James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Chapter 17
Concept Check Answers (continued)
where k is the spring constant and x is the distance the spring
is stretched (or compressed) from its natural length. If there
are external forces acting on the spring, then the differential
equation is modified.
Second-order linear differential equations are also used to
analyze electrical circuits involving an electromotive force, a
resistor, an inductor, and a capacitor in series.
See the discussion in Section 17.3 for additional details.
5. How do you use power series to solve a differential equation?
We first assume that the differential equation has a power
series solution of the form
y − ò c n x n − c 0 1 c 1 x 1 c 2 x 2 1 c 3 x 3 1 ∙ ∙ ∙
n−0
Differentiating gives
and
y9 − ò nc n x n21 − ò sn 1 1dc n11x n
n−1
n−0
y0 − ò nsn 2 1dc n x n22 − ò sn 1 2dsn 1 1dc n12 x n
n−2
n−0
We substitute these expressions into the differential equation
and equate the coefficients of x n to find a recursion relation
involving the constants c n. Solving the recursion relation gives
a formula for c n and then
y − ò c n x n
n−0
is the solution of the differential equation.
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