CBSE Questions for Class 12 Commerce Maths Differential Equations Quiz 7 - MCQExams.com

The general solution of the differential equation $$e^xdy+(ye^x+2x)dx=0$$ is
  • $$xe^x+x^2=C$$
  • $$xe^x-y^2=C$$
  • $$ye^x+x^2=C$$
  • $$ye^x-x^2=C$$
The solution of differential equation $$\cos x.\sin y dx+\sin x. \cos ydy=0$$ is 
  • $$\dfrac{\sin x}{\sin y}=c$$
  • $$\sin x. \sin y=c$$
  • $$\sin x+\sin y=c$$
  • $$\cos x.\cos y=c$$
Solution of the differential equation: $$\left( 2xcosy+{ y }^{ 2 }cosx \right) dx+\left( 2ysinx-{ x }^{ 2 }siny \right) dy=0$$ is :
  • $${ x }^{ 2 }sinx+y^{ 2 }cosx=c$$
  • $${ x }^{ 2 }siny+y^{ 2 }cosx=c$$
  • $${ x }^{ 2 }cosy+y^{ 2 }sinx=c$$
  • None of these
For the given differential equation find the general solution:
$$\dfrac { dy }{ dx } +2y=sin x$$
  • $$\dfrac 1 5 [2sin x -cosx ]+C e ^{-2x}$$
  • $$\dfrac 1 5 [2sin x -cosx ]+C $$
  • $$\dfrac 1 2 [5sin x -cosx ]+C e ^{-2x}$$
  • None of these
The differential equation of the system of circles touching the $$x$$-axis at origin is  
  • $$\left( { x }^{ 2 }-{ y }^{ 2 } \right) \dfrac { dy }{ dx } -2xy=0$$
  • $$\left( { x }^{ 2 }+{ y }^{ 2 } \right) \dfrac { dy }{ dx } +2xy=0$$
  • $$\left( { x }^{ 2 }-{ y }^{ 2 } \right) \dfrac { dy }{ dx } +2xy=0$$
  • $$\left( { x }^{ 2 }+{ y }^{ 2 } \right) \dfrac { dy }{ dx } -2xy=0$$
The solution of $$xdx+ydy=\frac { xdy-ydx }{ { x }^{ 2 }+{ y }^{ 2 } } $$ is ________________________.
  • $${ x }^{ 2 }+{ y }^{ 2 }=2\quad { tan }^{ -1 }\frac { y }{ x } +C$$
  • $${ x }^{ 2 }-{ y }^{ 2 }=2\quad { tan }^{ -1 }\frac { y }{ x } +C$$
  • $${ x }^{ 2 }+{ y }^{ 2 }=2\quad { tan }^{ -1 }\frac { x }{ y } +C$$
  • $${ x }^{ 2 }-{ y }^{ 2 }=2\quad { tan }^{ -1 }\frac { x }{ y } +C$$
Solution of differential equation $$ \cfrac {dy} {dx} + x{sin}^{2} y $$= $$sin y  \quad cos y  \quad $$is
  • $$ tany=(x-1)+ce-x $$
  • $$ coty=(x-1)+ce-x $$
  • $$ tany=(x-1)ex+c $$
  • $$ coty=(x-1)ex+c $$
Let y=y(x) be the solution of the differential equation $$sinx\dfrac { dy }{ dx } +ycosx=4x,x\in (0,\pi )$$. If $$y=\left( \dfrac { \pi  }{ 2 }  \right) =0,then\quad y\left( \dfrac { \pi  }{ 6 }  \right) $$ is equal to :
  • $$\dfrac { 4 }{ 9\sqrt { 3 } } { \pi }^{ 2 }$$
  • $$\dfrac { 8 }{ 9\sqrt { 3 } } { \pi }^{ 2 }$$
  • $$-\dfrac { 8 }{ 9 } { \pi }^{ 2 }$$
  • $$-\dfrac { 4 }{ 9 } { \pi }^{ 2 }$$
Let $$f\left( x \right) ={ cos }^{ -1 }\left( cosx \right)$$ then 
  • f\left( x \right) is differentable for all $$\quad x\epsilon R$$
  • f\left( x \right) is not differentable at$$ \quad x\epsilon 0$$
  • f\left( x \right) is not differentable if$$ x=n\pi .n\epsilon I$$
  • f\left( x \right) is not differentable if$$ x=\frac { n\pi }{ 2 } n\epsilon I$$
differential equation of all parabolas whose axis s y - axis.......
  • y$$\dfrac{dy}{dx} + x \dfrac{d^2y}{dx^2} = 1$$
  • x$$\dfrac{d^2y}{dx^2} - \dfrac{dy}{dx} = 1$$
  • $$y^2\dfrac{dy}{dx} + 2x \dfrac{dy}{dx} = 0$$
  • $$x\dfrac{dy}{dx} -x^2 \dfrac{d^2y}{dx^2} = 0$$
$$Let_y-y(x)$$ be the solution of the differential
equation $$sinx\dfrac{dy}{dx} -ycosx -4x,_x\epsilon (0,\pi ). if y\left(\frac{\pi}{2}\right)$$ = 0,
then y $$\left(\dfrac{\pi}{6}\right)$$ is equal to 
  • $$-\dfrac{4}{9}\pi^2$$
  • $$-\dfrac{4}{9\sqrt3}\pi^2$$
  • $$-\dfrac{-8}{9\sqrt3}\pi^2$$
  • None of these
Integrating factors of the differential equation $$\frac{{dy}}{{dx}} + y = \frac{{1 + y}}{x}$$ is 
  • $$x/{e^x}$$
  • $${e^x}/x$$
  • $$x{e^x}$$
  • $${e^x}$$
Solution of differential equation $$\dfrac{dy}{dx} + \dfrac{y}{x} = \dfrac{1}{(1 + \ell nx + \ell ny)}$$ is (where C is an integration constant)
  • $$2(1 + \ell n (xy)) = x^2 + C$$
  • $$xy \ell n (xy) = x^2 + C$$
  • $$(1 + \ell n (xy)) = x^2 + C$$
  • $$2xy \ell n (xy) = x^2 + C$$
The solution of the differential equation $$(x^2-yx^2)\dfrac{y^3}{x}=k+y^2+xy^2=0$$ is?
  • $$log\left(\dfrac{x}{y}\right)=\dfrac{1}{x}+\dfrac{1}{y}+c$$
  • $$log\left(\dfrac{y}{x}\right)=\dfrac{1}{x}+\dfrac{1}{y}+c$$
  • $$log(xy)=\dfrac{1}{x}+\dfrac{1}{y}+c$$
  • $$log(xy)+\dfrac{1}{x}+\dfrac{1}{y}=c$$
An integrating factor for the $$DE: (1+y^{2})dx-(\tan^{-1}y-x)dy=0$$ is 
  • $$\tan^{-1} y$$
  • $$e^{\tan^{-1}y}$$
  • $$\dfrac{1}{1+y^{2}}$$
  • $$\dfrac{1}{x(1+y^{2})}$$
Solution of differential equation $$\left( { 2y+xy }^{ 3 } \right) dx+\left( x{ +x }^{ 2 }{ y }^{ 2 } \right) dy=0$$
  • $${ xy }^{ 2 }+\dfrac { { x }^{ 3 }{ y }^{ 3 } }{ 3 } =c$$
  • $${ xy }^{ 2 }-\dfrac { { x }^{ 3 }{ y }^{ 3 } }{ 3 } =c$$
  • $${ x }^{ 2 }y+\dfrac { { x }^{ 4 }{ y }^{ 4 } }{ 3 } =c$$
  • None of these
Solution of the differential equation $$\left( { x }^{ 2 }+1 \right) y'+2xy=4{ x }^{ 2 }$$ is 
  • $$y\left( 1+{ x }^{ 2 } \right) =\dfrac { 4{ x }^{ 3 } }{ 3 } +C$$
  • $$y\left( 1-{ x }^{ 2 } \right) ={ x }^{ 3 }+C$$
  • $$y\left( 1-{ x }^{ 2 } \right) =\dfrac { { x }^{ 3 } }{ 2 } +C$$
  • None of these
Consider the differential equation, $$ydx-(x+y^{2})dy=0$$. If for $$y=1$$, x takes value $$1$$, then value of $$x$$ when $$y=4$$ is:
  • $$16$$
  • $$36$$
  • $$64$$
  • $$9$$
General solution of the differential equation $$\frac{{dy}}{{dx}} = 1 + xy$$ is
  • $$y = c \cdot \,{e^{ - x2/2}}$$
  • $$y = c \cdot \,{e^{ x2/2}}$$
  • $$y = (x + c) \cdot {e^{ - x2/2}}$$
  • None
The solution of the differentiable equation $$x^{2}\dfrac {dy}{dx}.\cos \dfrac {1}{x}-y\sin \dfrac {1}{x}=-1$$, where $$y\rightarrow -1$$ as $$x\rightarrow \infty$$ is
  • $$y=\sin \dfrac {1}{x}-\cos \dfrac {1}{x}$$
  • $$y=\dfrac {x+1}{x\sin \dfrac {1}{x}}$$
  • $$y=\cos \dfrac {1}{x}-\sin \dfrac {1}{x}$$
  • $$y=\dfrac {x+1}{x\cos \dfrac {1}{x}}$$
The population  $$p(t)$$  at time  $$t$$  of a certain mouse species satisfies the differential equation  $$\dfrac { d p ( t )  } { d t } = 0.5 p ( t ) - 450.$$  If  $$p ( 0 ) = 850 ,$$  then the time at which the population becomes zero is
