Preview Activity 8.6.1.
In Chapter 7, we learned some of the many important applications of differential equations, and learned some approaches to solve or analyze them. Here, we consider an important approach that will allow us to solve a wider variety of differential equations.
Let’s consider the familiar differential equation from exponential population growth given by
\begin{equation}
y' = ky\text{,}\tag{8.6.1}
\end{equation}
where \(k\) is the constant of proportionality. While we can solve this differential equation using methods we have already learned, we take a different approach now that can be applied to a much larger set of differential equations. For the rest of this activity, let’s assume that \(k=1\text{.}\) We will use our knowledge of Taylor series to find a solution to the differential equation (8.6.1).
To do so, we assume that we have a solution \(y=f(x)\) and that \(f(x)\) has a Taylor series that can be written in the form
\begin{equation*}
y = f(x) = \sum_{k=0}^{\infty} a_kx^k\text{,}
\end{equation*}
where the coefficients \(a_k\) are undetermined. Our task is to find the coefficients.
(a)
Assume that we can differentiate a power series term by term. By taking the derivative of \(f(x)\) with respect to \(x\) and substituting the result into the differential equation (8.6.1), show that the equation
\begin{equation*}
\sum_{k=1}^{\infty} ka_kx^{k-1} = \sum_{k=0}^{\infty} a_kx^{k}
\end{equation*}
must be satisfied in order for \(f(x) = \sum_{k=0}^{\infty} a_kx^k\) to be a solution of the DE.
(b)
Two series are equal if and only if they have the same coefficients on like power terms. Use this fact to find a relationship between \(a_1\) and \(a_0\text{.}\)
(c)
(d)
(e)
(f)
Observe that there is a pattern in (b)-(e). Find a general formula for \(a_k\) in terms of \(a_0\text{.}\)
(g)
Write the series expansion for \(y\) using only the unknown coefficient \(a_0\text{.}\) From this, determine what familiar functions satisfy the differential equation (8.6.1). (Hint: Compare to a familiar Taylor series.)
