8.6. POWER SERIES
519
given by
y
′ = k y,
(8.25)
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. We will use our knowledge of Taylor series to find a solution to the
differential equation (8.25).
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
y = f (x) =
∞
k=0
a k x
k ,
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 (8.25), show that the equation
∞
k=1
ka k x
k−1 =
∞
k=0
a k x
k
must be satisfied in order for f (x) =
∞
k=0 a k x 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 .
(c) Now write a 2 in terms of a 1 . Then write a 2 in terms of a 0 .
(d) Write a 3 in terms of a 2 . Then write a 3 in terms of a 0 .
(e) Write a 4 in terms of a 3 . Then write a 4 in terms of a 0 .
(f) Observe that there is a pattern in (b)-(e). Find a general formula for a k in terms of
a 0 .
(g) Write the series expansion for y using only the unknown coefficient a 0 . From this,
determine what familiar functions satisfy the differential equation (8.25). (Hint:
Compare to a familiar Taylor series.)
⊲⊳
519
given by
y
′ = k y,
(8.25)
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. We will use our knowledge of Taylor series to find a solution to the
differential equation (8.25).
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
y = f (x) =
∞
k=0
a k x
k ,
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 (8.25), show that the equation
∞
k=1
ka k x
k−1 =
∞
k=0
a k x
k
must be satisfied in order for f (x) =
∞
k=0 a k x 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 .
(c) Now write a 2 in terms of a 1 . Then write a 2 in terms of a 0 .
(d) Write a 3 in terms of a 2 . Then write a 3 in terms of a 0 .
(e) Write a 4 in terms of a 3 . Then write a 4 in terms of a 0 .
(f) Observe that there is a pattern in (b)-(e). Find a general formula for a k in terms of
a 0 .
(g) Write the series expansion for y using only the unknown coefficient a 0 . From this,
determine what familiar functions satisfy the differential equation (8.25). (Hint:
Compare to a familiar Taylor series.)
⊲⊳