For \(\frac{{{d^2}y}}{{d{x^2}}} + 4\frac{{dy}}{{dx}} + 3y = 3{e^{2x}}\), the particular integral is :

\({C_1}{e^{ - x}} + {C_2}{e^{ - 3x}}\)

General solution of the equation (x3 + 3xy2)dx + (3x2y + y3)dy = 0 is (c is a constant)

\(\frac{1}{5}\left( {8{x^4} + 6{x^3}y + {y^3}} \right) = c\)

\(\frac{1}{4}\left( {{x^4} + 6{x^2}{y^2} + {y^4}} \right) = c\)

\(\frac{1}{5}\left( {8{x^4} + 6{x^3}y + {y^3}} \right) = c\)

\(\frac{1}{{12}}\left( {{x^3} + 6x{y^2} + {y^4}} \right) = c\)

For the equation \(\frac{{dy}}{{dx}} + 7{x^2}y = 0\), if y(0) = \(\frac{{3}}{{7}}\), then the value of y(1) is

\(\frac{3}{7}{e^{ - \frac{3}{7}}}\)

\(\frac{7}{3}{e^{ - \frac{7}{3}}}\)

\(\frac{7}{3}{e^{ - \frac{3}{7}}}\)

\(\frac{3}{7}{e^{ - \frac{7}{3}}}\)

\(\frac{3}{7}{e^{ - \frac{3}{7}}}\)

\(\frac{{{\partial ^2}u}}{{\partial {t^2}}} = {c ^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}}\) represents the equation for

Vibration of a stretched string

Motion of a projectile in a gravitational field

Oscillation of a simple pendulum

A one-dimensional domain is discretized into N sub-domains of width Dx with node numbers i = 0, 1, 2, 3…………, N. If the time scale is discretized in steps of Dt, the forward-time and centered-space finite difference approximation at ith node and nth time step, for the partial differential equation \(\frac{{\partial v}}{{\partial t}} = \beta \frac{{{\partial ^2}v}}{{\partial {x^2}}}\) is

\(\frac{{V_i^{\left( n \right)} - V_i^{\left( n \right)}}}{{{\rm{\Delta }}t}} = \beta \left[ {\frac{{V_{i + 1}^{\left( n \right)} - 2V_i^{\left( n \right)} + V_{i - 1}^{\left( n \right)}}}{{{{\left( {{\rm{\Delta }}x} \right)}^2}}}} \right]\)

\(\frac{{V_{i + 1}^{\left( {n + 1} \right)} - V_i^{\left( N \right)}}}{{{\rm{\Delta }}t}} = \beta \left[ {\frac{{V_{i + 1}^{\left( n \right)} - 2V_i^{\left( n \right)} + V_{i - 1}^{\left( n \right)}}}{{2{\rm{\Delta }}x}}} \right]\)

\(\frac{{V_i^{\left( n \right)} - V_i^{\left( {n - 1} \right)}}}{{2{\rm{\Delta }}t}} = \beta \left[ {\frac{{V_{i + 1}^{\left( n \right)} - 2V_i^{\left( n \right)} + V_{i - 1}^{\left( n \right)}}}{{2{\rm{\Delta }}x}}} \right]\)

\(\frac{{V_i^{\left( {n + 1} \right)} - V_i^{\left( n \right)}}}{{{\rm{\Delta }}t}} = \beta \left[ {\frac{{V_{i + 1}^{\left( n \right)} - 2V_i^{\left( n \right)} + V_{i - 1}^{\left( n \right)}}}{{{{\left( {{\rm{\Delta }}x} \right)}^2}}}} \right]\)

\(\frac{{V_i^{\left( n \right)} - V_i^{\left( n \right)}}}{{{\rm{\Delta }}t}} = \beta \left[ {\frac{{V_{i + 1}^{\left( n \right)} - 2V_i^{\left( n \right)} + V_{i - 1}^{\left( n \right)}}}{{{{\left( {{\rm{\Delta }}x} \right)}^2}}}} \right]\)

If y = sin (ln x), then which one of the following is correct?

\({x^2}\frac{{{d^2}y}}{{d{x^2}}} - x\frac{{dy}}{{dx}} + y = 0\)

\(\frac{{{d^2}y}}{{d{x^2}}} + y = 0\)

\(\frac{{{d^2}y}}{{d{x^2}}} = 0\)

\({x^2}\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}} + y = 0\)

\({x^2}\frac{{{d^2}y}}{{d{x^2}}} - x\frac{{dy}}{{dx}} + y = 0\)

Consider the following differential equation:\(x\left( {ydx + xdy} \right)\cos \frac{y}{x} = y\left( {xdy - ydx} \right)\sin \frac{y}{x}\)Which of the following is the solution of the above equation (c is an arbitrary constant)?

\(\frac{x}{y}\cos \frac{y}{x} = c\)

\(\frac{x}{y}\sin \frac{y}{x} = c\)

A complete solution of partial differential equation\(x\;\frac{{\partial z}}{{\partial x}} + y\frac{{\partial z}}{{\partial y}} - z = \frac{{\partial z}}{{\partial x}}\frac{{\partial z}}{{\partial y}}\) will be:

z = x2 + y2 - 2ab, where ab b are arbitrary constants.

z = ax + by - ab, where a, b are arbitrary constants.

z = x2 + y2 - 2ab, where ab b are arbitrary constants.

z = ax2 + by2 + abxy, where a, b are arbitrary constants.

z = ax - by + ab, where a, b are arbitrary constants.

Euler’s equation is

\(\frac{{dp}}{{\rho}}-gdz\;+\;vdv=0\)

\(\frac{{dp}}{{\rho}}+gdz\;+\;vdv=0\)

General solution of the differential equation (D2 - 2D + 1) y = ex is

\(A{e^x} + B{e^{ - x}} + \frac{{{x^2}}}{2}{e^x}\)

\({e^x}\left( {A + Bx} \right) - \frac{{{x^2}}}{2}{e^x}\)

\(A{e^x} + B{e^{ - x}} - \frac{{{x^2}}}{2}{e^x}\)

\({e^x}\left( {A + Bx} \right) + \frac{{{x^2}}}{2}{e^x}\)

If \(U = \mathop e\nolimits^{\frac{{\mathop x\nolimits^2 }}{{\mathop y\nolimits^2 }}} + \mathop e\nolimits^{\frac{{\mathop y\nolimits^2 }}{{\mathop x\nolimits^2 }}} \)then \(x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}}\)is

\(\frac{1}{y}\mathop e\nolimits^{\frac{{\mathop x\nolimits^2 }}{{\mathop y\nolimits^2 }}} + \frac{1}{x}\mathop e\nolimits^{\frac{{\mathop y\nolimits^2 }}{{\mathop x\nolimits^2 }}} \)

\(\frac{{\mathop \partial \nolimits^2 u}}{{\partial x\partial y}}\)

Consider two differential equations:I: ey dx + (xey + 2y)dy = 0II: xdy + 2y2 dx = 0Which of the following statements is correct?

Both I and II are exact differential equation.

I is not exact but Ii is an exact differential equation.

Neither I nor II are exact differential equations

I is exact but II is not an exact differential equation.

In the following partial differential equation, θ is a function of t and z, and D and K are functions of θ\(D\left( \theta \right)\frac{{{\delta ^2}\theta }}{{\delta {z^2}}} + \frac{{\delta K\left( \theta \right)}}{{\delta z}} - \frac{{\delta \theta }}{{\delta t}} = 0\)The above equation is

a second degree non-linear equation

a second degree linear equation

a second order non-linear equation

a second degree non-linear equation

Consider the differential equation \(\frac{{dy}}{{dx}} = \left( {1 + {y^2}} \right)x\). The general solution with constant C is

\(y = \tan \frac{{{x^2}}}{2} + tanc\)

\(y = {\tan ^2}\left( {\frac{x}{2} + c} \right)\)

\(y = {\tan ^2}\left( {\frac{x}{2}} \right) + c\)

\(y = \tan \left( {\frac{{{x^2}}}{2} + c} \right)\)

If y = a cos 2x + b sin 2x, then

\(\frac{{{d^2}y}}{{d{x^2}}} + y = 0\)

\(\frac{{{d^2}y}}{{d{x^2}}} + 2y = 0\)

\(\frac{{{d^2}y}}{{d{x^2}}} - 4y = 0\)

\(\frac{{{d^2}y}}{{d{x^2}}} + 4y = 0\)

If u = xy choose the correct option

\(\frac{{{\partial ^3}u}}{{\partial x\partial {y^2}}} = \frac{{{\partial ^3}u}}{{\partial y\partial x\partial y}}\)

\(\frac{{{\partial ^3}u}}{{\partial {x^2}\partial y}} = \frac{{{\partial ^3}u}}{{\partial x\partial y\partial x}}\)

\(\frac{{{\partial ^3}u}}{{\partial x\partial {y^2}}} = \frac{{{\partial ^3}u}}{{\partial y\partial x\partial y}}\)

\(\frac{{{\partial ^3}u}}{{\partial x\partial {y^2}}} = \frac{{{\partial ^3}u}}{{\partial {x^2}\partial y}}\)

\(\frac{{{\partial ^2}u}}{{\partial {x^2}}} = \frac{{{\partial ^2}u}}{{\partial {y^2}}}\)

Consider the following equations\(\frac{{\partial V\left( {x,y} \right)}}{{\partial x}} = p{x^2} + {y^2} + 2xy\)\(\frac{{\partial V\left( {x,y} \right)}}{{\partial y}} = {x^2} + q{y^2} + 2xy\)Where p and q are constants. V (x, y) that satisfies the above equations is

\(p\frac{{{x^3}}}{3} + q\frac{{{y^3}}}{3} + 2xy + 6\)

\(p\frac{{{x^3}}}{3} + q\frac{{{y^3}}}{3} + 5\)

\(p\frac{{{x^3}}}{3} + q\frac{{{y^3}}}{3} + {x^2}y + x{y^2} + xy\)

\(p\frac{{{x^3}}}{3} + q\frac{{{y^3}}}{3} + {x^2}y + x{y^2}\)

Consider the differential equation \({x^2}\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}} - 4y = 0\) with the boundary conditions of y(0) = 0 and y(1) = 1. The complete solution of the differential equation is

\(\sin \left( {\frac{{\pi x}}{2}} \right)\)

\({e^x}\sin \left( {\frac{{\pi x}}{2}} \right)\)

\({e^{ - x}}\sin \left( {\frac{{\pi x}}{2}} \right)\)

A first order differential is given by \(\frac{{dy}}{{dx}} = {e^{ - 3x}}\) Find the possible solution out of the following assuming K as constant.

\(- \frac{1}{3}{e^{ - 3x}} + K\)

If \({I_n} = \mathop \smallint \limits_0^{\pi /4} {\tan ^n}\theta d\theta\) then the following reduction formula exists:

\({I_n} + {I_{n - 2}} + \frac{1}{{n + 1}} = 0\)

\({I_n} + {I_{n - 2}} = \frac{1}{{n - 1}}\)

\({I_n} + {I_{n - 2}} + \frac{1}{{n + 1}} = 0\)

\({I_n} - {I_{n - 2}} = \frac{1}{{n + 1}}\)

\({I_n} - {I_{n - 2}} + \frac{1}{{n - 1}} = 0\)

Determine the partial-fraction expansion for:\(TF,\;G\;\left( s \right) = \frac{{32}}{{s\left( {s + 4} \right)\left( {s + 8} \right)}}\)

\(\frac{1}{s} + \frac{{ - 1}}{{s + 4}} + \frac{2}{{s + 8\;}}\)

\(\frac{1}{s} + \frac{2}{{s + 4\;}} + \frac{1}{{s + 8\;}}\)

\(\frac{1}{s} + \frac{{ - 1}}{{s + 4}} + \frac{1}{{s + 8}}\)

\(\frac{1}{s} + \frac{{ - 2}}{{s + 4\;}} + \frac{1}{{s + 8\;}}\)

If integrating factor of x(1 - x2) dy + (2x2y - y - ax3) dx = 0 is \({e^{\int {Pdx} }}\), then P is equal to

\(\frac{{2{x^2} - a{x^3}}}{{x(1 - {x^2})}}\)

\(\frac{{2{x^2} - 1}}{{a{x^3}}}\)

\(\frac{{2{x^2} - 1}}{{x(1 - {x^2})}}\)

Order of the partial differential equation is

the order of the lowest derivative appears in it

the order of the highest derivative appears in it

the order of the lowest derivative appears in it

the order of any derivative appears in it

An ordinary differential equation is given below \(6\frac{{{d^2}y}}{{d{x^2}}} + \frac{{dy}}{{dx}} - y = 0\) The general solution of the above equation (with constants C1 and C2), is

\(y\left( x \right) = \;{C_1}x{e^{ - \frac{x}{3}}} + {C_2}{e^{\frac{x}{2}}}\)

\(y\left( x \right) = \;{C_1}{e^{ - \frac{x}{3}}} + {C_2}{e^{\frac{x}{2}}}\)

\(y\left( x \right) = \;{C_1}{e^{\frac{x}{3}}} + {C_2}{e^{ - \frac{x}{2}}}\)

\(y\left( x \right) = \;{C_1}x{e^{ - \frac{x}{3}}} + {C_2}{e^{\frac{x}{2}}}\)

\(y\left( x \right) = \;{C_1}{e^{ - \frac{x}{3}}} + {C_2}x{e^{\frac{x}{2}}}\)

If a particular integral of the differential equation \((D^2+2D-1)y=e^{ax}\) is \(\frac{{ - 4}}{7}{e^{ax}}\), then the value of 'a' is

\(\left( {\frac{1}{2}~or~\frac{{ - 3}}{2}} \right)\)

\(\left( {\frac{1}{2}~or~\frac{3}{2}} \right)\)

\(\left( {\frac{{ - 1}}{2}~or~\frac{{ - 3}}{2}} \right)\)

\(\left( {\frac{1}{2}~or~\frac{{ - 3}}{2}} \right)\)

\(\left( {\frac{{ - 1}}{2}~or~\frac{3}{2}} \right)\)

88 + Mcqs in Ordinary Differential Equations in BITSAT Page 1 McqOptions

1.	For \(\frac{{{d^2}y}}{{d{x^2}}} + 4\frac{{dy}}{{dx}} + 3y = 3{e^{2x}}\), the particular integral is :
A.	\(\frac{1}{{15}}{e^{2x}}\)
B.	\(\frac{1}{5}{e^{2x}}\)
C.	\(3{e^{2x}}\)
D.	\({C_1}{e^{ - x}} + {C_2}{e^{ - 3x}}\)
Answer» C. \(3{e^{2x}}\)

2.	Consider the differential equation given below:\(\dfrac{dy}{dx} + \dfrac{x}{1-x^2} y= x\sqrt{y}\)The integrating factor of the differential equation is:
A.	(1 - x2)-1/4
B.	(1 - x2)-1/2
C.	(1 - x2)-3/4
D.	(1 - x2)-3/2
Answer» B. (1 - x2)-1/2

5.	If \(\rm \dfrac{d^2y}{dx^2}+\sin x = 0\), then the solution of this differential equation is -
A.	\(\rm y=\sin x + c_1 x+c_2\)
B.	\(\rm y=\cos x+c_1 x^2+c_2\)
C.	\(\rm y=\tan x+ c\)
D.	\(\rm y=\log \sin x + cx\)
Answer» B. \(\rm y=\cos x+c_1 x^2+c_2\)

6.	\(\frac{{{\partial ^2}u}}{{\partial {t^2}}} = {c ^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}}\) represents the equation for
A.	Vibration of a stretched string
B.	Heat flow of a thin rod
C.	Motion of a projectile in a gravitational field
D.	Oscillation of a simple pendulum
Answer» B. Heat flow of a thin rod

7.	Consider the following partial differential equation for with the constant c > 1:Solution of this equation is \(\frac{{\partial u}}{{\partial y}} + c\frac{{\partial u}}{{\partial x}} = 0\)
A.	u(x,y) = f(x + cy)
B.	u(x, y) = f(x – cy)
C.	u(x, y) = f(cx + y)
D.	u(x,y) = f(cx – y)
Answer» C. u(x, y) = f(cx + y)

8.	Determine the degree and order of the given differential equation respectively. \({\rm{y}} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + {\left( {\frac{{{\rm{dx}}}}{{{\rm{dy}}}}} \right)^2}?\)
A.	1, 2
B.	2, 1
C.	1, 4
D.	4, 1
Answer» E.

9.	If the characteristic equation of the differential equation \(\frac{{{d^2}y}}{{d{x^2}}} + 2\alpha \frac{{dy}}{{dx}} + y = 0\) has two equal roots, then the values of α are
A.	± 1
B.	0, 0
C.	± j
D.	± 1/2
Answer» B. 0, 0

10.	A particle starts from origin with a velocity (in m/s) given by the equation \(\rm \frac{dx}{dt}=x+1\). The time (in seconds) taken by the particle to traverse a distance of 24 m is:
A.	ln 24
B.	ln 5
C.	2 ln 5
D.	2 ln 4
Answer» D. 2 ln 4

13.	Consider the following statements:1. The general solution of \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} = {\rm{f}}\left( {\rm{x}} \right) + {\rm{x}}\) is of the form y = g(x) + c, where c is an arbitrary constant.2. The degree of \({\left( {\frac{{{\rm{dx}}}}{{{\rm{dy}}}}} \right)^2} = {\rm{f}}\left( {\rm{x}} \right)\) is 2.Which of the above statements is/are correct?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» D. Neither 1 nor 2

14.	Consider the following in respect of the differential equation:\(\frac{{{d^2}y}}{{d{x^2}}} + 2{\left( {\frac{{dy}}{{dx}}} \right)^2} + 9y = x\)1. The degree of the differential equation is 1.2. The order of the differential equation is 2.Which of the above statements is/are correct?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» D. Neither 1 nor 2

15.	If \(u = \sin^{-1} \left(\dfrac{x}{y}\right) + \tan^{-1} \left(\dfrac{y}{x}\right)\), then the value of \(x \dfrac{\partial u}{\partial x} + y \dfrac{\partial u}{\partial y}\) is
A.	U
B.	2u
C.	3u
D.	0
Answer» E.

18.	It is given that y” + 2y’ + y = 0, y(0) = 0, y(1) = 0. What is y(0.5) ?
A.	0
B.	0.37
C.	0.62
D.	1.13
Answer» B. 0.37

19.	Euler’s equation is
A.	\(\frac{{dp}}{{\rho}}-gdz\;+\;vdv=0\)
B.	\(\frac{{dp}}{{\rho}}+gdz\;+\;vdv=0\)
C.	ρdp + gdz + vdv = 0
D.	none of these
Answer» C. ρdp + gdz + vdv = 0

26.	If \(3e^x\ tan\ ydx + (1-e^x)sec^2\ ydy = 0\),then y is
A.	tan y = c(1 + ex)3
B.	y = c(1 - ex)3
C.	log y = c(1 + ex)3
D.	tan y = c(1 - ex)3
Answer» E.

30.	Consider two solutions x(t) = x1(t) and x(t) = x2(t) of the differential equation \(\frac{{{d^2}x\left( t \right)}}{{d{t^2}}} + x\left( t \right) = 0,\;t > 0\), such that\(\ x_{1}(0)=1,\ {\left. {\frac{{d{x_1}\left( t \right)}}{{dt}}} \right\|_{t = 0}} = 0,\;{x_2}\left( 0 \right) = 0,{\left. {\frac{{d{x_2}\left( t \right)}}{{dt}}} \right\|_{t = 0}} = 1\)The Wronskian \(W\left( t \right) = \left\| {\begin{array}{*{20}{c}} {{x_1}\left( t \right)}&{{x_2}\left( t \right)}\\ {\frac{{d{x_1}\left( t \right)}}{{dt}}}&{\frac{{d{x_2}\left( t \right)}}{{dt}}} \end{array}} \right\|\) at \(t = \frac{\pi}{2}\) is
A.	1
B.	-1
C.	0
D.	π/2
Answer» B. -1

20.	Consider the following differential equation\((1 + y) \frac{dy}{dx} = y\)The solution of the equation that satisfies the condition y(1) = 1 is
A.	(1 + y)ey = 2ex
B.	yey = ex
C.	y2ey = ex
D.	2yey = ex + e
Answer» C. y2ey = ex

23.	If x dy = y dx + y2 dy, y > 0 and y (1) = 1, then what is y (-3) equal to?
A.	3 only
B.	-1 only
C.	Both -1 and 3
D.	Neither -1 nor 3
Answer» B. -1 only

25.	A function \(y\left( t \right)\) such that y(0) = 1 and y(1) = 3e-1, is a solution of the differential equation\(\frac{{{d^2}y}}{{d{t^2}}} + 2\frac{{dy}}{{dt}} + y = 0\;\). Then \(y\left( 2 \right)\) is
A.	\(5{e^{ - 1}}\)
B.	\(5{e^{ - 2}}\)
C.	\(7{e^{ - 1}}\)
D.	\(7{e^{ - 2}}\)
Answer» C. \(7{e^{ - 1}}\)

27.	Consider the differential equation \({x^2}\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}} - y = 0\). Which of the following is a solution to this differential equation for x > 0?
A.	ex
B.	x2
C.	1/x
D.	In x
Answer» D. In x

28.	Differential equation \(y^3\frac{dy}{dx}\;+\;x^3=0\), y(0) = 1 has a solution given by y. The value of y(-1) is:
A.	-1
B.	0
C.	1
D.	2
Answer» C. 1

29.	In the following partial differential equation, θ is a function of t and z, and D and K are functions of θ\(D\left( \theta \right)\frac{{{\delta ^2}\theta }}{{\delta {z^2}}} + \frac{{\delta K\left( \theta \right)}}{{\delta z}} - \frac{{\delta \theta }}{{\delta t}} = 0\)The above equation is
A.	a second order linear equation
B.	a second degree linear equation
C.	a second order non-linear equation
D.	a second degree non-linear equation
Answer» D. a second degree non-linear equation

31.	Consider a function u which depends on position x and time t. The partial differential Equation \(\frac{{\partial u}}{{\partial t}} = \frac{{{\partial ^2}u}}{{\partial {x^2}}}\) is known as the:
A.	Wave equation
B.	Heat equation
C.	Laplace equation
D.	Elasticity equation
Answer» C. Laplace equation

32.	For the Ordinary Differential Equation \(\frac{{{d^2}x}}{{d{t^2}}} - 5\frac{{dx}}{{dt}} + 6x = 0,\) with initial conditions x(0) = 0 and \(\frac{{dx}}{{dt}}\left( 0 \right) = 10,\) the solution is
A.	-5e2t +6e3t
B.	5e2t = + 6e3t
C.	-10e2t + 10e3t
D.	10e2t + 10e+3t
Answer» D. 10e2t + 10e+3t

36.	If \(\rm z = \cos \left( {\frac{x}{y}} \right) + \sin \left( {\frac{x}{y}} \right)\)then \(x\frac{{\partial z}}{{\partial x}} + y\frac{{\partial z}}{{\partial y}} = \)
A.	1
B.	cos x
C.	sin y
D.	0
Answer» E.

39.	A first order differential is given by \(\frac{{dy}}{{dx}} = {e^{ - 3x}}\) Find the possible solution out of the following assuming K as constant.
A.	e3x + K
B.	\(- \frac{1}{3}{e^{ - 3x}} + K\)
C.	\(- \frac{1}{3}{e^{3x}} + K\)
D.	-3e3x + K
Answer» C. \(- \frac{1}{3}{e^{3x}} + K\)

Explore topic-wise MCQs in BITSAT.

For \(\frac{{{d^2}y}}{{d{x^2}}} + 4\frac{{dy}}{{dx}} + 3y = 3{e^{2x}}\), the particular integral is :

Consider the differential equation given below:\(\dfrac{dy}{dx} + \dfrac{x}{1-x^2} y= x\sqrt{y}\)The integrating factor of the differential equation is:

General solution of the equation (x3 + 3xy2)dx + (3x2y + y3)dy = 0 is (c is a constant)

For the equation \(\frac{{dy}}{{dx}} + 7{x^2}y = 0\), if y(0) = \(\frac{{3}}{{7}}\), then the value of y(1) is

If \(\rm \dfrac{d^2y}{dx^2}+\sin x = 0\), then the solution of this differential equation is -

\(\frac{{{\partial ^2}u}}{{\partial {t^2}}} = {c ^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}}\) represents the equation for

Consider the following partial differential equation for with the constant c > 1:Solution of this equation is \(\frac{{\partial u}}{{\partial y}} + c\frac{{\partial u}}{{\partial x}} = 0\)

Determine the degree and order of the given differential equation respectively. \({\rm{y}} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + {\left( {\frac{{{\rm{dx}}}}{{{\rm{dy}}}}} \right)^2}?\)

If the characteristic equation of the differential equation \(\frac{{{d^2}y}}{{d{x^2}}} + 2\alpha \frac{{dy}}{{dx}} + y = 0\) has two equal roots, then the values of α are

A particle starts from origin with a velocity (in m/s) given by the equation \(\rm \frac{dx}{dt}=x+1\). The time (in seconds) taken by the particle to traverse a distance of 24 m is:

If y = sin (ln x), then which one of the following is correct?

Consider the following in respect of the differential equation:\(\frac{{{d^2}y}}{{d{x^2}}} + 2{\left( {\frac{{dy}}{{dx}}} \right)^2} + 9y = x\)1. The degree of the differential equation is 1.2. The order of the differential equation is 2.Which of the above statements is/are correct?

If \(u = \sin^{-1} \left(\dfrac{x}{y}\right) + \tan^{-1} \left(\dfrac{y}{x}\right)\), then the value of \(x \dfrac{\partial u}{\partial x} + y \dfrac{\partial u}{\partial y}\) is

Consider the following differential equation:\(x\left( {ydx + xdy} \right)\cos \frac{y}{x} = y\left( {xdy - ydx} \right)\sin \frac{y}{x}\)Which of the following is the solution of the above equation (c is an arbitrary constant)?

A complete solution of partial differential equation\(x\;\frac{{\partial z}}{{\partial x}} + y\frac{{\partial z}}{{\partial y}} - z = \frac{{\partial z}}{{\partial x}}\frac{{\partial z}}{{\partial y}}\) will be:

It is given that y” + 2y’ + y = 0, y(0) = 0, y(1) = 0. What is y(0.5) ?

Euler’s equation is

Consider the following differential equation\((1 + y) \frac{dy}{dx} = y\)The solution of the equation that satisfies the condition y(1) = 1 is

General solution of the differential equation (D2 - 2D + 1) y = ex is

If \(U = \mathop e\nolimits^{\frac{{\mathop x\nolimits^2 }}{{\mathop y\nolimits^2 }}} + \mathop e\nolimits^{\frac{{\mathop y\nolimits^2 }}{{\mathop x\nolimits^2 }}} \)then \(x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}}\)is

If x dy = y dx + y2 dy, y > 0 and y (1) = 1, then what is y (-3) equal to?

Consider two differential equations:I: ey dx + (xey + 2y)dy = 0II: xdy + 2y2 dx = 0Which of the following statements is correct?

A function \(y\left( t \right)\) such that y(0) = 1 and y(1) = 3e-1, is a solution of the differential equation\(\frac{{{d^2}y}}{{d{t^2}}} + 2\frac{{dy}}{{dt}} + y = 0\;\). Then \(y\left( 2 \right)\) is

If \(3e^x\ tan\ ydx + (1-e^x)sec^2\ ydy = 0\),then y is

Consider the differential equation \({x^2}\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}} - y = 0\). Which of the following is a solution to this differential equation for x > 0?

Differential equation \(y^3\frac{dy}{dx}\;+\;x^3=0\), y(0) = 1 has a solution given by y. The value of y(-1) is:

In the following partial differential equation, θ is a function of t and z, and D and K are functions of θ\(D\left( \theta \right)\frac{{{\delta ^2}\theta }}{{\delta {z^2}}} + \frac{{\delta K\left( \theta \right)}}{{\delta z}} - \frac{{\delta \theta }}{{\delta t}} = 0\)The above equation is

Consider a function u which depends on position x and time t. The partial differential Equation \(\frac{{\partial u}}{{\partial t}} = \frac{{{\partial ^2}u}}{{\partial {x^2}}}\) is known as the:

For the Ordinary Differential Equation \(\frac{{{d^2}x}}{{d{t^2}}} - 5\frac{{dx}}{{dt}} + 6x = 0,\) with initial conditions x(0) = 0 and \(\frac{{dx}}{{dt}}\left( 0 \right) = 10,\) the solution is

Consider the differential equation \(\frac{{dy}}{{dx}} = \left( {1 + {y^2}} \right)x\). The general solution with constant C is

If y = a cos 2x + b sin 2x, then

If u = xy choose the correct option

If \(\rm z = \cos \left( {\frac{x}{y}} \right) + \sin \left( {\frac{x}{y}} \right)\)then \(x\frac{{\partial z}}{{\partial x}} + y\frac{{\partial z}}{{\partial y}} = \)

Consider the following equations\(\frac{{\partial V\left( {x,y} \right)}}{{\partial x}} = p{x^2} + {y^2} + 2xy\)\(\frac{{\partial V\left( {x,y} \right)}}{{\partial y}} = {x^2} + q{y^2} + 2xy\)Where p and q are constants. V (x, y) that satisfies the above equations is

Consider the differential equation \({x^2}\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}} - 4y = 0\) with the boundary conditions of y(0) = 0 and y(1) = 1. The complete solution of the differential equation is

A first order differential is given by \(\frac{{dy}}{{dx}} = {e^{ - 3x}}\) Find the possible solution out of the following assuming K as constant.

If \({I_n} = \mathop \smallint \limits_0^{\pi /4} {\tan ^n}\theta d\theta\) then the following reduction formula exists:

Determine the partial-fraction expansion for:\(TF,\;G\;\left( s \right) = \frac{{32}}{{s\left( {s + 4} \right)\left( {s + 8} \right)}}\)

Every Cauchy sequence is

Consider the differential equation \(\left( {{t^2} - 81} \right)\frac{{dy}}{{dt}} + 5ty = \sin \left( t \right)\) with y(1) = 2π. There exists a unique solution for this differential equation when t belongs to the interval

If integrating factor of x(1 - x2) dy + (2x2y - y - ax3) dx = 0 is \({e^{\int {Pdx} }}\), then P is equal to

Find solution of the differential equation \(\rm xdy+ydx\over y\) = 0

Order of the partial differential equation is

An ordinary differential equation is given below \(6\frac{{{d^2}y}}{{d{x^2}}} + \frac{{dy}}{{dx}} - y = 0\) The general solution of the above equation (with constants C1 and C2), is

A simple mass-spring oscillatory system consists of a mass m, suspended from a spring of stiffness k. Considering z as the displacement of the system at any time t, the equation of motion for the free vibration of the system is \(m\ddot z + kz = 0\). The natural frequency of the system is

If a particular integral of the differential equation \((D^2+2D-1)y=e^{ax}\) is \(\frac{{ - 4}}{7}{e^{ax}}\), then the value of 'a' is

Find the solution of the differential equation \(\frac{dy}{dx}+2y=1\) such that y(0) = 0.

42.	Every Cauchy sequence is
A.	Unbounded
B.	Bounded
C.	Infinite
D.	None of these
Answer» C. Infinite

43.	Consider the differential equation \(\left( {{t^2} - 81} \right)\frac{{dy}}{{dt}} + 5ty = \sin \left( t \right)\) with y(1) = 2π. There exists a unique solution for this differential equation when t belongs to the interval
A.	(–2, 2)
B.	(–10, 10)
C.	(–10, 2)
D.	(0, 10)
Answer» B. (–10, 10)

44.	If integrating factor of x(1 - x2) dy + (2x2y - y - ax3) dx = 0 is \({e^{\int {Pdx} }}\), then P is equal to
A.	\(\frac{{2{x^2} - a{x^3}}}{{x(1 - {x^2})}}\)
B.	2x3 - 1
C.	\(\frac{{2{x^2} - 1}}{{a{x^3}}}\)
D.	\(\frac{{2{x^2} - 1}}{{x(1 - {x^2})}}\)
Answer» E.

45.	Find solution of the differential equation \(\rm xdy+ydx\over y\) = 0
A.	xy - c = 0
B.	y - cx = 0
C.	y - cx2 = 0
D.	x - cy2 = 0
Answer» B. y - cx = 0

48.	A simple mass-spring oscillatory system consists of a mass m, suspended from a spring of stiffness k. Considering z as the displacement of the system at any time t, the equation of motion for the free vibration of the system is \(m\ddot z + kz = 0\). The natural frequency of the system is
A.	\(\frac{k}{m}\)
B.	\(\sqrt {\frac{m}{k}}\)
C.	\(\sqrt {\frac{k}{m}}\)
D.	\(\frac{m}{k}\)
Answer» D. \(\frac{m}{k}\)

50.	Find the solution of the differential equation \(\frac{dy}{dx}+2y=1\) such that y(0) = 0.
A.	\(\frac{1-~{{e}^{-2x}}}{2}\)
B.	\(\frac{1+~{{e}^{-2x}}}{2}\)
C.	y = 1 + ex
D.	\(\frac{1-{{e}^{-x}}}{2}\)
Answer» B. \(\frac{1+~{{e}^{-2x}}}{2}\)