Interactive Differential Equations: A Step-by-Step Approach to Methods & Modeling

Section2.1What is a Solution?

Regardless what kind of equation you are working with, a solution is a value or function that “satisfies” the equation. The term satisfies simply means that when you plug the value into the equation, it simplifies to a statement that is undeniably true.
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This “undeniably true” statement is sometimes called an identity.
For example, suppose I want to check if $$y=2$$ and $$y = 0$$ are solutions to the equation
\begin{equation*} y^3 = 3y + 2 \text{.} \end{equation*}
To do this, we verify that both $$2$$ and $$0$$ satisfy the equation by separately plugging $$2$$ and $$0$$ in for each $$y\text{,}$$ simplify and see if we end up with an undeniably true statement, like so
\begin{align*} (2)^3 =\amp\ 3(2) + 2 \\ 8 =\amp\ 6 + 2 \\ 8 =\amp\ 8 \quad \leftarrow \text{true} \end{align*}
\begin{align*} (0)^3 =\amp\ 3(0) + 2 \\ 0 =\amp\ 0 + 2 \\ 0 =\amp\ 2 \quad \leftarrow \text{false} \end{align*}
Since $$y=2$$ yields an true statement we say it satisfies the equation and is a solution. In contrast, $$y=0$$ does not give a true statement, so it does not satisfy the equation and is not a solution.
The same idea applies to differential equations, except in that solutions to differential equations are functions instead of numbers. To see this, let’s verify if $$y = 3x$$ and $$y = e^{3x}$$ are solutions to the differential equation $$\displaystyle y^\prime = 3y \text{.}$$
Separately plugging $$3x$$ and $$e^{3x}$$ into the equation yields
\begin{align*} y^\prime =\amp\ 3y \\ \left[3x\right]^\prime =\amp\ 3(3x) \\ 3 =\amp\ 9x \quad \leftarrow \text{false} \end{align*}
\begin{align*} y^\prime =\amp\ 3y \\ \left[e^{3x}\right]^\prime =\amp\ 3e^{3x} \\ e^{3x} \cdot \left[3x\right]^\prime =\amp\ 3e^{3x} \\ e^{3x} \cdot 3 =\amp\ 3e^{3x} \\ 3e^{3x} =\amp\ 3e^{3x} \quad \leftarrow \text{true} \end{align*}
Since $$y=3x$$ results in a false statement, it does not satisfy the equation and is not a solution. However, $$y=e^{3x}$$ does satisfy the equation and is a solution.
To summarize, verifying a solution involves substituting the function into the differential equation and ensuring that the equality is satisfied.

2.What does it mean for a function to satisfy a differential equation?

What does it mean for a function to satisfy a differential equation?
• If you plug the function into the equation, you get a true statement.
• Yes, a function that satisfies a differential equation yields a true statement when plugged into the equation.
• If you plug the function into the equation, you get the solution.
• Incorrect. The function is being checked to see if it is a solution, you do not get the solution by plugging it in.
• If you take the derivative of the function, you get a true statement.
• Incorrect. Carefully read the section again.
• If you integrate the function, you get a true statement.
• Incorrect. Carefully read the section again.

3.The function, $$y = x^3\text{,}$$ satisfies the differential equation $$y' = 3y$$.

• True.

• $$y = x^3$$ is not a solution since
\begin{align*} y' =\amp\ 3y \\ \left[x^3\right]^{\prime} =\amp\ 3(x^3) \\ 3x^2 =\amp\ 3x^3 \quad \leftarrow \text{false} \end{align*}
• False.

• $$y = x^3$$ is not a solution since
\begin{align*} y' =\amp\ 3y \\ \left[x^3\right]^{\prime} =\amp\ 3(x^3) \\ 3x^2 =\amp\ 3x^3 \quad \leftarrow \text{false} \end{align*}

5.In general, a “solution” to a differential equation is a .

In general, a “solution” to a differential equation is a
• constant
• It is possible for a solution to be a constant, but not in general.
• function
• Yes, when you solve a differential equation, you get a function.
• number
• It is possible for a solution to be a number, but not in general.
• limit
• Sorry, no.

6.Which variable in $$\ds \frac{dP}{ds} + \frac{P}{s^2} = 17s$$ does the solution depend on?

Which variable in $$\ds \frac{dP}{ds} + \frac{P}{s^2} = 17s$$ does the solution depend on?
• dependent variable, $$s$$
• Incorrect. The solution depends on $$s\text{,}$$ but $$s$$ is not a dependent variable.
• independent variable, $$s$$
• Yes! the solution, $$P\text{,}$$ depends on the independent variable $$s\text{.}$$
• dependent variable, $$P$$
• Incorrect. $$P$$ is the solution, so it does not depend on $$P\text{.}$$
• independent variable, $$P$$
• Incorrect. The variable $$P$$ is not the independent variable.

7.What is the primary goal of solving a differential equation?

What is the primary goal of solving a differential equation?
• To find an unknown function that satisfies the equation.
• Correct! The goal of solving a differential equation is to find the function that meets the equation’s conditions.
• To find the derivative of a function.
• Incorrect. While derivatives are involved, the goal is to find the function, not just its derivative.
• To identify the constants in an equation.
• Incorrect. Identifying constants might be part of the process, but it is not the primary goal.
• To determine the independent variable.
• Incorrect. The independent variable is usually known; we solve for the dependent variable.