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

{\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

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

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

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) 2
• C) 1
• D) none of these.

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

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

• A) \[n\]
• B) \[n + 1\]
• C) \[n - 1\]
• D) \[{n^2}\]

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

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

The regular 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

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

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

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

• A) 2
• B) -2
• C) -1
• D) 1

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) \[{x_1}^\prime = {x_2},{x_2}^\prime = - 2{x_2} + \frac{5}{2}{x_1}\]

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) $$ \frac{1} {{(x - 2)(x + 2)}} $$
• B) $$ \frac{1} {{(x - 2)(x + 2)^3 }} $$
• C) $$ \frac{1} {{(x + 2)^2 }} $$
• D) None 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 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) regular singular
• B) irregular singular
• C) ordinary
• D) none of them

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

The irregular 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

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

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

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

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

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

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