The differential equation \[5\frac{{dx}}{{dt}} + 5x - \frac{{dy}}{{dt}} = {e^{2t}}\] in terms of differential operator is __________.

• A) \[(D + 1)5x - Dy = {e^t}\]
• B) \[(D - 1)5x - Dy = {e^{2t}}\]
• C) \[(D + 1)5x + Dy = {e^{2t}}\]
• D) \[(D + 1)5x - Dy = {e^{2t}}\]

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

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

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

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

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

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) 3
• B) none of these.
• C) 2
• D) 1

\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^{th}}
• C) n + 1
• D) n - 1

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

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

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) ordinary
• B) singular
• C)
• D)

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)

A second order linear differential equation of the form \[(1 - {x^2})y - 2xy' + n(n + 1)y = 0\] is called ______________ differential equation.

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

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

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

The linear normal form of 2\frac{{{d^2}y}}{{d{x^2}}} + 4\frac{{dy}}{{dx}} - 5y = 0 ,by using y = {x_1},y' = {x_1}^\prime {\text{and }}y'' = {x_2}^\prime , is__________.

• A) {x_1}^\prime = {x_2},{x_2}^\prime = - 2{x_2} - \frac{5}{2}{x_1}
• B) {x_1}^\prime = {x_2},{x_2}^\prime = 2{x_2} + \frac{5}{2}{x_1}
• C) {x_1}^\prime = {x_2},{x_2}^\prime = - 2{x_2} + \frac{5}{2}{x_1}
• D) {x_1}^\prime = {x_2},{x_2}^\prime = 2{x_2} - \frac{5}{2}{x_1}

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

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

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

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

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

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}

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

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

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

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