CBSE Questions for Class 12 Commerce Maths Differential Equations Quiz 10 - MCQExams.com

Solution of the differential equation $$\left\{ \dfrac { 1 }{ x } -\dfrac { { y }^{ 2 } }{ (x-y)^{ 2 } }  \right\} dy+\left\{ \dfrac { { y }^{ 2 } }{ (x-y)^{ 2 } } -\dfrac { 1 }{ y }  \right\} dx=0$$  is
  • $$\ln|\dfrac{x}{y}|+\dfrac{xy}{x-y}=c$$
  • $$\dfrac{xy}{x-y}=ce^{x/y}$$
  • $$\ln|xy|=c+\dfrac{xy}{x-y}$$
  • $$none\ of\ these$$
The solution of the differential equation $$x^2\dfrac{dy}{dx}\cos\left(\dfrac{1}{x}\right) - y \sin\left(\dfrac{1}{x}\right) = -1 ; where \  y \rightarrow -1 \ as \ x \rightarrow \infty$$ is 
  • $$y = \sin\left(\dfrac{1}{x}\right) - \cos\left(\dfrac{1}{x}\right) $$
  • $$y = \dfrac{x+1}{x\sin\left(\dfrac{1}{x}\right)}$$
  • $$y = \cos\left(\dfrac{1}{x}\right) + \sin\left(\dfrac{1}{x}\right) $$
  • $$y = \dfrac{x+1}{x\cos\left(\dfrac{1}{x}\right)}$$
The solution of the differential equation $$\dfrac{dy}{dx} = \dfrac{x^2 + y^2 + 1}{2xy}$$ satisfying $$y (1) = 1$$, is
  • a hyperbola
  • a circle
  • $$y^2 = x (1 + x) - 10$$
  • $$(x - 2)^2 + (y - 3)^2 = 5$$
The general solution of differential equation $$\dfrac{dy}{dx}=\sin^{3}{x}\cos^{2}{x}+xe^{x}$$
  • $$y=\dfrac{1}{5}\cos^{5}{x}+\dfrac{1}{3}csc^{3}{x}+(x+1)e^{x}+c$$
  • $$y=\dfrac{1}{5}\cos^{5}{x}-\dfrac{1}{3}csc^{3}{x}+(x-1)e^{x}+c$$
  • $$y=-\dfrac{1}{5}\cos^{5}{x}-\dfrac{1}{3}csc^{3}{x}-(x-1)e^{x}-c$$
  • None of these
The value of $$\displaystyle \lim_{x \rightarrow \infty}$$ y(x) obtained from the differential equation $$\dfrac{dy}{dx} = y - y^2$$, where y(0) = 2 is 
  • 1
  • -1
  • 0
  • $$\dfrac{2}{2-e}$$
The solution of the differential equation $$\dfrac{dy}{dx}+\dfrac{1}{x}\tan y=\dfrac{\tan y \sin y}{{x}^{2}}$$ is 
  • $$\dfrac{1}{x}\cos y = \dfrac{1}{{2x}^{2}}+K$$
  • $$\dfrac{1}{x}\cot y = \dfrac{1}{{2x}^{2}}+K$$
  • $$\dfrac{1}{x}\csc y = \dfrac{1}{{2x}^{2}}+K$$
  • $$\dfrac{1}{x}\cos x = \dfrac{1}{{2x}^{2}}+K$$
The solution of the differential equation $$(3xy+y^{2})dx+(x^{2}+xy)dy=0$$ is
  • $$x^2(2x+y)=3$$
  • $$y^2(2x+y)=3$$
  • $$x^2y(2x+y)=3$$
  • $$x^2y(2x+3)=9$$
The solution of the differential equation $$\left ({x}^{2}+1\right)\dfrac {dy}{dx}+{y}^{2}+1=0,$$ is  $$\left[ ify\left( 0 \right) =1 \right]$$
  • $$y=2+{x}^{2}$$
  • $$y=\dfrac { \left( 1+x \right) }{ \left( 1-x \right) }$$
  • $$y=x\left (x-1\right)$$
  • $$y=\dfrac { \left( 1-x \right) }{ \left( 1+x \right) }$$
The solution of differential equation $$\dfrac{dy}{dx} =\dfrac{x(2\ln{x+1})}{\sin{y}+y\cos{y}}$$ is
  • $$y\sin{y}=x\ln{x}+c$$
  • $$y\sin{y}=x^{2}\ln{x}+c$$
  • $$\sin{y}=x^{2}\ln{x}+c$$
  • $$y\cos{y}=x^{2}\ln{x}+c$$
The equation of a curve passing through the point (0, 0) and whose differential equation is $${y'} = {e^x}\sin x$$ is 
  • $$2y - 1 = {e^x}\left( {\sin x - \cos x} \right)$$
  • $$2y - 1 = {e^{ - x}}\left( {\cos x - \sin x} \right)$$
  • $$2y + 1 = {e^{ - x}}\left( {\cos x - \sin x} \right)$$
  • $$None\,of\,these$$
Solution of the differential equation $$ye^{x/y}dx=(xe^{\frac {x}{y}}+y^{2})dy(y \neq 0)$$ is ?
