DE-BOOK What is a Solution?

Section 2.1 What 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.

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

y^{3} = 3 y + 2 .

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To do this, we verify that both

2

and

0

satisfy the equation by separately plugging

2

and

0

in for each

y,

simplify and see if we end up with an undeniably true statement, like so

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\begin{aligned} (2)^{3} = & 3 (2) + 2 \\ 8 = & 6 + 2 \\ 8 = & 8 \leftarrow true \end{aligned}

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\begin{aligned} (0)^{3} = & 3 (0) + 2 \\ 0 = & 0 + 2 \\ 0 = & 2 \leftarrow false \end{aligned}

Since

y = 2

yields a 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.

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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 = 3 x

and

y = e^{3 x}

are solutions to the differential equation

y^{'} = 3 y .

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Separately plugging

3 x

and

e^{3 x}

into the equation yields

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\begin{aligned} y^{'} = & 3 y \\ {[3 x]}^{'} = & 3 (3 x) \\ 3 = & 9 x \leftarrow false \end{aligned}

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\begin{aligned} y^{'} = & 3 y \\ {[e^{3 x}]}^{'} = & 3 e^{3 x} \\ e^{3 x} \cdot {[3 x]}^{'} = & 3 e^{3 x} \\ e^{3 x} \cdot 3 = & 3 e^{3 x} \\ 3 e^{3 x} = & 3 e^{3 x} \leftarrow true \end{aligned}

Since

y = 3 x

results in a false statement, it does not satisfy the equation and is not a solution. However,

y = e^{3 x}

does satisfy the equation and is a solution.

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To summarize, verifying a solution involves substituting the function into the differential equation and ensuring that the equality is satisfied.

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Reading Questions Check-Point Questions

1. A solution to a differential equation is a function that the equation.

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, $y = x^{3},$ satisfies the differential equation $y^{'} = 3 y$ .

True.
$y = x^{3}$ is not a solution since

$\begin{aligned} y^{'} = & 3 y \\ {[x^{3}]}^{'} = & 3 (x^{3}) \\ 3 x^{2} = & 3 x^{3} \leftarrow false \end{aligned}$
False.
$y = x^{3}$ is not a solution since

$\begin{aligned} y^{'} = & 3 y \\ {[x^{3}]}^{'} = & 3 (x^{3}) \\ 3 x^{2} = & 3 x^{3} \leftarrow false \end{aligned}$

4. Which variable in equation $u^{″} + t^{2} u = 0$ represents the solution?

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 $\frac{d P}{d s} + \frac{P}{s^{2}} = 17 s$ does the solution depend on?

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

dependent variable, $s$
Incorrect. The solution depends on $s,$ but $s$ is not a dependent variable.
independent variable, $s$
Yes! the solution, $P,$ depends on the independent variable $s .$
dependent variable, $P$
Incorrect. $P$ is the solution, so it does not depend on $P .$
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.

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