Differential Equations Test 4

Total Questions:50 Total Time: 75 Min

Remaining:

 

Questions 1 of 50

Question:The order and degree of the differential equation \(y = x\frac{{dy}}{{dx}} + \sqrt {{a^2}{{\left( {\frac{{dy}}{{dx}}} \right)}^2} + {b^2}} \) are

Answers Choices:

1, 2

2, 1

1, 1

2, 2

Questions 2 of 50

Question:The order of the differential equation of a family of curves represented by an equation containing four arbitrary constants, will be

Answers Choices:

2

4

6

None of these

Questions 3 of 50

Question:Order and degree of differential equation \(\frac{{{d^2}y}}{{d{x^2}}} = {\left\{ {y + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right\}^{1/4}}\) are

Answers Choices:

4 and 2

1 and 2

1 and 4

2 and 4

Questions 4 of 50

Question:The degree of the differential equation \(\left( {\frac{{2 + \sin x}}{{1 + y}}} \right)\frac{{dy}}{{dx}} = - \cos ,x\;y(0) = 1,\) is

Answers Choices:

1

2

3

6

Questions 5 of 50

Question:Degree of the given differential equation \({\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^3} = {\left( {1 + \frac{{dy}}{{dx}}} \right)^{1/2}}\) , is

Answers Choices:

2

3

\(\frac{1}{2}\)

6

Questions 6 of 50

Question:The order and degree of the differential equation \(\frac{{{d^2}y}}{{d{x^2}}} + {\left( {\frac{{dy}}{{dx}}} \right)^{\frac{1}{3}}} + {x^{\frac{1}{4}}} = 0\) are respectively

Answers Choices:

2, 3

3, 3

2, 6

2, 4

Questions 7 of 50

Question:The degree and order of the differential equation of the family of all parabolas whose axis is x–axis, are respectively

Answers Choices:

2, 1

1, 2

3, 2

2, 3

Questions 8 of 50

Question:The degree of the differential equation \({\left( {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right)^{3/4}} = {\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^{1/3}}\) is

Answers Choices:

\(\frac{1}{3}\)

4

9

\(\frac{3}{4}\)

Questions 9 of 50

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

Answers Choices:

3, 2

2, 1

2, 2

2, 3

Questions 10 of 50

Question:The degree of the differential equation \(\frac{{{d^2}y}}{{d{x^2}}} - \sqrt {\frac{{dy}}{{dx}} - 3} = x\) is

Answers Choices:

2

1

2-Jan

3

Questions 11 of 50

Question:The differential equation whose solution is \(y = {c_1}\cos ax + {c_2}\sin ax\) is (Where \({c_1},\;{c_2}\) are arbitrary constants)

Answers Choices:

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

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

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

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

Questions 12 of 50

Question:The differential equation for the line \(y = mx + c\) is (where c is arbitrary constant)

Answers Choices:

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

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

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

None of these

Questions 13 of 50

Question:The differential equation of all straight lines passing through the point \((1,\, - 1)\) is

Answers Choices:

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

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

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

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

Questions 14 of 50

Question:If \({x^2} + {y^2} = 1\) then \(\left( {y' = \frac{{dy}}{{dx}},y'' = \frac{{{d^2}y}}{{d{x^2}}}} \right)\)

Answers Choices:

\(yy'' - 2{(y')^2} + 1 = 0\)

\(yy'' + {(y')^2} + 1 = 0\)

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

\(yy'' + 2{(y')^2} + 1 = 0\)

Questions 15 of 50

Question:Differential equation of \(y = \sec ({\tan ^{ - 1}}x)\) is

Answers Choices:

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

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

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

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

Questions 16 of 50

Question:The differential equation satisfied by the family of curves \(y = ax\cos \,\left( {\frac{1}{x} + b} \right)\), where a, b are parameters, is

Answers Choices:

\({x^2}{y_2} + y = 0\)

\({x^4}{y_2} + y = 0\)

\(x{y_2} - y = 0\)

(d) \({x^4}{y_2} - y = 0\)

Questions 17 of 50

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

Answers Choices:

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

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

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

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

Questions 18 of 50

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

Answers Choices:

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

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

\(y = 2cx\)

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

Questions 19 of 50

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

Answers Choices:

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

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

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

None of these

Questions 20 of 50

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

Answers Choices:

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

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

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

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

Questions 21 of 50

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

Answers Choices:

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

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

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

None of these

Questions 22 of 50

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

Answers Choices:

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

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

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

None of these

Questions 23 of 50

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

Answers Choices:

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

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

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

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

Questions 24 of 50

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

Answers Choices:

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

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

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

None of these

Questions 25 of 50

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

Answers Choices:

\({e^{5x}} + {e^{5y}} = c\)

\({e^{5x}} - {e^{5y}} = c\)

\({e^{5x + 5y}} = c\)

None of these

Questions 26 of 50

Question:Solution of differential equation \(\frac{{dy}}{{dx}} = 2xy\)is

Answers Choices:

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

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

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

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

Questions 27 of 50

Question:Solution of \(ydx - xdy = {x^2}ydx\)is

Answers Choices:

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

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

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

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

Questions 28 of 50

Question:For solving \(\frac{{dy}}{{dx}} = (4x + y + 1)\), suitable substitution is

Answers Choices:

\(y = vx\)

\(y = 4x + v\)

