JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II MATHEMATICS-III Previous Question Paper (March - 2021 - R15)

To find the series solution for the differential equation y'' + xy = 0, we assume a power series solution of the form:

Differentiating term by term:

Substituting these into the original equation:

Adjusting indices to equate powers of x to x^k:

1. For the first sum, let k = n - 2, so n = k + 2. When n=2, k=0.
2. For the second sum, let k = n + 1, so n = k - 1. When n=0, k=1.

The equation becomes:

Equating coefficients to zero:

1. a_2 = 0
2. a_{k+2} = -a_{k-1} / [(k+2)(k+1)] for k ≥ 1.

Using the recurrence relation:

1. If k=1: a_3 = -a_0 / (3·2)
2. If k=2: a_4 = -a_1 / (4·3)
3. If k=3: a_5 = -a_2 / (5·4) = 0 (since a_2 = 0)
4. If k=4: a_6 = -a_3 / (6·5) = a_0 / (6·5·3·2)

The general solution is y = a_0 y_1(x) + a_1 y_2(x), where:

The Bessel function J_n(x) satisfies the Bessel differential equation:

Let u = J_n(m₁x) and v = J_n(m₂x). These functions satisfy the following differential equations:

Dividing by x, we obtain:

Multiplying the first equation by v and the second by u, then subtracting the two equations, we get:

Integrating both sides from 0 to 1:

Evaluating the right-hand side at the limits:

1. At x = 0, the expression vanishes because x = 0 and J_n(0) is finite.
2. At x = 1, the expression becomes 1 * (m₁J_n'(m₁)J_n(m₂) - m₂J_n'(m₂)J_n(m₁)).

Since m₁ and m₂ are roots of J_n(x) = 0, we have J_n(m₁) = 0 and J_n(m₂) = 0. Thus, the right-hand side is zero. Since m₁ ≠ m₂, the integral must be zero: