# Differential Equations Test 1

Total Questions:50 Total Time: 75 Min

Remaining:

## Questions 1 of 50

Question:A differential equation of first order and first degree is

$$x{\left( {\frac{{dy}}{{dx}}} \right)^2} - x + a = 0$$

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

$$dy + dx = 0$$

None of these

## Questions 2 of 50

Question:The order and degree of the differential equation $$\sqrt {\frac{{dy}}{{dx}}} - 4\frac{{dy}}{{dx}} - 7x = 0$$ are

1 and ½

2 and 1

1 and 1

1 and 2

## Questions 3 of 50

Question:The order and degree of the differential equation $$\frac{{{d^2}y}}{{d{x^2}}} = \sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}}$$is

4, 2

1, 2

2, 2

$$2,\frac{1}{2}$$

## Questions 4 of 50

Question:The order and the degree of the differential equation $${\left( {\frac{{{d^2}s}}{{d{t^2}}}} \right)^2} + 3{\left( {\frac{{ds}}{{dt}}} \right)^3} + 4 = 0$$ are

$$2,\,2$$

$$2,\,3$$

$$3,2$$

None of these

## Questions 5 of 50

Question:Which of the following differential equations has the same order and degree

$$\frac{{{d^4}y}}{{d{x^4}}} + 8{\rm{ }}{\left( {\frac{{dy}}{{dx}}} \right)^6} + 5y = {e^x}$$

$$5{\rm{ }}{\left( {\frac{{{d^3}y}}{{d{x^3}}}} \right)^4} + 8{\rm{ }}{\left( {1 + \frac{{dy}}{{dx}}} \right)^2} + 5y = {x^8}$$

$${\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^3}} \right]^{2/3}} = 4\frac{{{d^3}y}}{{d{x^3}}}$$

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

## Questions 6 of 50

Question:The differential equation $$\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}} + \sin y + {x^2} = 0$$ is of the following type

Linear

Homogeneous

Order two

Degree one

3 and 4 are correct

## Questions 7 of 50

Question:The differential equation $$x{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^3} + {\left( {\frac{{dy}}{{dx}}} \right)^4} + y = {x^2}$$ is of

Degree 3 and order 2

Degree 1 and order 1

Degree 4 and order 3

Degree 4 and order 4

## Questions 8 of 50

Question:If m and n are the order and degree of the differential equation $${\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^5} + 4\frac{{{{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)}^3}}}{{\left( {\frac{{{d^3}y}}{{d{x^3}}}} \right)}} + \frac{{{d^3}y}}{{d{x^3}}} = {x^2} - 1$$ , then

m = 3 and n = 5

m = 3 and n = 1

m = 3 and n = 3

m = 3 and n = 2

## Questions 9 of 50

Question:The degree of the differential equation $$y(x) = 1 + \frac{{dy}}{{dx}} + \frac{1}{{1.2}}{\left( {\frac{{dy}}{{dx}}} \right)^2} + \frac{1}{{1.2.3}}{\left( {\frac{{dy}}{{dx}}} \right)^3} + ...$$ is

2

3

1

None of these

## Questions 10 of 50

Question:$$y = 4\sin 3x$$ is a solution of the differential equation

$$\frac{{dy}}{{dx}} + 8y = 0$$

$$\frac{{dy}}{{dx}} - 8y = 0$$

$$\frac{{{d^2}y}}{{d{x^2}}} + 9y = 0$$

$$\frac{{{d^2}y}}{{d{x^2}}} - 9y = 0$$

## Questions 11 of 50

Question:The differential equation of all the lines in the xy-plane is

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

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

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

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

## Questions 12 of 50

Question:The differential equation of the family of curves represented by the equation $${x^2} + {y^2} = {a^2}$$ is

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

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

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

None of these

## Questions 13 of 50

Question:The differential equation of all circles which passes through the origin and whose centre lies on y-axis, is

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

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

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

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

## Questions 14 of 50

Question:The differential equation of displacement of all "Simple harmonic motions" of given period $$2\pi /n$$ , is

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

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

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

$$\frac{{{d^2}x}}{{d{t^2}}} + \frac{1}{{{n^2}}}x = 0$$

## Questions 15 of 50

Question:The differential equation of all parabolas whose axes are parallel to y-axis is

$$\frac{{{d^3}y}}{{d{x^3}}} = 0$$

$$\frac{{{d^2}x}}{{d{y^2}}} = c$$

$$\frac{{{d^3}y}}{{d{x^3}}} + \frac{{{d^2}x}}{{d{y^2}}} = 0$$

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

## Questions 16 of 50

Question:The differential equation found by the elimination of the arbitrary constant K from the equation $$y = (x + K){e^{ - x}}$$ is

