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

Answers Choices:

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.

Answers Choices:

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

Answers Choices:

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

Answers Choices:

1

2

3

4

Questions 5 of 50

Question:The second order differential equation is

Answers Choices:

\({y'^2} + x = {y^2}\)

\(y'y'' + y = \sin x\)

\(y''' + y'' + y = 0\)

(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

Answers Choices:

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

Answers Choices:

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

Answers Choices:

1

2

3

4

Questions 9 of 50

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

Answers Choices:

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

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

\(y\frac{{dy}}{{dx}} = x\)

\(x\frac{{dy}}{{dx}} = y\)

Questions 10 of 50

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

Answers Choices:

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

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

\(\frac{{dy}}{{dx}} + y = 0\)

None of these

Questions 11 of 50

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

Answers Choices:

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

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

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

None of these

Questions 12 of 50

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

Answers Choices:

\(\frac{{dy}}{{dx}} = c\)

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

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

None of these

Questions 13 of 50

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

Answers Choices:

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

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

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

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

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

Answers Choices:

\(y''' - y' = 0\)

\(y''' - y'' + y' = 0\)

\(y''' + y'' = 0\)

\(y'' + y'' - y' = 0\)

Questions 15 of 50

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

Answers Choices:

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

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

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

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

Questions 16 of 50

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

Answers Choices:

\(y = \sec x + \tan x + c\)

\(y = \sec x + \cot x + c\)

\(y = \sec x - \tan x + c\)

None of these

Questions 17 of 50

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

Answers Choices:

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

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

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

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

Questions 18 of 50

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

Answers Choices:

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

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

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

None of these

Questions 19 of 50

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

Answers Choices:

\(x{y^2} = c\)

\({x^2}{y^2} = c\)

\({x^2}y = c\)

\(xy = c\)

Questions 20 of 50

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

Answers Choices:

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

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

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

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

Answers Choices:

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

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

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

None of these

Questions 22 of 50

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

Answers Choices:

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

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

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

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

Answers Choices:

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

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

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

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

Answers Choices:

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

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

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

None of these

Questions 25 of 50

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

Answers Choices:

\(y{\tan ^{ - 1}}x = c\)

\(x{\tan ^{ - 1}}y = c\)

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

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

Questions 26 of 50

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

Answers Choices:

\({x^3} - {y^3} = c\)

\({x^3} + {y^3} = c\)

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

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

Questions 27 of 50

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

Answers Choices:

\( - \exp (x)\)

\( - \exp ( - x)\)

1

\(\exp (x) - 2\)

Questions 28 of 50

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

Answers Choices:

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

Answers Choices:

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

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

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

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

Questions 30 of 50

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

Answers Choices:

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

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

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

None of these

Questions 31 of 50

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

Answers Choices:

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

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

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

None of these

Questions 32 of 50

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

Answers Choices:

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

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

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

None of these

Questions 33 of 50

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

Answers Choices:

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

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

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

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

Answers Choices:

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

\(y + \sqrt {{x^2} + {y^2}} = c{x^2}\)

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

None of these

Questions 35 of 50

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

Answers Choices:

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

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

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

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

Questions 36 of 50

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

Answers Choices:

\(\log x + \frac{x}{y}\)

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

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

\(\log xy - {y^3} = k\)

Questions 37 of 50

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

Answers Choices:

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

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

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

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

Questions 38 of 50

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

Answers Choices:

\({\sin ^{ - 1}}xy = c - x\)

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

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

\(y = x\sin x + c\)

Questions 39 of 50

Question:Which of the following equation is linear

Answers Choices:

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

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

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

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

Questions 40 of 50

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

Answers Choices:

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

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

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

None of these

Questions 41 of 50

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

Answers Choices:

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

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

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

None of these

Questions 42 of 50

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

Answers Choices:

\(y\sin x + \cos 2x = 2c\)

\(2y\sin x + \cos x = c\)

\(y\sin x + \cos x = c\)

\(2y\sin x + \cos 2x = c\)

Questions 43 of 50

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

Answers Choices:

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

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

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

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

Questions 44 of 50

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

Answers Choices:

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

Answers Choices:

\({x^2} + 1\)

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

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

None of these

Questions 46 of 50

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

Answers Choices:

\(y = 3 + c{e^{x/3}}\)

\(y = 3 + c{e^{ - x/3}}\)

\(3y = c + {e^{x/3}}\)

\(3y = c + {e^{ - x/3}}\)

Questions 47 of 50

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

Answers Choices:

\({x^2} + {y^2} = 1\)

\({x^2} - {y^2} = 1\)

\(2{x^2} + {y^2} = 2\)

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

Answers Choices:

\(6xy = 3{x^2} - 6x + 29\)

\(6xy = 3{x^3} - 29x + 6\)

\(6xy = 3{x^3} + 29x - 6\)

None of these

Questions 49 of 50

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

Answers Choices:

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

Answers Choices:

\(y = {x^2} + x + 1\)

\(xy = {x^2} + x + 1\)

\(xy = x + 1\)

None of these