  • $$e^{x/y}=x+C$$
  • $$e^{x/y}+x=C$$
  • $$e^{y/x}=x+C$$
  • $$e^{x/y}=y+C$$
Solution of differential equation $$xdy=(y-{x}^{2}-{y}^{2})dx$$ is (where $$c$$ is arbitrary constant)
  • $$y=x\cos{(c+x)}$$
  • $$y=x\tan{(x-c)}$$
  • $$y=x\csc{(c+x)}$$
  • none of these
Let $$y=y(x)$$ be the solution of the differential equation $$(1-x^2)\dfrac{dy}{dx}-xy=1,x\in(-1,1)$$ if $$y(0)=0$$, then $$y\left(\dfrac{1}{2}\right)$$ is equal to
  • $$\dfrac{\pi}{3\sqrt3}$$
  • $$\dfrac{\pi}{\sqrt3}$$
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{3}$$
If the general solution of the differential equation if $$y'=\frac { y }{ x } +\Phi \left( \frac { x }{ y }  \right) $$, for some function $$\Phi $$, is given by $$yin|cx|=x$$, where c is an arbitrary constant, then $$\Phi $$ (2) is equal to :
  • 4
  • $$\frac { 1 }{ 4 } $$
  • -4
  • $$-\frac { 1 }{ 4 } $$
$$(2y+xy^{3})dx+(x+x^{2}y^{2})dy=0$$ solution of differential equation is 
  • $$3x^{2}y+x^{3}y^{2}=c$$
  • $$3x^{2}y^{2}+xy^{2}=c$$
  • $$yx^{2}+\dfrac{xy}{3}=c$$
  • $$x^{2}y+\dfrac{x^{3}y^{3}}{3}=c$$
Solution of differential equation $${ x }^{ 6 }dy+3{ x }^{ 5 }ydx=xdy-2y\ dx$$ is
  • $${ x }^{ 3 }y=\dfrac { y }{ { x }^{ 2 } } +C$$
  • $${ x }^{ 3 }y=\dfrac { 2y }{ { x }^{ 2 } } +C$$
  • $${ x }^{ 3 }{ y }^{ 2 }=\dfrac { y }{ { x }^{ 2 } } +C$$
  • $${ x }^{ 3 }=\dfrac { y }{ { x }^{ 2 } } +C$$
The solution of equation $$\dfrac { dy }{ dx } =\dfrac { 1 }{ x+y+1 } $$
  • $$x={ ce }^{ y }-y-2$$
  • $$y=x+{ ce }^{ y }-2$$
  • $$x+{ ce }^{ y }-2-2=0$$
  • $$y=x$$
Let $$y=y(x)$$ be the solution of the differential equation $$\sin x\dfrac {dy}{dx}+y\cos x=4x,x\ \in\ (0,\pi)$$, if $$y\left(\dfrac {\pi}{2}\right)=0$$, then $$y\left(\dfrac {\pi}{6}\right)$$ is equal to
  • $$-\dfrac {4}{9}\pi^{2}$$
  • $$\dfrac {4}{9\sqrt {2}}\pi^{2}$$
  • $$\dfrac {-8}{9\sqrt {2}}\pi^{2}$$
  • $$-\dfrac {8}{9}\pi^{2}$$
Solution of the differential equation $$dr=a\left( r\sin\theta d\theta-\cos\theta dr \right )$$ is 
  • $$r\left( 1+a\cos { \theta } \right) =c$$
  • $$r\left( 1+a\cos { \theta } \right) =ac$$
  • $$r\left( 1-a\cos { \theta } \right) =c$$
