Section 2.3 Types of Solutions
At this point, you should comfortable with the idea that
a solution to a differential equation is a function that satisfies it. You have also seen that
a single differential equation can have more than one solution. However, this begs the question:
“How many solutions does a differential equation have?”
The answer depends on the context of the problem, and that’s what we will explore in this section. Let’s kick things off with an example.
Example 22.
\(\ \ \) Show that the functions
\begin{equation*}
y = e^{2x},\ \ y = 2e^{2x},\ \ y = 3e^{2x},\ \ y = -7e^{2x},\ \ y = 0.3e^{2x},\ \ y = \pi e^{2x}
\end{equation*}
are solutions to the differential equation \(\quad y' - 2y = 0\text{.}\) Solution.
Verifying each function individually would be repetitive because they all follow the same pattern:
\begin{equation*}
y = (\text{some number})\ e^{2x} \quad \text{or} \quad y = c e^{2x} \text{,}
\end{equation*}
where \(c\) can be any constant. Let’s prove that \(y = c e^{2x}\) is a solution.
First, compute \(y'\text{:}\)
\begin{align*}
y =\amp\ c e^{2x} \\
y' =\amp\ 2c e^{2x}
\end{align*}
Now, substitute \(y\) and \(y'\) into the differential equation:
\begin{align*}
y' - 2y =\amp\ 0 \\
2c e^{2x} - 2(c e^{2x}) =\amp\ 0 \\
2c e^{2x} - 2c e^{2x} =\amp\ 0 \\
0 =\amp\ 0
\end{align*}
Therefore, \(y = c e^{2x}\) is a solution, and the specific value of \(c\) does not matter.
From the above example, you can see that the function
\(y = c e^{2x}\) satisfies the differential equation for any value of
\(c\text{.}\) This leads to an important observation:
“How many choices for \(c\) are there?”
Of course, the answer is infinitely many! This means the equation \(y' - 2y = 0\) has an infinite number of solutions. This brings us to some key terms related to solutions of differential equations.
Note 23. Don’t assume a function that satisfies a DE is a general solution!
Definition 24. Types of Solutions.
- Family of Solutions
The collection of all possible solutions of a differential equation.
- General Solution
The common form of all the solutions in the family. This includes arbitrary constants that can take any value.
- Particular Solution
A single solution obtained by assigning specific values to the constants in the general solution.
Applying these terms to our previous example, we can summarize that:
\(y = c e^{2x}\) is the general solution of the equation \(y' - 2y = 0\text{.}\)
\(y = 5e^{2x}\) is one particular solution in the family of solutions.
The family of solutions consists of all functions of the form \(y = c e^{2x}\text{.}\)
Understanding the different types of solutions—general, particular, and families of solutions—is crucial as you continue to explore differential equations. In the upcoming sections, we’ll talk more about where particular solutions come from.
Reading Questions Check your Understanding
1. A family of solutions can be viewed as the collection of all particular solutions.
A family of solutions can be viewed as the collection of all particular solutions- True
- Correct! A family of solutions includes all possible particular solutions.
- False
- Incorrect. A family of solutions is a set of all possible solutions, not just one particular solution.
2. A particular solution can be viewed as a member of the family of solutions.
3. Which task is fundamentally different from the others?
4. What is the difference between a general solution and a particular solution?
What is the difference between a general solution and a particular solution?- A general solution is a specific solution to a differential equation, while a particular solution is a general form of the solution.
- Incorrect. A general solution is a form of the solution that includes arbitrary constants, while a particular solution is a specific member of the family of solutions.
- A general solution is a form of the solution that includes arbitrary constants, while a particular solution is a specific member of the family of solutions.
- Correct! The general solution represents all possible solutions, while a particular solution is one specific solution.
- A general solution is a specific member of the family of solutions, while a particular solution is a form of the solution.
- Incorrect. A general solution is a form of the solution that includes arbitrary constants, while a particular solution is a specific member of the family of solutions.
- A general solution is a specific solution to a differential equation, while a particular solution is a general form of the solution.
- Incorrect. A general solution is a form of the solution that includes arbitrary constants, while a particular solution is a specific member of the family of solutions.
5. Fill in the Blank.
6. What role does the general solution play in solving a differential equation?
What role does the general solution play in solving a differential equation?
- It provides the exact value of the function.
Incorrect. The general solution provides a family of functions, not just one specific value.
- It represents a family of functions that satisfy the differential equation.
Correct! The general solution includes all possible solutions, depending on the values of the constants.
- It simplifies the differential equation.
Incorrect. While solving the equation involves simplification, the general solution represents the functions that satisfy the equation.
- It eliminates the constants from the solution.
Incorrect. The general solution still includes arbitrary constants.
7. What is a family of solutions?
8. A general solution always includes arbitrary constants.
9. Fill in the Blank.
10. In Example 22, we verified that \(y = Ce^{2x}\) was a solution to
\begin{equation*}
y''' - 4y' = 0
\end{equation*}
Based on this, how many solutions must this equation have?
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