  • $$\dfrac { 1 } { 2 } \ln 18$$
  • $$\ln 18$$
  • $$2 \ln 18$$
  • $$\ln 9$$
The solution of differential equation  $$\cos ^ { 2 } x \dfrac { d y } { d x } - ( \tan 2 x ) y = \cos ^ { 4 } x , | x | < \dfrac { \pi } { 4 } ,$$  where $$y \left( \dfrac { \pi } { 6 } \right) = \dfrac { 3 \sqrt { 3 } } { 8 }$$
  • $$y = \tan 2 x \cos ^ { 2 } x$$
  • $$y = \cot 2 x \cos ^ { 2 } x$$
  • $$2 y = \tan 2 x \cos ^ { 2 } x$$
  • $$2 y = \cot 2 x \cos ^ { 2 } x$$
The order of the differential equation
$$2x^2\dfrac{d^2y}{dx^2}-3\dfrac{dy}{dx}+y=0$$ is
  • $$2$$
  • $$1$$
  • $$0$$
  • Not defined
The general solution of the differential equation $$\dfrac{ydx-xdy}{y}=0$$ is
  • $$xy=C$$
  • $$x=Cy^2$$
  • $$y=Cx$$
  • $$y=Cx^2$$
A differential equation associated with the primitive $$y=a+b\ e^{5x}+c\ e^{7x}$$ is
  • $$y_{3}+2y_{2}-y_{1}=0$$
  • $$y_{3}+2y_{2}-35y_{1}=0$$
  • $$4y_{3}+5y_{2}-20y_{1}=0$$
  • $$none\ of\ these$$
If $$\sqrt { \dfrac { \upsilon  }{ \mu  }  } +\sqrt { \dfrac { \mu  }{ \upsilon  }  } =6$$, then $$\dfrac { d\upsilon  }{ d\mu  } =$$
  • $$\dfrac { 17\mu -\upsilon }{ \mu -17\upsilon } $$
  • $$\dfrac { \mu -17\upsilon }{ 17\mu -\upsilon } $$
  • $$\dfrac { 17\mu +\upsilon }{ \mu -17\upsilon } $$
  • $$\dfrac { \mu +17\upsilon }{ 17\mu -\upsilon } $$
The number of arbitrary constant in the particular solution of a differential equation is
  • $$3$$
  • $$4$$
  • infinite
  • zero
Solution of differential equation $$\sin y.\dfrac {dy}{dx}+\dfrac {1}{x}\cos y=x^{4}\cos^{2}y$$ is
  • $$x\sec y=x^{6}+C$$
  • $$6x\sec y=x+C$$
  • $$6x\sec y=x^{6}+C$$
  • $$6x\sec y=6x^{6}+C$$
The general solution of the differential equation $$\dfrac{dy}{dx}=e^{x+y}$$  is : 
  • $$e^{-x}+e^{-y}=c$$
  • $$e^{x}+e^{-y}=c$$
  • $$e^{x}+e^{y}=c$$
  • $$e^{-x}+e^{y}=c$$
The solution of the differential equation $${ 2x }^{ 2 }y\dfrac { dy }{ dx } =tan\left( { x }^{ 2 }{ y }^{ 2 } \right) -{ 2xy }^{ 2 }$$ given $$y(1)=\sqrt { \dfrac { \pi  }{ 2 }  } $$ is
  • sin $$\left( { x }^{ 2 }{ y }^{ 2 } \right) -1=0$$
  • $$cos\left( \dfrac { \pi }{ 2 } +{ x }^{ 2 }{ y }^{ 2 } \right) +x=0$$
  • $$sin\left( { x }^{ 2 }{ y }^{ 2 } \right) ={ e }^{ x-1 }$$
  • $$sin\left( { x }^{ 2 }{ y }^{ 2 } \right) ={ e }^{ 2(x-1) }$$
The solution of the equation $$(x^2 +xy)dy=(x^2+y^2)dx $$is 
  • $$log x=(x-y)+\frac{y}{x}+c$$
  • $$logx=2log(x-y)+\frac{y}{x}+c$$
  • $$logx=log(x-y)+\frac{x}{y}+c$$
  • none of these above
Solve the given differential equation $$\dfrac{dy}{dx}=(cosx-sinx),$$
  • $$y=sinx+cosx+c$$
  • $$y=sinx-cosx+c$$
  • $$y=tanx+secx$$
  • $$\text{None  of  these}$$
The solution of the differential equation $$x\dfrac{dy}{dx}=y(log y-log x+1)$$ is
  • $$y=xe^{cx}$$
  • $$y+xe^{cx}=0$$
  • $$y+e^{x}$$
  • none of these
Which of the following functions is differentiable at x=0?