  • $$r\left( 1-a\cos { \theta } \right) =ac$$
The solutions of the differential equation  $$\frac{dy}{dx}= \frac{siny+x}{sin2y-x cos y }$$ is 

  • $$sin^{2}y = xsin y +\frac{x^{2}}{2}+c$$
  • $$sin^{2}y = xsin y -\frac{x^{2}}{2}+c$$
  • $$sin^{2}y = x+sin y +\frac{x^{2}}{2}+c$$
  • $$sin^{2}y = x-sin y +\frac{x^{2}}{2}+c$$
The general solution of the differential equation $$\dfrac{dy}{dx} + \sin \dfrac{x+y}{2} = \sin \dfrac{x-y}{2}$$ is
  • $$\log \left(\tan \dfrac{y}{2}\right) = c-2\sin x$$
  • $$\log\left( \tan \dfrac{y}{4}\right) = c-2\sin \left(\dfrac{x}{2}\right)$$
  • $$\log \left(\tan \left(\dfrac{y}{2} +\dfrac{\pi}{4}\right) \right)= c-2 \sin x$$
  • None of the above
If $$y\left(x\right)$$ is the solution of the differential equation $$\left(x+2\right)\dfrac{dy}{dx}=x^{2}+4x-9,x\neq -2$$ and $$y\left(0\right)=0$$, then $$y\left(-4\right)$$ is equal to 
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$2$$
The solution of differential equation,$$\left(x+\tan y\right)dy=\sin 2y\,dx$$ is
  • $$xcoty=\dfrac{1}{2}\log |cosec2y-cot2y|+c$$
  • $$x=\tan y+c\sqrt{\tan y}$$
  • $$x=\cot y+c\sqrt{\cot y}$$
  • $$None\ of\ these$$
If a curve $$y=f(x)$$ passes through the point $$(1,-1)$$ and satisfies the differential equation, $$y(1+xy)dx=xdy$$, then $$f\left(-\dfrac {1}{2}\right)$$ is equal to
  • $$-\dfrac {4}{5}$$
  • $$\dfrac {2}{5}$$
  • $$\dfrac {4}{5}$$
  • $$-\dfrac {2}{5}$$
General solution of differential equation $$x^{2}(x+y\frac{dy}{dx})+(x\frac{dy}{dx}-y)\sqrt{x^{2}+y^{2}}=0$$ is
  • $$\frac{1}{\sqrt{x^{2}+y^{2}}}+\frac{y}{x}=c$$
  • $$\sqrt{x^{2}+y^{2}}-\frac{y}{x}=c$$
  • $$\sqrt{x^{2}+y^{2}}+\frac{y}{x}=c$$
  • $$2\sqrt{x^{2}+y^{2}}+\frac{y}{x}=c$$
  • $$\frac{1}{\sqrt{x^{2}+y^{2}}}-\frac{y}{x}=c$$
Solution of the differential equation $$\dfrac{dy}{dx}=\dfrac{y(1+x)}{x(y-1)}$$ is
  • $$\log|xy|+x-y=C$$
  • $$\log|xy|-x+y=C$$
  • $$\log|xy|+x+y=C$$
  • $$\log|xy|-x-y=C$$
Solution of $$(x+y-1)dx+(2x+2y-3)dy=0$$ is 
  • $$y+x+log(x+y-2)=c$$
  • $$y+2x+log(x+y-2)=c$$
  • $$2y+x+log(x+y-2)=c$$
  • $$2y+2x+log(x+y-2)=c$$
The solution of $$\dfrac{xdy}{x^{2}+y^{2}}=\left(\dfrac{y}{x^{2}+y^{2}}-1\right)dx$$ is 
  • $$y=x\cot(c-x)$$
  • $$\cos^{-1}u/x=-x+C$$
  • $$y=x\tan (c-x)$$
  • $$y^{2}/x^{2}=x\tan (c-x)$$
General solution of differential equation $$x^2dy + y(x + y)dx = 0$$ is 
  • $$2x + y = Ax^2Y$$
  • $$2x - y = \dfrac{Ax^2}{y}$$
  • $$2x + y = \dfrac{Ax^2}{y}$$
  • $$x - 2y = Ax^2y$$
If $$y(t)$$ is solution of $$(t + 1) \dfrac{dy}{dt} - ty = 1, y(0) = -1$$, then $$y(1) =$$
  • $$\dfrac{1}{4}$$
  • $$-2$$
  • $$-\dfrac{1}{2}$$
  • $$\dfrac{1}{2}$$
Solution of the differential equation $$\cos{\,}x {\,}dy=y(\sin{\,}x-y)dx,0<x<\dfrac{\pi}{2}$$ is
  • y sec x$$=$$ tan x $$+$$ c
  • y sec x$$=$$ sec x $$+$$ c
  • y sec x$$=$$ (sec x $$+$$ c) y
  •  sec x$$=$$ y (tan x $$+$$ c) 
The solution of the differential equation $$ydx-xdy+3x^2y^2e^{x^3} dx=0$$ is
  • $$\dfrac{x}{y}+e^{x^3}=c$$
  • $$\dfrac{x}{y}-e^x=c$$
  • $$\dfrac{x}{y}+ex^{x^3}-x^2y=c$$
  • $$\dfrac{x}{y}-e^{x^3}=c$$
Solution of the equation $$(x+y)^2\dfrac{dy}{dx}=4,y(0)=0$$ is
  • $$y=2tan^{-1}\left(\dfrac{x+y}{2}\right)$$
  • $$y=4tan^{-1}\left(\dfrac{x+y}{4}\right)$$
  • $$y=4tan^{-1}\left(\dfrac{x+y}{2}\right)$$
  • $$y=2tan^{-1}\left(\dfrac{x+y}{4}\right)$$
The solution of the differential equation $$\log\left(\dfrac {dy}{dx}\right)=4x-2y-2,y=1$$ when $$x=1$$ is
  • $$2e^{2y+2}=e^{4x}+e^{2}$$
  • $$2e^{2y-2}=e^{4x}+e^{2}$$
  • $$2e^{2y+2}=e^{4x}+e^{4}$$
  • $$3e^{2y+2}=e^{3x}+e^{2}$$
the solution of $$ydx-xdy+3{ x }^{ 2 }{ y }^{ 2 }{ e }^{ { x }^{ 3 } }dx=0$$ is
  • $$\frac { x }{ y } +{ e }^{ { x }^{ 3 } }$$=C
  • $$\frac { x }{ y } -{ e }^{ { x }^{ 3 } }$$=0
  • $$-\frac { x }{ y } +{ e }^{ { x }^{ 3 } }$$=C
  • none of these
The solution of the differential equation $$\left(x+2y^{3}\right)\dfrac{dy}{dx}=y$$ is 
  • $$\dfrac{x}{y^{2}}=y+C$$
  • $$\dfrac{x}{y}=y^{2}+C$$
  • $$\dfrac{x^{2}}{y}=y^{2}+C$$
  • $$\dfrac{x}{y}=x^{2}+C$$
General solution of the differential equation (x+y-2) dy=(x+y) dx is 
  • y+x=log(y-x+1)+c
  • y-x log (x+y-1)+c
  • y-2x=log (x+y-1)+c
  • y+2x=log (x+y+1)+c
The solution of $$x^2dy-y^2dx+xy^2(x-y)dy=0$$ is?
  • $$ln\left|\dfrac{x-y}{xy}\right|=\dfrac{y^2}{2}+c$$
  • $$ln\left|\dfrac{xy}{x-y}\right|=\dfrac{x^2}{2}+c$$
  • $$ln\left|\dfrac{x-y}{xy}\right|=\dfrac{x^2}{2}+c$$
  • $$ln\left|\dfrac{x-y}{xy}\right|=x+c$$
Solution of $$\left(1+xy\right)ydx+\left(1-xy\right)xdy=0$$ is 
  • $$\log\dfrac{x}{y}+\dfrac{1}{xy}=c$$
  • $$\log\dfrac{x}{y}=c$$
  • $$\log\dfrac{x}{y}-\dfrac{1}{xy}=c$$
  • $$\log\dfrac{y}{x}-\dfrac{1}{xy}=c$$
The general solution of the differential equation $$y-x\dfrac{dy}{dx}=y^{2}\cos x\left(1-\sin x\right)$$ is
  • $$y\left(\sin x+c\right)=\tan x+\sec x$$
  • $$y\left(\sin x+c\right)=\tan x-\sec x$$
  • $$\dfrac{x}{y}=\sin x+\dfrac{1}{4}\cos 2x+c$$
  • $$\dfrac{x}{y}=\sin x+\dfrac{1}{8}\cos 2x+c$$
Solution of  $$\dfrac { d y } { d x } + 2 x y = y$$  is
  • $$y = c e ^ { x - x ^ { 2 } }$$
  • $$y = c e ^ { x ^ { 2 } } - x$$
  • $$y = c e ^ { x }$$
  • $$y = c e ^ { - x ^ { 2 } }$$
Let =y(x) be the solution of the differential equation, x$$\frac { dy }{ dx } +y=x{ log }_{ e }x,(x>1).$$ If $$2y(2)={ log }_{ e }4-1,$$ then y(e) is equal to :
  • $$-\dfrac { { e }^{ 2 } }{ 2 } $$
  • $$\dfrac  { e } { 4 } $$
  • $$\dfrac { { e }^{ 2 } }{ 4 } $$
  • $$-\dfrac { { e }^{ 2 } }{ 4} $$
The solution of the Differential Equation $$(x+y)(dx-dy)=dx+dy$$ is
  • $$l\ n(x-y)=x+y+C$$
  • $$l\ n(x+y)=x-y+C$$
  • $$l\ n(x-y)=x-y-C$$
  • $$none\ of\ these$$
The integrating factor of the equation $$y^{2}dx+(3xy-1)dy=0$$ is
  • $$y^{2}$$
  • $$y^{3}$$
  • $$\dfrac {1}{y^{2}}$$
  • $$\dfrac {1}{y^{3}}$$
The solution of the differential equation $$\dfrac{dy}{dx}=\sec (x+y)$$ is 
  • $$y-\tan\dfrac{x+y}{2}=c$$
  • $$y+\tan\dfrac{x+y}{2}=c$$
  • $$y+2\tan\dfrac{x+y}{2}=c$$
  • $$None\ of\ these$$
The solution of differential equation $$dy/dx -y/x+\tan y/x$$ is:
  • $$x=C'\cos (y/x)$$
  • $$x=C'\sin (y/x)$$
  • $$x=C'\csc (y/x)$$
  • $$none\ of\ these$$
The solution of the equation $$\dfrac{dy}{dx}=\dfrac{(1+x)y}{(y-1)x}$$ is:
  • $$\log xy+x+y=c$$
  • $$\log (\dfrac{x}{y})+x-y=c$$
  • $$y-x-\log xy=c$$
  • $$None\ of\ these$$
Solution of $$\left(\dfrac {dy}{dx}\right)^{2}+x\dfrac {dy}{dx}-y=0$$ is
  • $$y=3x^{2}+9$$
  • $$y=3x+9$$
  • $$y=\dfrac {4}{3}x^{2}$$
  • $$y=9x+3$$
Number of values of m $$\in$$ N, for which $$y=e^{mx}$$ is a solution of the differential equation $$D^3y-3D^2y-4Dy+12y=0$$, is?
  • $$0$$
  • $$1$$
  • $$2$$
  • More than $$2$$
The solution the differential equation $$\left(\dfrac{dy}{dx}\right)^{2}-\dfrac{dt}{dx}(e^{x}+e^{-x})+1=0$$ is/are 
  • $$y+e^{-x}=c$$
  • $$y-e^{-x}=c$$
  • $$y+e^{x}=c$$
  • $$y-e^{x}=c$$
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