Differential Equations Test 2

Total Questions:50 Total Time: 75 Min

Remaining:

Questions 1 of 50

Question:The order and the degree of differential equation $$\frac{{{d^4}y}}{{d{x^4}}} - 4\frac{{{d^3}y}}{{d{x^3}}} + 8\frac{{{d^2}y}}{{d{x^2}}} - 8\frac{{dy}}{{dx}} + 4y = 0$$ are respectively

4, 1

1, 4

1, 1

None of these

Questions 2 of 50

Question:The order of the differential equation $$y\left( {\frac{{dy}}{{dx}}} \right) = \frac{x}{{{\textstyle{{dy} \over {dx}}} + {{\left( {{\textstyle{{dy} \over {dx}}}} \right)}^3}}}$$ is.

1

2

3

4

Questions 3 of 50

Question:The order of the differential equation whose solution is $${x^2} + {y^2} + 2gx + 2fy + c = 0$$, is

1

2

3

4

Questions 4 of 50

Question:The order of the differential equation of all circles of radius r, having centre on y-axis and passing through the origin is

1

2

3

4

Questions 5 of 50

Question:The second order differential equation is

$${y'^2} + x = {y^2}$$

$$y'y'' + y = \sin x$$

$$y''' + y'' + y = 0$$

(d) $$y' = y$$

Questions 6 of 50

Question:The order and degree of the differential equation $$x\frac{{{d^2}y}}{{d{x^2}}} + {\left( {\frac{{dy}}{{dx}}} \right)^2} + {y^2} = 0$$ are respectively

2 and 2

1 and 1

2 and 1

1 and 2

Questions 7 of 50

Question:The order and degree of the differential equation $$\rho = \frac{{{{\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]}^{3/2}}}}{{{d^2}y/d{x^2}}}$$ are respectively

2, 2

2, 3

2, 1

None of these

Questions 8 of 50

Question:Order of the differential equation of the family of all concentric circles centered at (h, k) is

1

2

3

4

Questions 9 of 50

Question:$$y = \frac{x}{{x + 1}}$$ is a solution of the differential equation

$${y^2}\frac{{dy}}{{dx}} = {x^2}$$

$${x^2}\frac{{dy}}{{dx}} = {y^2}$$

$$y\frac{{dy}}{{dx}} = x$$

$$x\frac{{dy}}{{dx}} = y$$

Questions 10 of 50

Question:The differential equation whose solution is $$y = A\sin x + B\cos x,$$ is

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

$$\frac{{{d^2}y}}{{d{x^2}}} - y = 0$$

$$\frac{{dy}}{{dx}} + y = 0$$

None of these

Questions 11 of 50

Question:The differential equation of the family of curves $$y = a\cos (x + b)$$ is

$$\frac{{{d^2}y}}{{d{x^2}}} - y = 0$$

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

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

None of these

Questions 12 of 50

Question:Differential equation whose solution is $$y = cx + c - {c^3}$$ , is

$$\frac{{dy}}{{dx}} = c$$

$$y = x\frac{{dy}}{{dx}} + \frac{{dy}}{{dx}} - {\left( {\frac{{dy}}{{dx}}} \right)^3}$$

$$\frac{{dy}}{{dx}} = c - 3{c^2}$$

None of these

Questions 13 of 50

Question:Family of curves $$y = {e^x}(A\cos x + B\sin x)$$ , represents the differential equation

$$\frac{{{d^2}y}}{{d{x^2}}} = 2\frac{{dy}}{{dx}} - y$$

$$\frac{{{d^2}y}}{{d{x^2}}} = 2\frac{{dy}}{{dx}} - 2y$$

$$\frac{{{d^2}y}}{{d{x^2}}} = \frac{{dy}}{{dx}} - 2y$$

$$\frac{{{d^2}y}}{{d{x^2}}} = 2\frac{{dy}}{{dx}} + y$$

Questions 14 of 50

Question:The elimination of the arbitrary constants A, B and C from $$y = A + Bx + C{e^{ - x}}$$leads to the differential equation

$$y''' - y' = 0$$

$$y''' - y'' + y' = 0$$

$$y''' + y'' = 0$$

$$y'' + y'' - y' = 0$$

Questions 15 of 50

Question:The solution of the differential equation $$\frac{{dy}}{{dx}} + \frac{{1 + {x^2}}}{x} = 0$$ is

