Free Undamped motion mass-spring system with increase in spring constant 𝑥 is modelled by
the following differential equation 𝑦′′ + 𝑥𝑦 = 0. Find power series solution of the differential
equation about ordinary point 𝑥 = 0.
To find the power series solution of the differential equation about the ordinary point x = 0, we assume the solution y(x) can be written as a power series:
y(x) = ∑ (n=0 to ∞) a_n x^n
Taking the first and second derivatives of y(x):
y'(x) = ∑ (n=1 to ∞) n*a_n x^(n-1)
y''(x) = ∑ (n=2 to ∞) n*(n-1)*a_n x^(n-2)
Now substitute y(x), y'(x), and y''(x) into the differential equation:
∑ (n=2 to ∞) n*(n-1)*a_n x^(n-2) + x*∑ (n=0 to ∞) a_n x^n = 0
Simplify the expression:
∑ (n=0 to ∞) n*(n+1)*a_(n+2) x^n + ∑ (n=0 to ∞) a_n x^(n+1) = 0
Set up the indicial equation by equating the coefficients of x^0 to zero:
1*2*a_2 + a_0 = 0
a_2 = -a_0/2
Equating the coefficients of x^n to zero for n > 0 gives the recurrence relation:
n*(n+1)*a_(n+2) + a_n = 0
a_(n+2) = -a_n / (n*(n+1))
Using the recurrence relation and the initial condition a_0 = 1, we can calculate the coefficients of the power series solution:
a_2 = -1/2
a_4 = 1 / (2*3*4) = 1/24
a_6 = -1 / (4*5*6) = -1/120
and so on
The power series solution of the differential equation about the ordinary point x = 0 is:
y(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ...