Preview Activity 7.4.1.
In this preview activity, we explore whether certain differential equations are separable or not, and then revisit some key ideas from earlier work in integral calculus.
(a)
Which of the following differential equations are separable? If the equation is separable, write the equation in the revised form \(g(y) \frac{dy}{dt} = h(t)\text{.}\)
-
\(\displaystyle \frac{dy}{dt} = -3y\text{.}\)
-
\(\displaystyle \frac{dy}{dt} = ty - y\text{.}\)
-
\(\displaystyle \frac{dy}{dt} = t + 1\text{.}\)
-
\(\displaystyle \frac{dy}{dt} = t^2 - y^2\text{.}\)
(b)
Explain why any autonomous differential equation is guaranteed to be separable.
(c)
Why do we include the term “\(+C\)” in the expression
\begin{equation*}
\int x~dx =
\frac{x^2}{2} + C?
\end{equation*}
(d)
Suppose we know that a certain function \(f\) satisfies the equation
\begin{equation*}
\int f'(x)~dx = \int x~dx\text{.}
\end{equation*}
What can you conclude about \(f\text{?}\)
