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

Answers Choices:

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

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

\(dy + dx = 0\)

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

Answers Choices:

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

Answers Choices:

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

Answers Choices:

\(2,\,2\)

\(2,\,3\)

\(3,2\)

None of these

Questions 5 of 50

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

Answers Choices:

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

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

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

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

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

Answers Choices:

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

Answers Choices:

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

Answers Choices:

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

Answers Choices:

2

3

1

None of these

Questions 10 of 50

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

Answers Choices:

\(\frac{{dy}}{{dx}} + 8y = 0\)

\(\frac{{dy}}{{dx}} - 8y = 0\)

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

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

Questions 11 of 50

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

Answers Choices:

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

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

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

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

Questions 12 of 50

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

Answers Choices:

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

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

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

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

Answers Choices:

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

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

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

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

Questions 14 of 50

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

Answers Choices:

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

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

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

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

Questions 15 of 50

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

Answers Choices:

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

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

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

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

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

Answers Choices:

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

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

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

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

Questions 17 of 50

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

Answers Choices:

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

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

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

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

Questions 18 of 50

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

Answers Choices:

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

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

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

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

Questions 19 of 50

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

Answers Choices:

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

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

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

None of these

Questions 20 of 50

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

Answers Choices:

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

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

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

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

Questions 21 of 50

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

Answers Choices:

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

\(cx = \sec y + \tan y\)

\(cy = \sec x\tan x\)

\(cy = \sec x + \tan x\)

Questions 22 of 50

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

Answers Choices:

\(y = 1 + cxy\)

\(y = \log \{ cxy\} \)

\(y + 1 = cxy\)

\(y = c + xy\)

Questions 23 of 50

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

Answers Choices:

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

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

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

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

Questions 24 of 50

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

Answers Choices:

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

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

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

\(y = x + A\)

Questions 25 of 50

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

Answers Choices:

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

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

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

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

Questions 26 of 50

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

Answers Choices:

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

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

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

None of these

Questions 27 of 50

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

Answers Choices:

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

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

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

None of these

Questions 28 of 50

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

Answers Choices:

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

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

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

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

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

Answers Choices:

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

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

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

None of these

Questions 30 of 50

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

Answers Choices:

\(y + \log x + c = 0\)

\(y = \log x + c\)

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

None of these

Questions 31 of 50

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

Answers Choices:

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

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

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

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

Questions 32 of 50

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

Answers Choices:

\(\cos x = c\,\cos {\rm{ec}}y\)

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

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

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

Questions 33 of 50

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

Answers Choices:

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

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

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

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

Answers Choices:

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

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

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

None of these

Questions 35 of 50

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

Answers Choices:

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

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

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

\(x + y = c\)

Questions 36 of 50

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

Answers Choices:

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

\(\sec y - 2\cos x = c\)

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

\(\tan y - 2\sec y = c\)

Questions 37 of 50

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

Answers Choices:

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

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

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

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

Questions 38 of 50

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

Answers Choices:

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

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

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

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

Questions 39 of 50

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

Answers Choices:

1

3

5

1

Questions 40 of 50

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

Answers Choices:

\(\frac{x}{y} + \frac{1}{{xy}} = k\)

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

\(\frac{x}{y} + \frac{1}{{xy}} = k\)

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

Questions 41 of 50

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

Answers Choices:

\(\log y = cx\)

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

\( - \frac{1}{{xy}} - \log y = c\)

\(\frac{1}{{xy}} + \log y = c\)

Questions 42 of 50

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

Answers Choices:

\(4xy = {x^4} + c\)

\(xy = {x^4} + c\)

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

\(xy = 4{x^4} + c\)

Questions 43 of 50

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

Answers Choices:

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

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

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

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

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

Answers Choices:

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

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

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

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

Questions 45 of 50

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

Answers Choices:

\(x\)

\(\log x\)

\( - x\)

\({e^x}\)

Questions 46 of 50

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

Answers Choices:

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

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

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

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

Questions 47 of 50

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

Answers Choices:

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

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

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

None of these

Questions 48 of 50

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

Answers Choices:

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

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

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

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

Answers Choices:

\(y = {x^2}\)

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

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

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

Answers Choices:

\(y({x^2} + 1) = 4\)

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

\(y({x^2} - 1) = 4\)

None of these