Definition 23.
A separable differential equation is a first-order differential equation that can be written in the form
\begin{equation*}
\frac{dy}{dx}=g(x)f(y)\text{.}
\end{equation*}
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Handout Lesson 4, Separable Differential Equations and Applications
Textbook Section(s).
This lesson is based on Section 1.4 of your textbook by Edwards, Penney, and Calvis.
What is a separable differential equation?
Solving separable differential equations.
\begin{gather}
\frac{dy}{dx}=k(y)g(x)\tag{✶}
\end{gather}
Claim For \(k(y) \neq 0\text{,}\)
\begin{equation*}
\int \frac{1}{k(y)} \, dy = \int g(x) \, dx
\end{equation*}
is a solution to (✶).
Justification:
Applications.
Several applications have DE models of the form
\begin{equation*}
\frac{dx}{dt}=kx
\end{equation*}
where \(k\) is a constant.
-
If \(k\) is positive, then \(x\) is .
-
If \(k\) is negative, then \(x\) is .
Here are some examples,
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Populations with constant birth and death rates follow the model:
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If an amount of money is accruing continuously compounded interest, the model\begin{equation*} \frac{dA}{dt}=rA \end{equation*}applies where
-
\(t\) is ,
-
\(r\) is , and
-
\(A\) is .
-
-
If a radioactive element is decaying, then the amount, \(N\text{,}\) of the original element remaining at time \(t\) follows the model:
Theorem 26.
The solutions of
\begin{equation*}
\frac{dx}{dt}=kx
\end{equation*}
for \(x(t) \neq 0\) are
\begin{equation*}
x(t)=x_0e^{kt}
\end{equation*}
Definition 27.
A quantity \(x\) grows exponentially if it satisfies the differential equation
\begin{equation*}
\frac{dx}{dt}=kx
\end{equation*}
and \(k\) is .
A quantity \(x\) decays exponentially if it satisfies the differential equation
\begin{equation*}
\frac{dx}{dt}=kx
\end{equation*}
and \(k\) is .
Newton’s Law of Cooling.
If \(T\) is the temperature of an object at time \(t\) in a room with a constant temperature \(A\text{,}\) then the temperature of the object is modeled by
\begin{equation*}
\frac{dT}{dt}=k(A-T)
\end{equation*}
