Differential Equations Test 3

Total Questions:50 Total Time: 75 Min

Remaining:

 

Questions 1 of 50

Question:The order of the differential equation whose solution is \(y = a\cos x + b\sin x + c{e^{ - x}}\) is

Answers Choices:

3

2

1

None of these

Questions 2 of 50

Question:The differential equation of all circles of radius a is of order

Answers Choices:

2

3

4

None of these

Questions 3 of 50

Question:The differential equation of all circles in the first quadrant which touch the coordinate axes is of order

Answers Choices:

1

2

3

None of these

Questions 4 of 50

Question:The degree of differential equation \(\frac{{{d^2}y}}{{d{x^2}}} + {\left( {\frac{{dy}}{{dx}}} \right)^3} + 6y = 0\) is

Answers Choices:

1

3

2

5

Questions 5 of 50

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

Answers Choices:

2 and 4

3 and 2

4 and 5

2 and 3

Questions 6 of 50

Question:\(\frac{{{d^3}y}}{{d{x^3}}} + 2\,\left[ {1 + \frac{{{d^2}y}}{{d{x^2}}}} \right] = 1\) has degree and order as

Answers Choices:

1, 3

2, 3

3, 2

3, 1

Questions 7 of 50

Question:The differential equation for all the straight lines which are at a unit distance from the origin is

Answers Choices:

\({\left( {y - x\frac{{dy}}{{dx}}} \right)^2} = 1 - {\left( {\frac{{dy}}{{dx}}} \right)^2}\)

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

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

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

Questions 8 of 50

Question:If \(y = c{e^{{{\sin }^{ - 1}}x}}\) , then corresponding to this the differential equation is

Answers Choices:

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

\(\frac{{dy}}{{dx}} = \frac{1}{{\sqrt {1 - {x^2}} }}\)

\(\frac{{dy}}{{dx}} = \frac{x}{{\sqrt {1 - {x^2}} }}\)

None of these

Questions 9 of 50

Question:The differential equation of the family of curves represented by the equation \({x^2}y = a\), is

Answers Choices:

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

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

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

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

Questions 10 of 50

Question:The differential equation corresponding to primitive \(y = {e^{cx}}\)is or The elimination of the arbitrary constant m from the equation \(y = {e^{mx}}\) gives the differential equation

Answers Choices:

\(\frac{{dy}}{{dx}} = \left( {\frac{y}{x}} \right)\log x\)

\(\frac{{dy}}{{dx}} = \left( {\frac{x}{y}} \right)\log y\)

\(\frac{{dy}}{{dx}} = \left( {\frac{y}{x}} \right)\log y\)

\(\frac{{dy}}{{dx}} = \left( {\frac{x}{y}} \right)\log x\)

Questions 11 of 50

Question:The differential equation obtained on eliminating A and B from the equation \(y = A\cos \omega t + B\sin \omega t\) is

Answers Choices:

\(y'' = - {\omega ^2}y\)

\(y'' + y = 0\)

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

\(y'' - {\omega ^2}y = 0\)

Questions 12 of 50

Question:If \(y = a{x^{n + 1}} + b{x^{ - n}},\) then \({x^2}\frac{{{d^2}y}}{{d{x^2}}}\) equals to

Answers Choices:

\(n(n - 1)y\)

\(n(n + 1)y\)

ny

n2y

Questions 13 of 50

Question:The differential equation of all straight lines passing through the origin is

Answers Choices:

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

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

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

None of these

Questions 14 of 50

Question:\(y = a{e^{mx}} + b{e^{ - mx}}\) satisfies which of the following differential equations

Answers Choices:

\(\frac{{dy}}{{dx}} - my = 0\)

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

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

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

Questions 15 of 50

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

Answers Choices:

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

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

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

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

Questions 16 of 50

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

Answers Choices:

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

\(y - 2\sin y = c\)

\(x = \cot y + c\)

\(y = \cot x + c\)

