# 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

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

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

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

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

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

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

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

$$\frac{1}{3}$$

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

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

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)

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

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

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

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

## Questions 12 of 50

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

$$\frac{{dy}}{{dx}} = m$$

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

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

None of these

## Questions 13 of 50

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

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

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

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

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

## Questions 14 of 50

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

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

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

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

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

## Questions 15 of 50

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

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

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

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

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

## 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

$${x^2}{y_2} + y = 0$$

$${x^4}{y_2} + y = 0$$

$$x{y_2} - y = 0$$

(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

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

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

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

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

## Questions 18 of 50

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

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

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

$$y = 2cx$$

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

## Questions 19 of 50

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

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

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

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

None of these

## Questions 20 of 50

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

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

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

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

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

## Questions 21 of 50

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

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

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

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

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

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

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

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

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

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

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

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

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

## Questions 24 of 50

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

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

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

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

None of these

## Questions 25 of 50

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

$${e^{5x}} + {e^{5y}} = c$$

$${e^{5x}} - {e^{5y}} = c$$

$${e^{5x + 5y}} = c$$

None of these

## Questions 26 of 50

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

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

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

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

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

## Questions 27 of 50

Question:Solution of $$ydx - xdy = {x^2}ydx$$is

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

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

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

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

## Questions 28 of 50

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

$$y = vx$$

$$y = 4x + v$$

$$y = 4x$$

$$y + 4x + 1 = v$$

## Questions 29 of 50

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

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

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

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

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

## Questions 30 of 50

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

$$y = \cos x$$

$$x = c\sec y$$

$$x = \sin y$$

$$y = \sin x$$

## Questions 31 of 50

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

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

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

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

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

## Questions 32 of 50

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

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

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

$$y = \log y + 1$$

$$y = xy + c$$

## Questions 33 of 50

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

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

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

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

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

## Questions 34 of 50

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

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

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

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

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

## Questions 35 of 50

Question:Which of the following equation is non-linear

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

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

$$dx + dy = 0$$

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

## Questions 36 of 50

Question:Which of the following equation is linear

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

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

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

$$y\frac{{dy}}{{dx}} - 4 = x$$

## Questions 37 of 50

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

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

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

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

None of these

## Questions 38 of 50

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

$${\tan ^{ - 1}}y$$

$${e^{{{\tan }^{ - 1}}y}}$$

$$\frac{1}{{1 + {y^2}}}$$

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

## Questions 39 of 50

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

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

$$y{e^{ax}} = m{e^{mx}} + c$$

$$y = {e^{mx}} + c{e^{ - ax}}$$

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

## Questions 40 of 50

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

$$\log x$$

$$\log (\log x)$$

$${e^x}$$

$$x$$

## Questions 41 of 50

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

$${e^{\sin x}}$$

$$\frac{1}{{\sin x}}$$

$$\frac{1}{{\cos x}}$$

$${e^{\cos x}}$$

## Questions 42 of 50

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

$$y = \log x + c$$

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

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

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

## Questions 43 of 50

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

$$\int {P\,dx}$$

$$\int {Q\,dx}$$

$${e^{\int {P\,dx} }}$$

$${e^{\int {Q\,dx} }}$$

## 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

$$v = \frac{1}{{{y^n}}}$$

$$v = \frac{1}{{{y^{n - 1}}}}$$

$$v = {y^n}$$

$$v = {y^{n - 1}}$$

## Questions 45 of 50

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

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

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

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

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

## 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

$$y = \sin x + {\rm{cosec }}x$$

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

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

None of these

## Questions 47 of 50

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

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

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

$$x = \frac{1}{4}$$

$$x = \frac{3}{4}$$

$$x = \frac{1}{2}$$

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

$$2({x^2} - {y^2}) = 3x$$
$$2({x^2} - {y^2}) = 6y$$
$$x({x^2} - {y^2}) = 6$$
$$x({x^2} + {y^2}) = 10$$