$$y = - \frac{1}{2}{\tan ^{ - 1}}x + c$$

$$y + \log x + \frac{{{x^2}}}{2} + c = 0$$

$$y = \frac{1}{2}{\tan ^{ - 1}}x + c$$

$$y - \log x - \frac{{{x^2}}}{2} = c$$

Questions 16 of 50

Question:The solution of the differential equation $$\frac{{dy}}{{dx}} = \sec x(\sec x + \tan x)$$is

$$y = \sec x + \tan x + c$$

$$y = \sec x + \cot x + c$$

$$y = \sec x - \tan x + c$$

None of these

Questions 17 of 50

Question:The solution of the differential equation $$(1 + {x^2})\frac{{dy}}{{dx}} = x$$ is

$$y = {\tan ^{ - 1}}x + c$$

$$y = - {\tan ^{ - 1}}x + c$$

$$y = \frac{1}{2}{\log _e}(1 + {x^2}) + c$$

$$y = - \frac{1}{2}{\log _e}(1 + {x^2}) + c$$

Questions 18 of 50

Question:The general solution of the equation $$({e^y} + 1)\cos xdx + {e^y}\sin xdy = 0$$ is

$$({e^y} + 1)\cos x = c$$

$$({e^y} - 1)\sin x = c$$

$$({e^y} + 1)\sin x = c$$

None of these

Questions 19 of 50

Question:The solution of the differential equation$${x^2}dy = - 2xydx$$ is

$$x{y^2} = c$$

$${x^2}{y^2} = c$$

$${x^2}y = c$$

$$xy = c$$

Questions 20 of 50

Question:The solution of the differential equation $$\frac{{dy}}{{dx}} = (a{e^{bx}} + c\cos mx)$$ is

$$y = \frac{{a{e^x}}}{b} + \frac{c}{m}\sin mx + k$$

$$y = a{e^x} + c\sin mx + k$$

$$y = \frac{{a{e^{bx}}}}{b} + \frac{c}{m}\sin mx + k$$

None of these

Questions 21 of 50

Question:The solution of the differential equation $$x({e^{2y}} - 1)dy + ({x^2} - 1){e^y}dx = 0$$is

$${e^y} + {e^{ - y}} = \log x - \frac{{{x^2}}}{2} + c$$

$${e^y} - {e^{ - y}} = \log x - \frac{{{x^2}}}{2} + c$$

$${e^y} + {e^{ - y}} = \log x + \frac{{{x^2}}}{2} + c$$

None of these

Questions 22 of 50

Question:The solution of $$\frac{{dy}}{{dx}} = \sin (x + y) + \cos (x + y)$$is

$$\log \left[ {1 + \tan \left( {\frac{{x + y}}{2}} \right)} \right] + c = 0$$

$$\log \left[ {1 + \tan \left( {\frac{{x + y}}{2}} \right)} \right] = x + c$$

$$\log \left[ {1 - \tan \left( {\frac{{x + y}}{2}} \right)} \right] = x + c$$

None of these

Questions 23 of 50

Question:The solution of the differential equation $$\frac{{dy}}{{dx}} = \frac{{x - y + 3}}{{2(x - y) + 5}}$$ is

$$2(x - y) + \log (x - y) = x + c$$

$$2(x - y) - \log (x - y + 2) = x + c$$

$$2(x - y) + \log (x - y + 2) = x + c$$

None of these

Questions 24 of 50

Question:The solution of the equation $$\frac{{dy}}{{dx}} = \frac{{{y^2} - y - 2}}{{{x^2} + 2x - 3}}$$ is

$$\frac{1}{3}\log \left| {\frac{{y - 2}}{{y + 1}}} \right| = \frac{1}{4}\log \left| {\frac{{x + 3}}{{x - 1}}} \right| + c$$

$$\frac{1}{3}\log \left| {\frac{{y + 1}}{{y - 2}}} \right| = \frac{1}{4}\log \left| {\frac{{x - 1}}{{x + 3}}} \right| + c$$

$$4\log \left| {\frac{{y - 2}}{{y + 1}}} \right| = 3\log \left| {\frac{{x - 1}}{{x + 3}}} \right| + c$$

None of these

Questions 25 of 50

Question:The general solution of the differential equation $$ydx\, + (1 + {x^2}){\tan ^{ - 1}}xdy = 0,$$ is

$$y{\tan ^{ - 1}}x = c$$

$$x{\tan ^{ - 1}}y = c$$

$$y + {\tan ^{ - 1}}x = c$$

$$x + {\tan ^{ - 1}}y = c$$

Questions 26 of 50

Question:The general solution of the differential equation $$\frac{{dy}}{{dx}} = \frac{{{x^2}}}{{{y^2}}}$$ is

$${x^3} - {y^3} = c$$

$${x^3} + {y^3} = c$$

$${x^2} + {y^2} = c$$

$${x^2} - {y^2} = c$$

Questions 27 of 50

Question:The solution of $$y' - y = 1,\;y(0) = - 1$$ is given by $$y(x) =$$

$$- \exp (x)$$

$$- \exp ( - x)$$

1

$$\exp (x) - 2$$

Questions 28 of 50

Question:The number of solutions of $$y' = \frac{{y + 1}}{{x - 1}},\,y(1) = 2$$ is

None

One

Two

Infinite

Questions 29 of 50

Question:The general solution of the differential equation $$\frac{{dy}}{{dx}} + \sin \left( {\frac{{x + y}}{2}} \right) = \sin \left( {\frac{{x - y}}{2}} \right)$$ is

$$\log \tan \left( {\frac{y}{2}} \right) = c - 2\sin x$$

$$\log \tan \,\left( {\frac{y}{4}} \right) = c - 2\sin \left( {\frac{x}{2}} \right)$$

$$\log \tan \,\left( {\frac{y}{2} + \frac{\pi }{4}} \right) = c - 2\sin x$$

$$\log \tan \left( {\frac{y}{4} + \frac{\pi }{4}} \right) = c - 2\sin \left( {\frac{x}{2}} \right)$$

Questions 30 of 50

Question:The solution of the differential equation $${(x + y)^2}\frac{{dy}}{{dx}} = {a^2}$$ is

$${(x + y)^2} = \frac{{{a^2}}}{2}x + c$$

$${(x + y)^2} = {a^2}x + c$$

$${(x + y)^2} = 2{a^2}x + c$$

None of these

Questions 31 of 50

Question:The solution of the differential equation $$({x^2} + {y^2})dx = 2xydy$$ is

$$x = c({x^2} + {y^2})$$

$$x = c({x^2} - {y^2})$$

$$x + c({x^2} - {y^2}) = 0$$

None of these

Questions 32 of 50

Question:The solution of the equation $$\frac{{dy}}{{dx}} = \frac{{x + y}}{{x - y}}$$is

$$c{({x^2} + {y^2})^{1/2}} + {e^{{{\tan }^{ - 1}}(y/x)}} = 0$$

$$c{({x^2} + {y^2})^{1/2}} = {e^{{{\tan }^{ - 1}}(y/x)}}$$

$$c({x^2} - {y^2}) = {e^{{{\tan }^{ - 1}}(y/x)}}$$

None of these

Questions 33 of 50

Question:The solution of the differential equation $$(3xy + {y^2})dx + ({x^2} + xy)dy = 0$$ is

$${x^2}(2xy + {y^2}) = {c^2}$$

$${x^2}(2xy - {y^2}) = {c^2}$$

$${x^2}({y^2} - 2xy) = {c^2}$$

None of these

Questions 34 of 50

Question:The solution of the differential equation $$x\,dy - y\,dx = (\sqrt {{x^2} + {y^2})} dx$$is

$$y - \sqrt {{x^2} + {y^2}} = c{x^2}$$

$$y + \sqrt {{x^2} + {y^2}} = c{x^2}$$

$$y + \sqrt {{x^2} + {y^2}} + c{x^2} = 0$$

None of these

Questions 35 of 50

Question:Solution of $$(xy\cos xy + \sin xy)dx + {x^2}\cos xy\,dy = 0$$ is

$$x\sin (xy) = k$$

$$xy\sin (xy) = k$$

$$\frac{x}{y}\sin (xy) = k$$

$$x\sin (xy) = k$$

Questions 36 of 50

Question:The solution of $$(x - {y^3})dx + 3x{y^2}dy = 0$$ is

$$\log x + \frac{x}{y}$$

$$\log x + \frac{{{y^3}}}{x} = k$$

$$\log x - \frac{x}{{{y^3}}} = k$$

$$\log xy - {y^3} = k$$

Questions 37 of 50

Question:The solution of $$y{e^{ - x/y}}dx - (x{e^{ - x/y}} + {y^3})dy = 0$$ is

$$\frac{{{y^2}}}{2} + {e^{ - x/y}} = k$$

$$\frac{{{x^2}}}{2} + {e^{ - x/y}} = k$$

$$\frac{{{x^2}}}{2} + {e^{x/y}} = k$$

$$\frac{{{y^2}}}{2} + {e^{x/y}} = k$$

Questions 38 of 50

Question:The solution of the differential equation $$x\,dy + y\,dx - \sqrt {1 - {x^2}{y^2}} dx = 0$$ is

