CBSE Questions for Class 12 Commerce Maths Differential Equations Quiz 6 - MCQExams.com

If $$y = cos^{-1} x$$ then $$(1 - x^2)y'' -xy'=?$$ where $$y'= \dfrac{dy}{dx}$$
  • $$1$$
  • $$0$$
  • $$2x$$
  • $$2y$$
Solution of $${y^2}dx + ({x^2} - xy + {y^2})dy = 0$$ 
  • $$\tan{^{ - 1}}\left( {\frac{x}{y}} \right) + \log y + c = 0$$
  • $$2\tan{^{ - 1}}\left( {\frac{x}{y}} \right) + \log y + c = 0$$
  • $$\log y\left( {y + \sqrt {{x^2} + {y^2}} } \right) + \log y + c = 0$$
  • $$\log y\left( {y - \sqrt {{x^2} + {y^2}} } \right) + \log y + c = 0$$
If $$y = \sqrt{\dfrac{1-x}{1+x}}$$ then find $$(1 - x^2) \dfrac{dy}{dx} + y$$ =
  • $$1$$
  • $$-1$$
  • $$2$$
  • $$0$$

If tangent at point p, with parameter t, on the curve $$x=4t^2+3$$,$$y=8t^3-1,t \epsilon R,$$ meets the curve again at point Q, then the coordinates of Q are:- 
  • $$t^2$$+3,-$$t^3$$−1
  • 4$$t^2$$+3,−8$$t^2$$−1
  • $$t^2$$+3,$$t^3$$−1
  • 16$$t^2$$+3,−64$$t^3$$−1
The solution of $$x \dfrac{dy}{dx} + y logy = xy e^x$$ is 
  • $$ logy = (x - 1)e^x + c$$
  • $$(x -1) logy = xe^x + c$$
  • $$x logy = (x + 1)e^x + c$$
  • $$x logy = (x - 1)e^x + c$$
The real value of n for which substitution $$y = {u^n}$$ will transform differential equation $$2{x^4}y\frac{{dy}}{{dx}} + {y^4} = 4{x^6}$$ into homogeneous equation
  • $$\frac{1}{2}$$
  • $$1$$
  • $$\frac{3}{2}$$
  • $$4$$
The solution of the differential equation $$x\dfrac{dy}{dx}=y+x\tan \dfrac{y}{x}$$ is :
  • $$\sin \dfrac{x}{y}=cx$$
  • $$\sin \dfrac{y}{x}=cx$$
  • $$\sin \dfrac{x}{y}=cy$$
  • $$\sin \dfrac{y}{x}=cy$$
Soluation of D.E. $$\dfrac{{dy}}{{dx}} = \dfrac{{3x + 4y + 3}}{{12x + 16y - 4}}$$ is 
  • $$y = 4x + \ell n\left| {3x + 4y} \right| + C$$
  • $$4y = x + \ell n\left| {3x + 4y} \right| + C$$
  • $$y = \ell n\left| {3x + 4y} \right| + C$$
  • $$x + y = \ell n\left| {3x + 4y} \right| + C$$
Solution of the differential equation $$\tan{y}\sec ^{ 2 }{ x } dx+\tan { x } \sec ^{ 2 }{ y } dy=0$$ is:
  • $$\tan{x}+\tan{y}=k$$
  • $$\tan{x}-\tan{y}=k$$
  • $$\cfrac{\tan{x}}{\tan{y}}=k$$
  • $$\tan{x}.\tan{y}=k$$
The solution of the differential equation $$ydx + (x+x^2y)dy = 0$$ is-
  • $$\dfrac{1}{xy} + \log y = c$$
  • $$\log y = cx$$
  • $$-\dfrac{1}{xy} = c$$
  • $$-\dfrac{1}{xy} + \log y = c$$
The general solution of the differential equation $$\dfrac { dy }{ dx } + y\cot { x } = \csc { x }$$, is
  • $$x + y sin { x } = C$$
  • $$x + y cos { x } = C$$
  • $$y+x(\sin { x + \cos { x } } )$$
  • $$y\sin { x = x + C }$$
The solution of $$\cfrac { dy }{ dx } =\left( \cfrac { ax+b }{ cy+d }  \right)  $$ represents a parabola if:- 
  • a=0,c=0
  • a=1,c=2
  • a=0,$$c\neq0$$
  • a=1,c=1
The solution of the differential equation $$x \, dx + y\, dy = x^2y\, dy - y^2\, x\, dx$$ is
  • $$x^2-1 = C(1+y^2)$$
  • $$x^2+1 = C(1-y^2)$$
  • $$x^3-1 = C(1+y^3)$$
  • $$x^3+1 = C(1-y^3)$$
Solve differential equation $$dx=\dfrac{xdv}{v^2-a^2}$$.