Questions 17 of 50

Question:The solution of the differential equation \((\sin x + \cos x)dy + (\cos x - \sin x)dx = 0\)is

Answers Choices:

\({e^x}(\sin x + \cos x) + c = 0\)

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

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

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

Questions 18 of 50

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

Answers Choices:

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

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

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

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

Questions 19 of 50

Question:The solution of the differential equation \((1 + \cos x)dy = (1 - \cos x)dx\) is

Answers Choices:

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

\(y = 2\tan x + x + c\)

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

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

Questions 20 of 50

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

Answers Choices:

\(\log xy + x + y = c\)

\(\log \left( {\frac{x}{y}} \right) + x - y = c\)

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

None of these

Questions 21 of 50

Question:The solution of the equation \({\sin ^{ - 1}}\left( {\frac{{dy}}{{dx}}} \right) = x + y\)is

Answers Choices:

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

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

\(\tan (x + y) + \sec (x + y) + x + c = 0\)

None of these

Questions 22 of 50

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

Answers Choices:

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

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

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

None of these

Questions 23 of 50

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

Answers Choices:

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

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

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

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

Questions 24 of 50

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

Answers Choices:

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

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

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

None of these

Questions 25 of 50

Question:The solution of \(({\rm{cosec}}\,x\log y)dy + ({x^2}y)dx = 0\) is

Answers Choices:

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

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

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

None of these

Questions 26 of 50

Question:The solution of \(\frac{{dy}}{{dx}} = \frac{{{e^x}({{\sin }^2}x + \sin 2x)}}{{y(2\log y + 1)}}\) is

Answers Choices:

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

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

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

None of these

Questions 27 of 50

Question:If \(\frac{{dy}}{{dx}} = {e^{ - 2y}}\) and \(y = 0\) when \(x = 5,\) the value of x for \(y = 3\) is

Answers Choices:

\({e^5}\)

\({e^6} + 1\)

\(\frac{{{e^6} + 9}}{2}\)

\({\log _e}6\)

Questions 28 of 50

Question:The solution of differential equation \(dy - \sin x\sin ydx = 0\) is

Answers Choices:

\({e^{\cos x}}\tan \frac{y}{2} = c\)

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

\(\cos x\tan y = c\)

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

Questions 29 of 50

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

Answers Choices:

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

\(({e^y} + 1){\rm{cosec}}\,x = K\)

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

None of these

Questions 30 of 50

Question:Solution of differential equation \(\frac{{dy}}{{dx}} = \sin x + 2x\), is

Answers Choices:

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

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

\(y = \cos x + 2\)

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

Questions 31 of 50

Question:Solution of the differential equation \(\frac{{dx}}{x} + \frac{{dy}}{y} = 0\) is

Answers Choices:

\(xy = c\)

\(x + y = c\)

\(\log x\,\,\log y = c\)

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

Questions 32 of 50

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

Answers Choices:

\((x + a)(x + ay) = cy\)

\((x + a)(1 - ay) = cy\)

\((x + a)(1 - ay) = c\)

None of these

Questions 33 of 50

Question:The solution of \(\log \,\left( {\frac{{dy}}{{dx}}} \right) = ax + by\) is

Answers Choices:

\(\frac{{{e^{by}}}}{b} = \frac{{{e^{ax}}}}{a} + c\)

\(\frac{{{e^{ - by}}}}{{ - b}} = \frac{{{e^{ax}}}}{a} + c\)

\(\frac{{{e^{ - by}}}}{a} = \frac{{{e^{ax}}}}{b} + c\)

None of these

Questions 34 of 50

Question:The solution of \(\frac{{dy}}{{dx}} = {\left( {\frac{y}{x}} \right)^{1/3}}\) is

Answers Choices:

\({x^{2/3}} + {y^{2/3}} = c\)

\({x^{1/3}} + {y^{1/3}} = c\)

\({y^{2/3}} - {x^{2/3}} = c\)

\({y^{1/3}} - {x^{1/3}} = c\)

