# 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

3

2

1

None of these

## Questions 2 of 50

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

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

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

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

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

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

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

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

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

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

## Questions 8 of 50

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

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

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

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

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

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

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

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

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

## 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

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

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

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

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

## 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

$$y'' = - {\omega ^2}y$$

$$y'' + y = 0$$

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

$$y'' - {\omega ^2}y = 0$$

## 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

$$n(n - 1)y$$

$$n(n + 1)y$$

ny

n2y

## Questions 13 of 50

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

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

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

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

None of these

## Questions 14 of 50

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

$$\frac{{dy}}{{dx}} - my = 0$$

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

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

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

## Questions 15 of 50

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

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

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

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

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

## Questions 16 of 50

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

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

$$y - 2\sin y = c$$

$$x = \cot y + c$$

$$y = \cot x + c$$

## Questions 17 of 50

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

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

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

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

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

## Questions 18 of 50

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

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

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

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

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

## Questions 19 of 50

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

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

$$y = 2\tan x + x + c$$

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

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

## Questions 20 of 50

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

$$\log xy + x + y = c$$

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

$$\log xy + x - y = c$$

None of these

## Questions 21 of 50

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

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

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

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

None of these

## Questions 22 of 50

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

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

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

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

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

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

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

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

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

## Questions 24 of 50

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

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

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

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

None of these

## Questions 25 of 50

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

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

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

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

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

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

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

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

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

$${e^5}$$

$${e^6} + 1$$

$$\frac{{{e^6} + 9}}{2}$$

$${\log _e}6$$

## Questions 28 of 50

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

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

$${e^{\cos x}}\tan y = c$$

$$\cos x\tan y = c$$

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

## Questions 29 of 50

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

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

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

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

None of these

## Questions 30 of 50

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

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

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

$$y = \cos x + 2$$

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

## Questions 31 of 50

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

$$xy = c$$

$$x + y = c$$

$$\log x\,\,\log y = c$$

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

## Questions 32 of 50

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

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

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

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

None of these

## Questions 33 of 50

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

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

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

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

None of these

## Questions 34 of 50

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

$${x^{2/3}} + {y^{2/3}} = c$$

$${x^{1/3}} + {y^{1/3}} = c$$

$${y^{2/3}} - {x^{2/3}} = c$$

$${y^{1/3}} - {x^{1/3}} = c$$

## Questions 35 of 50

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

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

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

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

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

## Questions 36 of 50

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

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

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

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

$$y - x = c + \frac{x}{{y - x}}$$

## Questions 37 of 50

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

$$a{y^2} = {e^{{x^2}/{y^2}}}$$

$$ay = {e^{x/y}}$$

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

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

## Questions 38 of 50

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

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

$$y = x + c$$

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

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

## Questions 39 of 50

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

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

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

$$2y - {y^2}x = \lambda y$$

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

$$y = cx\,{e^{ - x}}$$

$$x = cy{e^{ - x}}$$

$$y + {e^x} = cx$$

$$y{e^x} = cx$$

## Questions 41 of 50

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

$$\sqrt 3 e$$

$$\sqrt {{e^2} - \frac{1}{2}}$$

$$\sqrt {\frac{{{e^2} - 1}}{2}}$$

$$\sqrt {\frac{{{e^2} + 1}}{2}}$$

## Questions 42 of 50

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

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

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

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

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

## Questions 43 of 50

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

$$\tan x$$

$$\sec x$$

$$- \sec x$$

$$\cot x$$

## Questions 44 of 50

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

$$\cos x$$

$$\tan x$$

$$\sec x$$

$$\sin x$$

## Questions 45 of 50

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

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

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

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

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

## Questions 46 of 50

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

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

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

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

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

## Questions 47 of 50

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

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

$$y = x{e^x}$$

$$y = x{e^{ - x}} + 1$$

$$y = x{e^{ - x}}$$

## 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

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

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

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

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

$${x^2} = y + 5$$

$${y^2} = x - 5$$

$${y^2} = x + 5$$

$${x^2} = y - 5$$

## 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

$${\log _{10}}e$$
$$2{\log _e}10$$
$$2{\log _{10}}e$$
$$\frac{1}{2}{\log _{10}}e$$