  • $$x^{2a}=\dfrac{v-a}{v+a}\times c$$
  • $$x^{2a}=\dfrac{v+a}{v-a}\times c$$
  • $$x=\dfrac{v-a}{v+a}\times c$$
  • None of these
Solution for $$\dfrac{x + y \dfrac{dy}{dx}}{y - x \dfrac{dy}{dx}} = x^2 + 2y^2 + \dfrac{y^4}{x^2}$$ is 
  • $$\dfrac{y}{x} - \dfrac{1}{x^2 + y^2} = c$$
  • $$\dfrac{y}{x} + \dfrac{1}{x^2}{y^2} = c$$
  • $$\dfrac{x}{y} + \dfrac{1}{x^2 + y^2 } = c$$
  • $$\dfrac{x}{y} - \dfrac{1}{x^2 + y^2} = c$$
The particular solution of the differential equation $$y(1+\log x)\dfrac{dx}{dy} -x =0$$ when $$x=e, y=e^2$$ is
  • $$y =ex \log x$$
  • $$ey = x \log x$$
  • $$xy = e \log x$$
  • $$y \log x = ex$$
Solution of differential equation $$xdy-yx=0$$ represents:
  • rectangular hyperbola
  • straight line passing through origin
  • parabola whose vertex is at origin
  • circle whose center is at origin
The solution of $$\cos \, y \, \log (\sec \, x + \tan \, x) dx = \cos \, x \, \log (\sec \, y + \tan \, y) dy$$ is 
  • $$[\log ( \sec \, x + \tan \, x)]^2 - [\log (\sec \, y + \tan \, y)]^2 = c$$
  • $$[\log ( sec \, x + \tan \, x)]^2 + [\log (\sec \, y + \tan \, y)]^2 = c$$
  • $$[\log ( \sec \, x - \tan \, x)]^2 - [\log (\sec \, y - \tan \, y)]^2 = c$$
  • $$[\log ( \sec \, x - \tan \, x)]^2 + [\log (\sec \, y - \tan \, y)]^2 = c$$
The solution of $$\dfrac{{dy}}{{dx}} = \dfrac{{x{{{\mathop{\rm log x}\nolimits} }^2} + x}}{{\sin y + y\cos y}}$$
  • $$y\sin y = {x^2}\log x + c$$
  • $$y\sin y = {x^2} + c$$
  • $$y\sin y = {x^2} + \log x + c$$
  • $$y\sin y = x \log x + c$$
The solution to the differential equation $$\cos\ xdy=y(\sin x-y)dx$$, $$0<x<\dfrac{\pi}{2}$$, is
  • $$\tan x=(\sec x+c)y$$
  • $$\sec x=(\tan x+c)y$$
  • $$y\sec x=\tan x+c$$
  • $$y\tan x=\sec x+c$$
The solution of $$\dfrac{dy}{dx}=\dfrac{y}{x}+\tan\left(\dfrac{y}{x}\right)$$.
  • $$\sin\left(\dfrac{y}{x}\right)=cx$$
  • $$\sin\left(\dfrac{y}{x}\right)=cy$$
  • $$\cos\left(\dfrac{y}{x}\right)=cx$$
  •  $${ \sec  \left( \dfrac { y }{ x }\right) +\tan \left( \dfrac { y }{ x }\right) }=xc$$ 
If $$c$$ is any arbitrary constant, then the general solution  of differential equation $$ydx-xdy=xydx$$ is  given by -
  • $$y=cxe^{x}$$
  • $$y=cx e^{-x}$$
  • $$y+e^{x}=cx$$
  • $$ye^{x}=cx$$
Solve :

$$\displaystyle\int (x^x)^x(2xlog_ex+x)dx$$ 
  • $$x^{(x^x)}+C$$
  • $$(x^x)^x+C$$
  • $$x^x\cdot log_ex+C$$
  • $$(x^x)+C$$
The solution of the equation $$\dfrac {dy}{dx}+\sqrt {\dfrac { {1-y}^{2} }{ {1-x}^{2} }}=0$$ 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
The solution of $$\dfrac{dy}{dx}=\dfrac{\sqrt{x^2-y^2}+y}{x}$$.