$${\sin ^{ - 1}}xy = c - x$$

$$xy = \sin (x + c)$$

$$\log (1 - {x^2}{y^2}) = x + c$$

$$y = x\sin x + c$$

Questions 39 of 50

Question:Which of the following equation is linear

$$\frac{{dy}}{{dx}} + x{y^2} = 1$$

$${x^2}\frac{{dy}}{{dx}} + y = {e^x}$$

$$\frac{{dy}}{{dx}} + 3y = x{y^2}$$

$$x\frac{{dy}}{{dx}} + {y^2} = \sin x$$

Questions 40 of 50

Question:The solution of the equation $$x\frac{{dy}}{{dx}} + 3y = x$$ is

$${x^3}y + \frac{{{x^4}}}{4} + c = 0$$

$${x^3}y = \frac{{{x^4}}}{4} + c$$

$${x^3}y + \frac{{{x^4}}}{4} = 0$$

None of these

Questions 41 of 50

Question:The solution of the differential equation $$\frac{{dy}}{{dx}} + y = \cos x$$is

$$y = \frac{1}{2}(\cos x + \sin x) + c{e^{ - x}}$$

$$y = \frac{1}{2}(\cos x - \sin x) + c{e^{ - x}}$$

$$y = \cos x + \sin x + c{e^{ - x}}$$

None of these

Questions 42 of 50

Question:The solution of the differential equation $$\frac{{dy}}{{dx}} + y\cot x = 2\cos x$$ is

$$y\sin x + \cos 2x = 2c$$

$$2y\sin x + \cos x = c$$

$$y\sin x + \cos x = c$$

$$2y\sin x + \cos 2x = c$$

Questions 43 of 50

Question:Solution of the differential equation $$\frac{{dy}}{{dx}} + y{\sec ^2}x = \tan x{\sec ^2}x$$ is

$$y = \tan x - 1 + c{e^{ - \tan x}}$$

$${y^2} = \tan x - 1 + c{e^{\tan x}}$$

$$y{e^{\tan x}} = \tan x - 1 + c$$

$$y{e^{ - \tan x}} = \tan x - 1 + c$$

Questions 44 of 50

Question:An integrating factor of the differential equation $$(1 - {x^2})\frac{{dy}}{{dx}} - xy = 1,$$ is

x

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

$$\sqrt {(1 - {x^2})}$$

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

Questions 45 of 50

Question:Integrating factor of equation $$({x^2} + 1)\frac{{dy}}{{dx}} + 2xy = {x^2} - 1$$ is

$${x^2} + 1$$

$$\frac{{2x}}{{{x^2} + 1}}$$

$$\frac{{{x^2} - 1}}{{{x^2} + 1}}$$

None of these

Questions 46 of 50

Question:The solution of $$\frac{{dy}}{{dx}} + \frac{y}{3} = 1$$ is

$$y = 3 + c{e^{x/3}}$$

$$y = 3 + c{e^{ - x/3}}$$

$$3y = c + {e^{x/3}}$$

$$3y = c + {e^{ - x/3}}$$

Questions 47 of 50

Question:Equation of curve through point $$(1,\,0)$$which satisfies the differential equation $$(1 + {y^2})dx - xydy = 0$$, is

$${x^2} + {y^2} = 1$$

$${x^2} - {y^2} = 1$$

$$2{x^2} + {y^2} = 2$$

None of these

Questions 48 of 50

Question:Equation of curve passing through (3, 9) which satisfies the differential equation $$\frac{{dy}}{{dx}} = x + \frac{1}{{{x^2}}}$$, is

$$6xy = 3{x^2} - 6x + 29$$

$$6xy = 3{x^3} - 29x + 6$$

$$6xy = 3{x^3} + 29x - 6$$

None of these

Questions 49 of 50

Question:The differential equation $$y\frac{{dy}}{{dx}} + x = a$$(a is any constant) represents

A set of circles having centre on the y-axis

A set of circles centre on the x-axis

A set of ellipses

None of these

Questions 50 of 50

Question:The equation of a curve passing through $$\left( {2,\frac{7}{2}} \right)$$ and having gradient $$1 - \frac{1}{{{x^2}}}$$at$$(x,\,y)$$is

$$y = {x^2} + x + 1$$
$$xy = {x^2} + x + 1$$
$$xy = x + 1$$