Bessel functions of index half an _________ integer are called Spherical Bessel functions.

• A) even
• B) odd
• C)
• D)

$$\begin{gathered} {\text{Every}}\,\,{n^{th}}\,\,{\text{order}}\,\,{\text{differential}}\,\,{\text{equation}},\,\,{y^{(n)}} = F\left( {t,\,y,\,y',\,...,\,{y^{(n - 1)}}} \right),\,\,{\text{as}}\,\,{\text{well}}\,\,{\text{as}}\,\,{\text{most}}\,\,\,{\text{systems}}\,\,{\text{of}}\,\, \hfill \\ {\text{differential}}\,\,{\text{equations,}}\,\,{\text{could}}\,\,{\text{be}}\,\,{\text{reduced}}\,\,{\text{to}}\,\,{\text{the}}\,\,\_\_\_\_\_\_\_\_\,\,{\text{order}}\,\,{\text{system}}{\text{.}}\,\, \hfill \\\ \end{gathered}$$

• A) $${\text{infinite}}$$
• B) $$n - 1$$
• C) $${n^{th}}$$
• D) $$n + 1$$

Solution of Legendre’s differential equation is called

• A) Legendre function.
• B) Legendre differential.
• C) Legendre polynomial.
• D) Recurrence relation.

$${\text{The}}\,\,{\text{equation}},\,\,({x^2} + 1)\frac{{{d^2}y}}{{d{x^2}}} + 2x\frac{{dy}}{{dx}} + 6y = 0,\,\,{\text{has}}\,\,{\text{the}}\,\,{\text{singular}}\,\,{\text{point(s)}}\,\,{\text{at}}\,\,x = \,\,\_\_\_\_\_\_\_\_\_.$$

• A) $$(III)\,\,\,\,\,\,\, - \,\,i$$
• B) $$(II)\,\,\,\,\,\,\,\,i$$
• C) $$(IV)\,\,\,\,\,\,\,{\text{Both}}\,\,{\text{(II)}}\,\,{\text{and}}\,\,{\text{(III)}}.$$
• D) $$(I)\,\,\,\,\,\,\, \pm \,1$$

A function f is said to be analytic at a point a if it can be represented by a power series in (x-a) with a positive radius of________.

• A) divergence
• B) convergence
• C)
• D)

If determinant $$\left| {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {D + 1} \\\ 0 \end{array}}&{\begin{array}{*{20}{c}} 2 \\\ {D - 1} \end{array}} \end{array}} \right| = 0$$ then __________.

• A) $${D^2} - 1$$
• B) $$(D + 1)(D - 1) + 2$$
• C) $${D^2} + 1$$
• D) $$(D + 1)(D - 1) - 2$$

By converting the equation $$(x^2 + 4x + 4)y^{//} + (x - 2)y^/ + y = 0$$ into general form $$y^{//} + P(x)y^/ + Q(x)y = 0$$ ; Q(x) is

• A) None of these.
• B) $$\frac{1} {{(x - 2)(x + 2)}}$$
• C) $$\frac{1} {{(x - 2)(x + 2)^3 }}$$
• D) $$\frac{1} {{(x + 2)^2 }}$$

$$\begin{gathered} {\text{A}}\,\,{\text{solution}}\,\,{\text{of}}\,{\text{a}}\,\,{\text{system}}\,\,{\text{of}}\,{\text{differential}}\,\,{\text{equations}}\,\,{\text{is}}\,{\text{a}}\,\,{\text{set}}\,\,{\text{of}}\,\,{\text{differentiable}}\,\,{\text{functions}},\,\, \hfill \\ x = f(t),\,\,y = g(t),\,\,x = h(t),...,\,\,{\text{those}}\,\,{\text{satisfy}}\,\,\_\_\_\_\_\_\_\_\_\,\,{\text{equation(s)}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{system}}\,\,{\text{on}}\,\,{\text{some}}\,\,{\text{interval}}\,\,I. \hfill \\\ \end{gathered}$$

• A) $${\text{one}}$$
• B) $${\text{each}}$$
• C) $${\text{at}}\,\,{\text{least}}\,\,{\text{one}}$$
• D) $${\text{two}}$$

The irregular singular point of the differential equation $${({x^2} - 4)^2}y + (x - 2)y' + y = 0$$ is _______.

• A) 1
• B) -2
• C) 2
• D) -1

$${\text{Legendres}}\,\,{\text{polynomials}}\,\,{\text{are}}\,\,{\text{specific}}\,\,\_\_\_\_\_\_\_\_\_\,\,{\text{degree}}\,\,{\text{polynomials}}.$$

• A) $$n + 1$$
• B) $$n - 1$$
• C) $$n$$
• D) $${n^2}$$

The solution of Bessel equation is denoted by ___________.

• A) $${L_v}(x)$$
• B) $${P_v}(x)$$
• C) $${B_v}(x)$$
• D) $${J_v}(x)$$

A point that is not an ordinary point is said to be singular point of the equation.

• A) True
• B) False
• C)
• D)

It is NOT always possible to solve the given system for the highest-order derivative of each dependent variable.

• A) False
• B) True
• C)
• D)

A second order differential equation of the form $${x^2}\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}} + ({x^2} - {v^2})y = 0$$ is called ________ differential equation.

• A) Legendre
• B) Picard
• C) Bessel
• D) Bernoulli

$${\text{The}}\,\,{\text{equation}},\,\,{({x^2} - 4)^2}\frac{{{d^2}y}}{{d{x^2}}} + (x - 2)\frac{{dy}}{{dx}} + y = 0,\,\,{\text{has}}\,\,{\text{the}}\,\,{\text{singular}}\,\,{\text{point(s)}}\,\,{\text{at}}\,\,x = \,\,\_\_\_\_\_\_\_\_\_.$$

• A) $$\pm \,\,i$$
• B) $$\pm \,1$$
• C) $$\pm \,2$$
• D) $$0$$

If n is positive integer, then the solution of Legendre’s differential equation is called a Legendre’s __________ of degree n.

• A) function
• B) polynomial
• C)
• D)

The Differential Equation $$(x^2 - 16)y'' + 2xy' + 16y = 0$$ has singularity at

• A) $$x = - 16$$
• B) $$x = \pm 4$$
• C) None of these.
• D) $$x = 16$$

The regular singular point of the differential equation $${({x^2} - 4)^2}y + (x - 2)y' + y = 0$$ is ________.

• A) 1
• B) -2
• C) -1
• D) 2

A point $${x_0}$$ is said to be a _______point of a differential equation $${a_2}(x){y^{''}} + {a_1}(x)y' + {a_0}(x)y = 0$$ if both P(x) and Q(x) are analytic at $${x_0}$$ .

• A) singular
• B) ordinary
• C)
• D)

Legendre polynomials are normal with respect to the weight function

• A) $$w(x) = \frac{{2n + 1}} {{n^2 }}$$
• B) $$w(x) = \frac{{n + 1}} {{2n}}$$
• C) $$w(x) = \frac{{2n^2 + 1}} {n}$$
• D) $$w(x) = \frac{{2n + 1}} {n}\,$$