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

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

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

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

## Questions 17 of 50

Question:The solution of the differential equation $$3{e^x}\tan ydx + (1 - {e^x}){\sec ^2}ydy = 0$$ is

$$\tan y = c{(1 - {e^x})^3}$$

$${(1 - {e^x})^3}\tan y = c$$

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

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

## Questions 18 of 50

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

$$1 + xy + c(y + x) = 0$$

$$x + y = c(1 - xy)$$

$$y - x = c(1 + xy)$$

$$1 + xy = c(x + y)$$

## Questions 19 of 50

Question:The solution of the differential equation $$x\cos ydy = (x{e^x}\log x + {e^x})dx$$ is

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

$$\sin y + {e^x}\log x + c = 0$$

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

None of these

## Questions 20 of 50

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

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

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

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

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

## Questions 21 of 50

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

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

$$cx = \sec y + \tan y$$

$$cy = \sec x\tan x$$

$$cy = \sec x + \tan x$$

## Questions 22 of 50

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

$$y = 1 + cxy$$

$$y = \log \{ cxy\}$$

$$y + 1 = cxy$$

$$y = c + xy$$

## Questions 23 of 50

Question:If $$\frac{{dy}}{{dx}} + \frac{1}{{\sqrt {1 - {x^2}} }} = 0$$, then

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

$${y^2} + 2{\sin ^{ - 1}}x + c = 0$$

$$x + {\sin ^{ - 1}}y = 0$$

$${x^2} + 2{\sin ^{ - 1}}y = 1$$

## Questions 24 of 50

Question:If $$\frac{{dy}}{{dx}} = \frac{{xy + y}}{{xy + x}}$$, then the solution of the differential equation is

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

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

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

$$y = x + A$$

## Questions 25 of 50

Question:Solution of the equation $$({e^x} + 1)ydy = (y + 1){e^x}dx$$ is

$$c(y + 1)({e^x} + 1) + {e^y} = 0$$

$$c(y + 1)({e^x} - 1) + {e^y} = 0$$

$$c(y + 1)({e^x} - 1) - {e^y} = 0$$

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

## Questions 26 of 50

Question:Solution of the equation $$(1 - {x^2})dy + xydx = x{y^2}dx$$ is

$${(y - 1)^2}(1 - {x^2}) = 0$$

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

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

None of these

## Questions 27 of 50

Question:The solution of the equation $$\sqrt {a + x} \frac{{dy}}{{dx}} + x = 0$$ is

$$3y + 2\sqrt {a + x} .(x - 2a) = 3c$$

$$3y + 2\sqrt {x + a} .(x + 2a) = 3c$$

$$3y + \sqrt {x + a} .(x + 2a) = 3c$$

None of these

## Questions 28 of 50

Question:Solution of the equation $$\cos x\cos y\frac{{dy}}{{dx}} = - \sin x\sin y$$is

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

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

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

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

## Questions 29 of 50

Question:The solution of the differential equation $$\cos y\log (\sec x + \tan x)dx = \cos x\log (\sec y + \tan y)dy$$ is

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

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

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

None of these

## Questions 30 of 50

Question:The solution of $$\frac{{dy}}{{dx}} = \frac{1}{x}$$ is

$$y + \log x + c = 0$$

$$y = \log x + c$$

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

None of these

## Questions 31 of 50

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

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

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

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

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

## Questions 32 of 50

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

$$\cos x = c\,\cos {\rm{ec}}y$$

$$\sin x = c\sec y$$

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

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

## Questions 33 of 50

Question:The solution of $$\cos (x + y)\,dy = \,\,dx$$ is

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

$$y + {\cos ^{ - 1}}\left( {\frac{y}{x}} \right) = c$$

$$y = x\,\,\sec \left( {\frac{y}{x}} \right) + c$$

None of these

## Questions 34 of 50

Question:The solution of $$\frac{{dy}}{{dx}} + \sqrt {\,\left( {\frac{{1 - {y^2}}}{{1 - {x^2}}}} \right)} \, = \,0$$is

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

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

$${\sec ^{ - 1}}x + {\rm{cose}}{{\rm{c}}^{ - 1}}x = c$$

None of these

## Questions 35 of 50

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

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

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

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

$$x + y = c$$

## Questions 36 of 50

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

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

$$\sec y - 2\cos x = c$$

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

$$\tan y - 2\sec y = c$$

## Questions 37 of 50

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

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

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

$${\sin ^{ - 1}}\left( {\frac{y}{x}} \right) = \log x + c$$

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

## Questions 38 of 50

Question:Solution of differential equation $$2xy\frac{{dy}}{{dx}} = {x^2} + 3{y^2}$$ is

$${x^3} + {y^2} = p{x^2}$$

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

$${x^2} + {y^3} = p{x^2}$$

$${x^2} + {y^2} = p{x^3}$$

## Questions 39 of 50

Question:If $$xdy = y\,(dx + ydy),\,y > 0$$ and $$y(1) = 1,$$ then $$y( - 3)$$ is equal to

1

3

5

1

## Questions 40 of 50

Question:The solution of $$(1 + xy)y\,dx + (1 - xy)x\,dy = 0$$ is

$$\frac{x}{y} + \frac{1}{{xy}} = k$$

$$\log \left( {\frac{x}{y}} \right) = \frac{1}{{xy}} + k$$

$$\frac{x}{y} + \frac{1}{{xy}} = k$$

$$\log \left( {\frac{x}{y}} \right) = xy + k$$

## Questions 41 of 50

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

$$\log y = cx$$

$$- \frac{1}{{xy}} + \log y = c$$

$$- \frac{1}{{xy}} - \log y = c$$

$$\frac{1}{{xy}} + \log y = c$$

## Questions 42 of 50

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

$$4xy = {x^4} + c$$

$$xy = {x^4} + c$$

$$\frac{1}{4}xy = {x^4} + c$$

$$xy = 4{x^4} + c$$

## Questions 43 of 50

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

$$xy = \frac{{{x^3}}}{3} + \frac{3}{2}{x^2} + 2x + c$$

$$xy = \frac{{{x^4}}}{4} + {x^3} + {x^2} + c$$

$$xy = \frac{{{x^4}}}{4} + \frac{{{x^3}}}{3} + {x^2} + c$$

$$xy = \frac{{{x^4}}}{4} + {x^3} + {x^2} + cx$$

## Questions 44 of 50

Question:The solution of the differential equation $$\frac{{dy}}{{dx}} + \frac{{3{x^2}}}{{1 + {x^3}}}y = \frac{{{{\sin }^2}x}}{{1 + {x^3}}}$$ is

$$y(1 + {x^3}) = x + \frac{1}{2}\sin 2x + c$$

$$y(1 + {x^3}) = cx + \frac{1}{2}\sin 2x$$

$$y(1 + {x^3}) = cx - \frac{1}{2}\sin 2x$$

$$y(1 + {x^3}) = \frac{x}{2} - \frac{1}{4}\sin 2x + c$$

## Questions 45 of 50

Question:Integrating factor of $$\frac{{dy}}{{dx}} + \frac{y}{x} = {x^3} - 3$$is

$$x$$

$$\log x$$

$$- x$$

$${e^x}$$

## Questions 46 of 50

Question:Solution of $$\cos x\frac{{dy}}{{dx}} + y\sin x = 1$$is

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

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

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

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

## Questions 47 of 50

Question:The solution of $$\frac{{dy}}{{dx}} + 2y\,\tan x = \sin x$$, is

$$y\,{\sec ^3}x = {\sec ^2}x + c$$

$$y\,{\sec ^2}x = \sec x + c$$

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

None of these

## Questions 48 of 50

Question:Solution of the equation $$(x + \log y)dy + y\,dx = 0$$ is

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

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

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

None of these

## Questions 49 of 50

Question:The equation of the curve which passes through the point (1, 1) and whose slope is given by $$\frac{{2y}}{x}$$, is

$$y = {x^2}$$

$${x^2} - {y^2} = 0$$

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

None of these

## Questions 50 of 50

Question:The equation of the curve that passes through the point $$(1,\,2)$$ and satisfies the differential equation $$\frac{{dy}}{{dx}} = \frac{{ - 2xy}}{{({x^2} + 1)}}$$is

$$y({x^2} + 1) = 4$$
$$y({x^2} + 1) + 4 = 0$$
$$y({x^2} - 1) = 4$$