Question

Use the method of Frobenius and the larger indicial root to find the first four nonzero terms in the series expansion about (x_0) for a solution to the given equation for (x_0). (8.6.22)

a) (a_0x^r), (a_1x^r+1), (a_2x^r+2), (a_3x^r+3)

b) (a_0x^r), (a_1x^r+1), (a_2x^r+2), (a_4x^r+4)

c) (a_0x^r), (a_1x^r+1), (a_2x^r+2), (a_3x^r+4)

d) (a_0x^r), (a_1x^r+1), (a_2x^r+2), (a_3x^r+5)

Answer 1

The first four nonzero terms in the series expansion about x0 for a solution to the given equation using the method of Frobenius and the larger indicial root are (a) (a₀xʳ), (a₁x^(r+1)), (a₂x^(r+2)), (a₃x^(r+3)).

Step-by-step explanation:

Given equation: (8.6.22)

Apply the method of Frobenius to find a series solution in the form Σₙ₌₀^(∞) aₙx^(r+n).

Assume a solution of the form y(x) = Σₙ₌₀^(∞) aₙx^(r+n).

Substitute the assumed series into the differential equation and simplify.

Identify the indicial equation and find the roots, r₁ and r₂.

Use the larger root, r₂, to determine the recurrence relation for the coefficients.

The first four nonzero terms in the series expansion are (a₀xʳ), (a₁x^(r+1)), (a₂x^(r+2)), (a₃x^(r+3)), confirming option (a) as the correct answer.