  • $${ e }^{ -\left| x \right| }-\left| x \right| $$
  • $${ e }^{ \left| x \right| }+\left| x \right| $$
  • $$\left| x \right| -{ e }^{ \left| x \right| }$$
  • $$\left| x \right| -{ e }^{ -\left| x \right| }$$
Solution of the differential equation of $${ (y }^{ 2 }-{ x }^{ 3 })dx-xydy=0\quad $$ is
  • $${ y }^{ 2 }+2{ x }^{ 3 }+c{ x }^{ 2 }=0\quad $$
  • $${ y }^{ 2 }-2{ x }^{ 3 }+c{ x }^{ 2 }=0$$
  • $${ y }^{ 2 }+2{ x }^{ 3 }-c{ x }^{ 2 }=0$$
  • $${ y }^{ 2 }+3{ x }^{ 3 }+c{ x }^{ 2 }=0 $$
The integrating factor (I.F) of differential equation $$\cfrac { dy }{ dx } \left( 1+x \right) -xy=1-x$$ is _____
  • $$\left( 1+x \right) { e }^{ x }$$
  • $$\left( x-1 \right) { e }^{ x }\quad $$
  • $$\left( 1+x \right) { e }^{ -x }$$
  • $$\left( 1-x \right) { e }^{ -x }\quad $$
If $$\cos { x } \cfrac { dy }{ dx } -y\sin { x } =6x,(0<x<\cfrac { \pi  }{ 2 } )$$ and $$\quad y\left( \cfrac { \pi  }{ 3 }  \right) =0\quad $$ then $$y\left( \cfrac { \pi  }{ 6 }  \right) $$ is equal to:
  • $$-\cfrac { { \pi }^{ 2 } }{ 4\sqrt { 3 } } $$
  • $$-\cfrac { { \pi }^{ 2 } }{ 2 } $$
  • $$-\cfrac { { \pi }^{ 2 } }{ 2\sqrt { 3 } } $$
  • $$\cfrac { { \pi }^{ 2 } }{ 2\sqrt { 3 } } $$
Let $$f(x)$$ be a function such that $$f(0)=f'(0)=0, f''(x)=\sec^{4}x+4$$, then the function is
  • $$\log (\sin x)+\dfrac{1}{3}\tan^{3}x+xb$$
  • $$\dfrac{2}{3}\log (\sec^2 x)+\dfrac{1}{6}\tan^{2}x+2x^{2}$$
  • $$\log (\cos x)+\dfrac{1}{6}\cos^{2}x+\dfrac{x^{2}}{5}$$
  • $$None\ of\ these$$
if $$ y = y(x) $$ and $$ \dfrac{ 2 + sinx }{y + 1}(\dfrac{dy}{dx}) = -cosx, y(0) = 1, then y(\pi/2) $$ equals
  • 1/3
  • 2/3
  • -1/3
  • 1
If $$ x\dfrac{dy}{dx} = y(\log y - \log x + 1) $$, then the solution of the equation is 
  • $$ \log \dfrac{x}{y} = cy $$
  • $$ \log \dfrac{y}{x} = cy $$
  • $$ \log \dfrac{x}{y} = cx $$
  • None of these
Solution of $$ \dfrac{dy}{dx} + 2xy = y $$ is 
  • $$ y = ce^{x-x^{2}} $$
  • $$ y = ce^{x^{2}-x} $$
  • $$ y = ce^{x} $$
  • $$ y = ce^{-x^{2}} $$
The solution of differential equation $$ yy' = x\big(\dfrac{y^{2}}{x^{2}} + \dfrac{f (y^{2}/x^{2})}{f' (y^{2}/x^{2})}\big) $$ is
  • $$ f(y^{2}/x^{2}) = cx^{2} $$
  • $$ x^{2}f(y^{2}/x^{2}) = c^{2}y^{2} $$