  • $$\tan^{-1}\left(\dfrac{y}{x}\right)=log(cx)$$
  • $$\sin^{-1}\left(\dfrac{y}{x}\right)=log(cx)$$
  • $$\cos^{-1}\left(\dfrac{y}{x}\right)=log(cy)$$
  • $$\sec^{-1}\dfrac{y}{x}=log(cy)$$
The solution of the differential equations $$\dfrac{dy}{dx}=\dfrac{x-2y+1}{2x-4y}$$ is?
  • $$(x-2y)^2+2x=c$$
  • $$(x-2y)^2+x=c$$
  • $$(x-2y)+2x^2=c$$
  • $$(x-2y)+x^2=c$$
The solution of $$x\sqrt{1+y^2}dx+y\sqrt{1+x^2}dy=0$$.
  • $$\sin h^{-1}x+\sin h^{-1}y=c$$
  • $$\sqrt{1+x^2}+\sqrt{1+y^2}=c$$
  • $$(1+x^2)(1+y^2)=c$$
  • $$\sqrt{\dfrac{1+x^2}{1+y^2}}=c$$
The solution of $$\dfrac{dy}{dx}=(1+y^2)(1+x^2)^{-1}$$ is?
  • $$y-x=c(1+xy)$$
  • $$y+x=c(1+xy)$$
  • $$y=(1+2x)c$$
  • $$xy=x^2+x+x$$
The solution of $$\dfrac{dy}{dx}=\dfrac{y^2}{xy-x^2}$$.
  • $$y=ce^{xy}$$
  • $$y=\dfrac{e^{\frac{y}{x}}}{c}$$
  • $$log y=xy+c$$
  • $$log x=xy+c$$
If $$f:R \rightarrow R$$ be a continuous   function such that $$f(x)=\displaystyle \int^{x}_{1}2tf(t)dt$$, then which of the following does not hold(s) good?
  • $$f(\pi)=e^{\pi^{2}}$$
  • $$f(1)=e$$
  • $$f(0)=1$$
  • $$f(2)=-2$$
The solution of the differential equation $${x}^{2}dy=-2xydx$$ is
  • $$x{y}^{2}=c$$
  • $${x}^{2}{y}^{2}=c$$
  • $${x}^{2}y=c$$
  • $$xy=c$$
Solution of the differential equation $$\dfrac{dy}{dx}+y\sec x=\tan x\left(\le x< \dfrac{\pi}{2}\right)$$ is 
  • $$y\left( \sec { x } -\tan { x } \right) =\left( \sec { x } +\tan { x } \right) -x+C$$
  • $$y\left( \sec { x } +\tan { x } \right) =\left( \sec { x } -\tan { x } \right) -x+C$$
  • $$y\left( \sec { x } +\tan { x } \right) =\left( \sec { x } +\tan { x } \right) -x+C$$
  • $$none of these $$
Solution of the differential equation $$xdy-ydx=\sqrt {x^{2}+y^{2}}dx$$ is
  • $$y+\sqrt {x^{2}+y^{2}}=cx$$
  • $$y+\sqrt {x^{2}+y^{2}}=cx^{2}$$
  • $$y+\sqrt {x^{2}+y^{2}}=C$$
  • $$none\ of\ these$$
The solution of the differential equation $$\cfrac{dy}{dx}=1+x+y+xy$$ is
  • $$\log{(1+y)}=x+\cfrac{{x}^{2}}{2}+c$$
  • $${(1+y)}^{2}=x+\cfrac{{x}^{2}}{2}+c$$
  • $$\log{(1+y)}=\log{(1+x)}+c$$
  • None of these
A continuously differentiable function $$\phi(x)$$ in $$(0,\pi)$$ satisfying $$y'=1+{y}^{2},y(0)=0=y(\pi)$$ is
  • $$\tan{x}$$
  • $$x(x-\pi)$$
  • $$(x-\pi)(1-{e}^{x})$$
  • Not possible
Solution of the differential equation $$(1+x^{2})dy+2xy dx=\cot x dx$$ is
  • $$y=\log|\sin x|(1+x^{2})+C(1+x^{2})^{-1}$$
