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Section 3.1 Solutions to Equations
In algebra, we say that a value
satisfies an equation if, when we substitute that value in place of the variable, it makes both sides of the equation equal. This gives a true statement, like
\(0=0\) or
\(1=1\text{.}\)
For example, consider the equation:
\begin{equation*}
y^3 = 3y + 2.
\end{equation*}
Let’s check whether
\(y = 2\) and
\(y = 0\) are solutions by substituting them into the equation. Plugging them in, we get:
\begin{align*}
(2)^3 \amp = 3(2) + 2 \\
8 \amp = 6 + 2 \\
8 \amp = 8 \quad
\end{align*}
\begin{align*}
(0)^3 \amp = 3(0) + 2 \\
0 \amp = 0 + 2 \\
0 \amp \ne 2 \quad
\end{align*}
Since
\(8 = 8\) is a true statement,
\(y = 2\) satisfies the equation and is a solution. On the other hand,
\(y = 0\) leads to a false statement (
\(0 = 2\) ), showing that it is not a solution.
Subsection Solutions to Differential Equations
In differential equations, solutions are
functions rather than single values. To verify a solution, we substitute the function into the dependent variable in the equation. If both sides simplify to the same expression, then the function is a solution.
For instance, consider the differential equation:
\begin{equation*}
y' = 3y.
\end{equation*}
To check whether
\(y = x^3\) or
\(y = e^{3x}\) is a solution, we substitute each into the equation and compare both sides. In this case, we get:
\begin{align*}
\left[x^3\right]' \amp = 3(x^3) \\
3x^2 \amp \ne 3x^3 \quad
\end{align*}
\begin{align*}
\left[e^{3x}\right]' \amp = 3(e^{3x}) \\
3e^{3x} \amp = 3e^{3x} \quad
\end{align*}
For differential equations, a solution satisfies the equation if both sides simplify to the
exact same function. Since this only happened for the second function, we can conclude that
\(y = e^{3x}\) is a solution, while
\(y = x^3\) is not.
Checkpoint 22 .
(a)
To verify that a function satisfies a differential equation, you
plug the function into the equation,
simplify both sides of the equation, and
solve the simplified equation for \(x\text{.}\)
True.
Statements (1) and (2) are true, but (3) is not. Statement (3) should instead read:
(3) compare the functions on the left and right-hand sides of the simplified equation.
False.
Statements (1) and (2) are true, but (3) is not. Statement (3) should instead read:
(3) compare the functions on the left and right-hand sides of the simplified equation.
(b)
The function, \(y = x^2 + 3\text{,}\) is a solution to the equation
\begin{equation*}
\frac{dy}{dx} - 3 = 2x\text{.}
\end{equation*}
True.
The function \(y = x^2 + 3\) is not a solution since
\begin{align*}
\frac{dy}{dx} - 3 \amp = 2x \\
\frac{d}{dx}\left[x^2 + 3\right] - 3 \amp = 2x \\
2x - 3 \amp = 2x \quad \leftarrow \text{false}
\end{align*}
False.
The function \(y = x^2 + 3\) is not a solution since
\begin{align*}
\frac{dy}{dx} - 3 \amp = 2x \\
\frac{d}{dx}\left[x^2 + 3\right] - 3 \amp = 2x \\
2x - 3 \amp = 2x \quad \leftarrow \text{false}
\end{align*}
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