  • $$ x^{2}f(y^{2}/x^{2}) = c $$
  • $$ f(y^{2}/x^{2}) = cy/x $$
The solution of $$ \dfrac{dv}{dt} + \dfrac{k}{m}v = -g $$ is
  • $$ v = ce^{-\dfrac{k}{m}t} - \dfrac{mg}{k} $$
  • $$ v = c - \dfrac{mg}{k}e^{-\dfrac{k}{m}t} $$
  • $$ ve^{-\dfrac{k}{m}t} = c - \dfrac{mg}{k} $$
  • $$ ve^{-\dfrac{k}{m}t} = c - \dfrac{mg}{k} $$
The solution of the equation $$ (x^{2}y + x^{2})dx + y^{2}(x-1)dy = 0 $$ is given by
  • $$ x^{2} + y^{2} + 2(x-y) + 2 ln \dfrac{(x-1)(y+1)}{c} = 0 $$
  • $$ x^{2} + y^{2} + 2(x-y) + ln \dfrac{(x-1)(y+1)}{c} = 0 $$
  • $$ x^{2} + y^{2} + 2(x-y) - 2 ln \dfrac{(x-1)(y+1)}{c} = 0 $$
  • None of these
Solution of $$ 2y sin x \frac {dv}{dx}= 2 sin x cos x -y^2 cos x, $$ for $$x = \frac { \pi}{2} , y = 1 $$ is 
  • $$ y^2 = sin x $$
  • $$ y = sin^2 x $$
  • $$ y^2 = 1 +cos x $$
  • None of these
If $$ \phi(x) =\int \left\{ \phi (x) \right\}^{-2} $$dx and $$ \phi ( 1) =0 $$ then $$ \phi (x) = $$
  • $$ \left\{ 2(x -1) \right\}^{1/4} $$
  • $$ \left\{ 5(x -2) \right\}^{1/5} $$
  • $$ \left\{ 3(x -1) \right\}^{1/3} $$
  • None of these
Solution of differential equation $$ dy - \sin x \sin y dx = 0 $$ is
  • $$ e^{\cos x}\tan \dfrac{y}{2} = c $$
  • $$ e^{\cos x}\tan y = c $$
  • $$ \cos x \tan y = c $$
  • $$ \cos x \sin y = c $$
if integrating factor of $$ x(1-x^{2})dy + (2x^{2}y - y -ax^{3})dx=0 $$ is $$ e^{\int pdx} $$, then P is equal to 
  • $$ \dfrac{2x^{2} - ax^{3}}{x(1-x^{2})} $$
  • $$ 2x^{3} - 1 $$
  • $$ \dfrac{2x^{2} - a}{ax^{3}} $$
  • $$ \dfrac{2x^{2} - 1}{x(1-x^{2})} $$
The solution of differentiation equation $$(2y+xy^{3})dx+(x+x^{2}y^{2})dy=o $$ is 
  • $$ x^{2}y+\dfrac{x^{3}y^{3}}{3}= c $$
  • $$ xy^{2}+\dfrac{x^{3}y^{3}}{3}= c $$
  • $$ x^{2}y+\dfrac{x^{4}y^{4}}{4}= c $$
  • None of these
The solution of differential equation $$ \dfrac{x+y\dfrac{dy}{dx}}{y-x\dfrac{dy}{dx}} = \dfrac{x\cos^{2}(x^{2} + y^{2})}{y^{3}} $$ is
  • $$ \tan(x^{2}+y^{2})=\dfrac{x^{2}}{y^{2}}+c $$
  • $$ \cot(x^{2}+y^{2})=\dfrac{x^{2}}{y^{2}}+c $$
  • $$ \tan(x^{2}+y^{2})=\dfrac{y^{2}}{x^{2}}+c $$
  • $$ \tan(x^{2}+y^{2})=\dfrac{y^{2}}{x^{2}}+c $$
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