Jawaharlal Nehru Technological University Hyderabad B.Tech II Year MATHEMATICS-III Previous Question Paper (July/August - 2021 - R15)

To solve the differential equation

, we observe that x = 0 is a regular singular point. We seek a power series solution of the form , where .

Differentiating term by term:

Substituting into the original equation

:

Adjusting indices to align powers of

:

The indicial equation is obtained by setting the coefficient of the lowest power

to zero:

The recurrence relation for

is:

For

:

This yields

. The first solution is .

For

, the recurrence relation leads to a second linearly independent solution involving a logarithmic term due to the integer difference between roots . The general solution is , where .

To find the series solution of the differential equation

, we assume a power series solution of the form for n=0 to ∞. Differentiating term by term, we get and .

Substituting these into the differential equation:

Combining the first two sums:

Shifting the index of the first sum by letting k = n-1, so n = k+1:

Equating the coefficients of x^k to zero gives the recurrence relation:

Calculating the coefficients in terms of a₀:

• For k=0: a₁ = -4a₀ / 1² = -4a₀
• For k=1: a₂ = -4a₁ / 2² = -4(-4a₀) / 4 = 4a₀
• For k=2: a₃ = -4a₂ / 3² = -4(4a₀) / 9 = -16a₀ / 9

The general solution is

, which can be expressed using the Bessel function of the first kind of order zero, .