The differential equation $$y'' + (\cos x)y = 0$$ has ordinary point at ______.

• A) x=1
• B) x=0
• C) x=-1
• D) none of them

Recurrence relation for polynomials relates

• A) Equal polynomials.
• B) Polynomials of same degrees.
• C) Polynomials of different degrees.
• D) All options are true.

The second order linear differential equation $$x^2 \frac{{d^2 y}} {{dx^2 }} + x\frac{{dy}} {{dx}} + (x^2 - 49) = 0$$ is a Bessel equation of degree

• A) 1
• B) 3
• C) none of these.
• D) 2

The singular points need not to be _________ number.

• A) real
• B) whole
• C) complex
• D) natural

Operator method or systematic elimination method of solution of a system of linear homogeneous or linear non-homogeneous differential equations provides us _________ differential equation in one of the dependent variables that has not been eliminated.

• A) single
• B) many
• C)
• D)

$${\text{The}}\,\,{\text{Legendre}}\,\,{\text{polynomials}}\,\,{\text{can}}\,\,{\text{also}}\,\,{\text{be}}\,{\text{generated}}\,\,{\text{by}}\,\,{\text{Rodrigues}}\,\,{\text{formula}},\,\,{P_n}(x) = \frac{1}{{{2^n}\,n!}}\frac{{{d^n}}}{{d{x^n}}}{({x^2} - 1)^n}.$$

• A) $${\text{True}}$$
• B) $${\text{False}}$$
• C)
• D)

$${\text{The}}\,\,{\text{equation}},\,\,(1 - {x^2})\frac{{{d^2}y}}{{d{x^2}}} - 2x\frac{{dy}}{{dx}} + 30y = 0,\,\,{\text{has}}\,\,{\text{the}}\,\,{\text{singular}}\,\,{\text{point(s)}}\,\,{\text{at}}\,\,x = \,\,\_\_\_\_\_\_\_\_\_.$$

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

Operator method is based on process of

• A) None of these.
• B) Obtaining new dependent variables.
• C) Elimination of dependent variables.
• D) Addition of dependent variables.

Operator method or systematic elimination method of solution of a system of linear homogeneous or linear non-homogeneous differential equations is based on the process of systematic elimination of the _________ variables.

• A) dependent
• B) independent
• C)
• D)

To solve a differential equation $${a_2}(x){y^{''}} + {a_1}(x)y' + {a_0}(x)y = 0$$ about a regular singular point we employ the __________ theorem.

• A) none of them
• B) Bessel
• C) Frobenius
• D) Legendre

The differential equation $$(x^2 - 4)y^{//} + 2xy + y = 0$$ has singular point at

• A) x=0
• B) x=2 and x = -2
• C) x=1
• D) x=2

A singular point $$x = {x_0}$$ of the differential equation $${a_2}(x){y^{''}} + {a_1}(x)y' + {a_0}(x)y = 0$$ is said to be a __________ point if both $$(x - {x_0})P(x)$$ and $${(x - {x_0})^2}Q(x)$$ are analytical at $${x_0}$$ .

• A) none of them
• B) ordinary
• C) regular singular
• D) irregular singular

The simultaneous ordinary differential equations involve two or more equations that contain derivatives of two or more unknown functions of ____________ independent variable.

• A) 1
• B) 2
• C) 2
• D) infinite

Bessel equation occurs frequently in advances studies in ___________.

• A) applied mathematics
• B) engineering
• C) physics
• D) All of these

The linear normal form of $$4\frac{{{d^3}y}}{{d{t^3}}} + y = {e^t}$$ ,by using $$y = {x_1},y' = {x_1}^\prime {\text{, }}y'' = {x_2}^\prime {\text{and }}y''' = {x_3}^\prime$$ , is__________.

• A) $${x_1}^\prime = {x_2},{x_2}^\prime = {x_3},{x_3}^\prime = \frac{1}{4}{x_1} + \frac{1}{4}{e^t}$$
• B) $${x_1}^\prime = {x_2},{x_2}^\prime = {x_3},{x_3}^\prime = - \frac{1}{4}{x_1} - \frac{1}{4}{e^t}$$
• C) $${x_1}^\prime = {x_2},{x_2}^\prime = {x_3},{x_3}^\prime = \frac{1}{4}{x_1} - \frac{1}{4}{e^t}$$
• D) $${x_1}^\prime = {x_2},{x_2}^\prime = {x_3},{x_3}^\prime = \frac{{ - 1}}{4}{x_1} + \frac{1}{4}{e^t}$$

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 _________ radius of convergence.

• A) positive
• B) negative
• C)
• D)

x=0 is a(an) -------- point of the differential equation:$y'' + xy' + 2y = 0.$

• A) Irregular singular
• B) Regular singular
• C) Singular
• D) Ordinary

The solution of Legendre differential equation is denoted by ___________.

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

Spherical Bessel functions are used in many physical problems.

• A) True
• B) False
• C)
• D)

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

• A) complex
• B) singular
• C) constant
• D) analytic