Questions 35 of 50

Question:The general solution of the differential equation \((x + y)dx + xdy = 0\) is

Answers Choices:

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

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

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

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

Questions 36 of 50

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

Answers Choices:

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

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

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

\(y - x = c + \frac{x}{{y - x}}\)

Questions 37 of 50

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

Answers Choices:

\(a{y^2} = {e^{{x^2}/{y^2}}}\)

\(ay = {e^{x/y}}\)

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

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

Questions 38 of 50

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

Answers Choices:

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

\(y = x + c\)

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

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

Questions 39 of 50

Question:Solution of the differential equation, \(y\,dx - x\,dy + x{y^2}dx = 0\) can be

Answers Choices:

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

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

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

None of these

Questions 40 of 50

Question:If c is any arbitrary constant, then the general solution of the differential equation \(ydx - xdy = xy\,dx\) is given by

Answers Choices:

\(y = cx\,{e^{ - x}}\)

\(x = cy{e^{ - x}}\)

\(y + {e^x} = cx\)

\(y{e^x} = cx\)

Questions 41 of 50

Question:\(({x^2} + {y^2})dy = xydx\). If \(y({x_0}) = e\), \(y(1) = 1\), then value of \({x_0} = \)

Answers Choices:

\(\sqrt 3 e\)

\(\sqrt {{e^2} - \frac{1}{2}} \)

\(\sqrt {\frac{{{e^2} - 1}}{2}} \)

\(\sqrt {\frac{{{e^2} + 1}}{2}} \)

Questions 42 of 50

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

Answers Choices:

\(y(1 - xy) = Ax\)

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

\(x(1 - xy) = Ay\)

\(x(1 + xy) = Ay\)

Questions 43 of 50

Question:The integrating factor of the differential equation \(\frac{{dy}}{{dx}} = y\tan x - {y^2}\sec x,\)is

Answers Choices:

\(\tan x\)

\(\sec x\)

\( - \sec x\)

\(\cot x\)

Questions 44 of 50

Question:Integrating factor of differential equation \(\cos x\frac{{dy}}{{dx}} + y\sin x = 1\)is

Answers Choices:

\(\cos x\)

\(\tan x\)

\(\sec x\)

\(\sin x\)

Questions 45 of 50

Question:\(y + {x^2} = \frac{{dy}}{{dx}}\) has the solution

Answers Choices:

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

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

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

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

Questions 46 of 50

Question:The solution of \(\frac{{dy}}{{dx}} + p(x)y = 0\) is

Answers Choices:

\(y = c{e^{\int {p\,d\,x} }}\)

\(x = c{e^{ - \int {p\,d\,y} }}\)

\(y = c{e^{ - \int {P\,d\,x} }}\)

\(x = c{e^{\int {p\,d\,y} }}\)

Questions 47 of 50

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

Answers Choices:

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

\(y = x{e^x}\)

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

\(y = x{e^{ - x}}\)

Questions 48 of 50

Question:The equation of the curve through the point (1,0) and whose slope is \(\frac{{y - 1}}{{{x^2} + x}}\)is

Answers Choices:

\((y - 1)(x + 1) + 2x = 0\)

\(2x(y - 1) + x + 1 = 0\)

\(x(y - 1)(x + 1) + 2 = 0\)

None of these

Questions 49 of 50

Question:The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes though the point (4, 3). The equation of the curve is

Answers Choices:

\({x^2} = y + 5\)

\({y^2} = x - 5\)

\({y^2} = x + 5\)

\({x^2} = y - 5\)

Questions 50 of 50

Question:A particle moves in a straight line with a velocity given by \(\frac{{dx}}{{dt}} = x + 1\)(x is the distance described). The time taken by a particle to traverse a distance of 99 metre is

Answers Choices:

\({\log _{10}}e\)

\(2{\log _e}10\)

\(2{\log _{10}}e\)

\(\frac{1}{2}{\log _{10}}e\)