\(y = 4x\)

\(y + 4x + 1 = v\)

Questions 29 of 50

Question:The solution of the equation \((2y - 1)\,\,dx - (2x + 3)\,dy = 0\) is

Answers Choices:

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

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

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

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

Questions 30 of 50

Question:The differential equation \(\cot y\,\,dx = x\,\,dy\) has a solution of the form

Answers Choices:

\(y = \cos x\)

\(x = c\sec y\)

\(x = \sin y\)

\(y = \sin x\)

Questions 31 of 50

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

Answers Choices:

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

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

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

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

Questions 32 of 50

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

Answers Choices:

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

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

\(y = \log y + 1\)

\(y = xy + c\)

Questions 33 of 50

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

Answers Choices:

\({\log _e}({x^2} + {y^2}) + 2{\tan ^{ - 1}}\frac{y}{x} + c = 0\)

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

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

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

Questions 34 of 50

Question:If \(y' = \frac{{x - y}}{{x + y}}\), then its solution is

Answers Choices:

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

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

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

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

Questions 35 of 50

Question:Which of the following equation is non-linear

Answers Choices:

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

\(y\frac{{dy}}{{dx}} + 4x = 0\)

\(dx + dy = 0\)

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

Questions 36 of 50

Question:Which of the following equation is linear

Answers Choices:

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

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

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

\(y\frac{{dy}}{{dx}} - 4 = x\)

Questions 37 of 50

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

Answers Choices:

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

\(my = ({x^m} + c)\cos x\)

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

None of these

Questions 38 of 50

Question:An integrating factor for the differential equation \((1 + {y^2})dx - ({\tan ^{ - 1}}y - x)dy = 0\)

Answers Choices:

\({\tan ^{ - 1}}y\)

\({e^{{{\tan }^{ - 1}}y}}\)

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

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

Questions 39 of 50

Question:Solution of differential equation \(\frac{{dy}}{{dx}} + ay = {e^{mx}}\) is

Answers Choices:

\((a + m)\,y = {e^{mx}} + c\)

\(y{e^{ax}} = m{e^{mx}} + c\)

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

\((a + m)y = {e^{mx}} + c{e^{ - ax}}(a + m)\)

Questions 40 of 50

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

Answers Choices:

\(\log x\)

\(\log (\log x)\)

\({e^x}\)

\(x\)

Questions 41 of 50

Question:Integrating factor of the differential equation \(\frac{{dy}}{{dx}} + y\tan x - \sec x = 0\) is

Answers Choices:

\({e^{\sin x}}\)

\(\frac{1}{{\sin x}}\)

\(\frac{1}{{\cos x}}\)

\({e^{\cos x}}\)

Questions 42 of 50

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

Answers Choices:

\(y = \log x + c\)

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

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

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

Questions 43 of 50

Question:Integrating factor of the differential equation \(\frac{{dy}}{{dx}} + P(x)y = Q(x)\) is

Answers Choices:

\(\int {P\,dx} \)

\(\int {Q\,dx} \)

\({e^{\int {P\,dx} }}\)

\({e^{\int {Q\,dx} }}\)

Questions 44 of 50

Question:To reduce the differential equation \(\frac{{dy}}{{dx}} + P(x)y = Q(x).{y^n}\) to the linear form, the substitution is

Answers Choices:

\(v = \frac{1}{{{y^n}}}\)

\(v = \frac{1}{{{y^{n - 1}}}}\)

\(v = {y^n}\)

\(v = {y^{n - 1}}\)

Questions 45 of 50

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

Answers Choices:

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

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

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

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

Questions 46 of 50

Question:The solution of \(dy = \cos x(2 - y\cos {\rm{ec}}x)dx\) where \(y = 2\) when \(x = \frac{\pi }{2}\) is

Answers Choices:

\(y = \sin x + {\rm{cosec }}x\)

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

\(y = \frac{1}{{\sqrt 2 }}\sec \frac{x}{2} + \sqrt 2 \cos \frac{x}{2}\)

None of these

Questions 47 of 50

Question:Solution of differential equation \(x\,dy - y\,dx = 0\) represents

Answers Choices:

Rectangular hyperbola

Straight line passing through origin

Parabola whose vertex is at origin

Circle whose centre is at origin

Questions 48 of 50

Question:Integral curve satisfying \(y' = \frac{{{x^2} + {y^2}}}{{{x^2} - {y^2}}},\;y(1) = 2\) has the slope at the point (1, 0) of the curve, equal to

Answers Choices:

5/3

1

1

5/3

Questions 49 of 50

Question:A particle starts at the origin and moves along the x-axis in such a way that its velocity at the point (x, 0) is given by the formula \(\frac{{dx}}{{dt}} = {\cos ^2}\pi x.\) Then the particle never reaches the point on

Answers Choices:

\(x = \frac{1}{4}\)

\(x = \frac{3}{4}\)

\(x = \frac{1}{2}\)

x = 1

Questions 50 of 50

Question:The slope of the tangent at (x, y) to a curve passing through a point (2, 1) is \(\frac{{{x^2} + {y^2}}}{{2xy}}\), then the equation of the curve is

Answers Choices:

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

\(2({x^2} - {y^2}) = 6y\)

\(x({x^2} - {y^2}) = 6\)

\(x({x^2} + {y^2}) = 10\)