all possible particular solutions, not just one.
- The intersection of all the possible solution curves.
- Incorrect. A family of solutions includes all possible solutions, not just their intersections.
- The curve of the general solution.
- Incorrect. The general solution is a framework for all possible solutions, but it is not the family of solutions itself. Also, since a general solution includes constants, doesn’t have “a curve”.
6. Which of the following is the best visual description of a particular solution?
- A curve that represents the solution with a specific constant value.
- Correct! A particular solution is a single curve that satisfies the differential equation with a specific constant value, distinguishing it from the general solution.
- A family of curves that represents all possible solutions to the differential equation.
- Incorrect. This description fits a general solution, which encompasses all particular solutions by varying the parameter.
- A straight line that intersects all solutions to the differential equation.
- Incorrect. A particular solution is not necessarily a straight line, and it does not intersect all other solutions. It’s a unique curve based on specific conditions.
- A set of discrete points that satisfy the differential equation for various parameter values.
- Incorrect. A particular solution is typically represented by a continuous curve, not a set of discrete points.
7. The solution \(y = ce^{x^2} + 3\) where \(c = -1\) will pass through the origin \((0, 0)\).
- True
- Incorrect. Try recalculating \(y(0)\) when \(c = -1\text{.}\) Which point does it pass through on the \(y\)-axis?
- False
- Correct. If \(c = -1\text{,}\) then \(y(0) = -1 + 3 = 2\text{,}\) which passes through \((0,2)\text{.}\)
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