Solution of the D.Equation $$ 4y^{//} + y = 0. $$ is

• A) $$ y(x) = c_1 \cos \frac{x} {2} $$
• B) $$ y(x) = c_1 Sin\frac{x} {2} $$
• C) $$ y(x) = c_1 \cos \frac{x} {2} + c_2 Sin\frac{x} {2} $$
• D) None of them.

Consider a mathematical statement, V = IR, where V be the constant of proportionality and it represents the voltage. The equation is called _________.

• A) Ohm’s law
• B) Coulomb’s law
• C) Newtown’s law
• D) Hooke’s law

Consider a power series \[x - \frac{{{x^2}}}{2} + \frac{{{x^3}}}{3} - ....\] represents _______.

• A) e
• B) ln (1+x)
• C) sin x
• D) cos x

The damping force is __________to the instantaneous velocity \[\frac{{dx}}{{dt}}\].

• A) None of these
• B) Constant
• C) Inverse proportional
• D) Proportional

The conversion of Cauchy Euler equation $${x^2}\frac{{{d^2}y}}{{d{x^2}}} - x\frac{{dy}}{{dx}} + y = \ln x$$ after putting $$x = {e^t}$$ becomes

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

A power series in (x-2) is an infinite series of the form \[\sum\limits_{n = 0}^\infty {{c_n}} {(x - 2)^n} = {c_0} + {c_1}(x - 2) + {c_2}{(x - 2)^2} + ....\] the number 2 is called _______.

• A) Center of power series
• B) Radius of power series
• C) none of these
• D) Base of power series

$$\begin{gathered} {\text{In}}\,\,{\text{the}}\,\,{\text{study}}\,\,{\text{of}}\,\,{\text{mechanics,}}\,\,{\text{cosider}}\,\,{\text{the}}\,\,{\text{damping}}\,\,{\text{force}}\,\,{\text{acting}}\,\,{\text{on}}\,\,{\text{a}}\,\,{\text{body}}\,\,{\text{i}}{\text{.e}}{\text{.}}\,\,\, - \beta \,{\left( {\frac{{dx}}{{dt}}} \right)^2},\,{\text{where}}\,\,\beta \,\,{\text{is}}\,\,{\text{a}}\,\, \hfill \\ {\text{damping}}\,\,{\text{constant}}\,\,{\text{and}}\,\,{\text{negative}}\,\,{\text{sign}}\,\,{\text{indicates}}\,\,{\text{that}}\,\,{\text{the}}\,\,{\text{damping}}\,\,{\text{force}}\,{\text{acts}}\,\,{\text{in}}\,\,{\text{a}}\,{\text{direction}}\,\,\_\_\_\_\_\_\_\,\, \hfill \\ {\text{to}}\,\,{\text{the}}\,\,{\text{direction}}\,\,{\text{of}}\,{\text{motion}}{\text{.}}\,\,\,\, \hfill \\\ \end{gathered}$$

• A) $${\text{opposite}}$$
• B) $${\text{same}}$$
• C)
• D)

The time interval between two successive maxima of x(t) is called ________.

• A) quasi period
• B) quasi frequency
• C)
• D)

The nature of the roots of the differential equation \[{x^2}\frac{{{d^2}y}}{{d{x^2}}} - 2x\frac{{dy}}{{dx}} - 4y = 0\] is __________.

• A) Real and distinct
• B) Conjugate complex
• C) none of these
• D) Real and repeated

Consider a power series \[\sum\limits_{n = 1}^\infty {\frac{{{{( - 1)}^{n + 1}}}}{{{n^2}}}} {x^n} = x - \frac{{{x^2}}}{{{2^2}}} + \frac{{{x^3}}}{{{3^2}}} + ....\] , then the center of series is _______.

• A) \[{x^2}\]
• B) x
• C) 1
• D) 0

If $${\lambda ^2} - {\omega ^2} = 0$$ and $$\beta = k$$ then the system is said to be _______ damped.

• A) under
• B) none of these
• C) over
• D) critically

\[{\text{The}}\,\,{\text{quantity}}\,\,X = L\gamma - \frac{1}{{C\gamma }}\,\,{\text{is}}\,\,{\text{called}}\,\,\_\_\_\_\_\_\_\_\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{circuit}}{\text{.}}\]

• A) \[{\text{impedance}}\]
• B) \[{\text{reactance}}\]
• C)
• D)

The D.E $$r\frac{{{d^2}u}}{{d{r^2}}} + 2\frac{{du}}{{dr}} = 0$$ where the variable $$r > 0$$ represents the radial distance measured ________ from the center of the sphere.

• A) inward
• B) outward
• C) origin
• D) center

The ________ force is proportional to the instantaneous velocity \[\frac{{dx}}{{dt}}\] .

• A) retarding
• B) restoring
• C) umdamped
• D) damped

The infinite series \sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n + 2} }} {{n^3 }}(x + 3)^n } . is a power series in x centered at

• A) 2
• B) -2
• C) 3
• D) -3

The periodic time is given by

• A) \frac{\omega }{{2\pi }}
• B) \frac{\pi }{\omega }
• C) 2\pi * \omega
• D) \frac{{2\pi }}{\omega }

The hook’s law states that the force F is proportional to the __________.

• A) Length
• B) None of these
• C) Weight
• D) Elongation

Consider a power series \sum\limits_{n = 1}^\infty {\frac{{{{( - 1)}^{n + 1}}}}{{{n^2}}}} {x^n} = x - \frac{{{x^2}}}{{{2^2}}} + \frac{{{x^3}}}{{{3^2}}} + .... , then the center of series is _______.

• A) {x^2}
• B) 1
• C) x
• D) 0

\begin{gathered} {\text{In}}\,\,{\text{the}}\,\,{\text{study}}\,\,{\text{of}}\,\,{\text{mechanics,}}\,\,{\text{cosider}}\,\,{\text{the}}\,\,{\text{damping}}\,\,{\text{force}}\,\,{\text{acting}}\,\,{\text{on}}\,\,{\text{a}}\,\,{\text{body}}\,\,{\text{i}}{\text{.e}}{\text{.}}\,\,\, - \beta \,{\left( {\frac{{dx}}{{dt}}} \right)^2},\,{\text{where}}\,\,\beta \,\,{\text{is}}\,\,{\text{a}}\,\, \hfill \\ {\text{damping}}\,\,{\text{constant}}\,\,{\text{and}}\,\,{\text{negative}}\,\,{\text{sign}}\,\,{\text{indicates}}\,\,{\text{that}}\,\,{\text{the}}\,\,{\text{damping}}\,\,{\text{force}}\,{\text{acts}}\,\,{\text{in}}\,\,{\text{a}}\,{\text{direction}}\,\,\_\_\_\_\_\_\_\,\, \hfill \\ {\text{to}}\,\,{\text{the}}\,\,{\text{direction}}\,\,{\text{of}}\,{\text{motion}}{\text{.}}\,\,\,\, \hfill \\\ \end{gathered}

• A) {\text{opposite}}
• B) {\text{same}}
• C)
• D)

If {\lambda ^2} - {\omega ^2} = 0 and \beta = k then the system is said to be _______ damped.

• A) over
• B) under
• C) critically
• D) none of these