  • $$y=\log|\sin x|(1+x^{2})^{-1}+C(1+x^{2})$$
  • $$y=\log|\sin x|(1+x^{2})+C(1+x)$$
  • $$y=\log|\sin x|(1+x^{2})^{-1}+C(1+x^{2})^{-1}$$
The solution of the differential equation $$x^3 \dfrac{dy}{dx} + 4x^2 \tan y = e^x \sec y$$ satisfying $$y(1) = 0$$ is 
  • $$\tan \, y = (x - 2) e^x \log \, x$$
  • $$\sin \, y = e^x (x - 1) x^{-4}$$
  • $$\tan \, y = (x - 1) e^x x^{-3}$$
  • $$\sin \, y = e^x (x - 1) x^{-3}$$
The solution of $$\cfrac{dy}{dx}=\cfrac{1+{y}^{2}}{\sec{x}}$$ is
  • $$\tan ^{ -1 }{ y } =\cos { x } +c$$
  • $$\tan ^{ -1 }{ y } =\sin { x } +c$$
  • $${ y }^{ 3 }=co\sec { x } +c$$
  • $$\log { \left( 1+{ y }^{ 2 } \right) } =\cos { x } +c$$
The solution of $$ydx-xdy=0$$ is
  • $${y}^{2}=cx$$
  • $$y=c{x}^{3}$$
  • $$y=cx$$
  • $${x}^{2}=cy$$
Solution of differential equation $$\dfrac{dy}{dx}-2xy=x$$ is
  • $$y=Ce^{x^2}-\dfrac {1}{2}$$
  • $$y=Ce^{x^2}+\dfrac {1}{2}$$
  • $$y=Cx^2-\dfrac {1}{2}$$
  • None
If $$2x = {y^{\dfrac{1}{5}}} + {y^{\dfrac{{ - 1}}{5}}}{\text{and}}\left( {{x^2} - 1} \right)\dfrac{{{d^2}y}}{{d{x^2}}} + \lambda x\dfrac{{dy}}{{dx}} + {\text{ky}} = 0,\;{\text{then}}\;\lambda  + {\text{K}}$$ is equal to.
  • 26
  • -24
  • -23
  • -26
The solution of $$y d x - x d y + 3 x ^ { 2 } y ^ { 2 } e ^ { x ^ { 3 } } d x = 0$$
  • $$\frac { x } { y } + e ^ { x ^ { 3 } } = c$$
  • $$\frac { x } { y } - e ^ { x ^ { 3 } } = c$$
  • $$\frac { x } { y } = - e ^ { x ^ { 3 } } + c$$
  • none of these
The solution of, $$\dfrac{xdy}{x^2 + y^2} = \left(\dfrac{y}{x^2 + y^2} - 1 \right)dx$$, is given by
  • $$\tan^{-1} \left(\dfrac{x}{y}\right) + x = C$$
  • $$\tan^{-1} \left(\dfrac{y}{x}\right) + x = C$$
  • $$\tan^{-1} \left(\dfrac{y}{x}\right) + xy = C$$
  • $$\tan^{-1} \left(\dfrac{y}{x}\right) + x^2 = C$$
The solution of the differential equation $$\operatorname { xdy } \left( y ^ { 2 } e ^ { x y } + e ^ { \tfrac { x } { y } } \right) = y d x \left( e ^ { \frac { x } { y } } - y ^ { 2 } e ^ { x y } \right)$$ is-
  • $$x y = \ln \left| e ^ { x / y } + C \right|$$
  • $$e ^ { x y } = \ln ( x y + C )$$
  • $$x y = e ^ { x / y } + C$$
  • $$x y = e ^ { x / y } + \frac { x } { y }$$
The solution of the differential equation $$( x \cot y + \log \cos x ) d y$$ $$+ ( \log \sin y - y \tan x ) d x = 0$$ is:-
  • $$( \sin x ) ^ { y } ( \cos y ) ^ { x } = c$$
  • $$( \sin y ) ^ { x } ( \cos x ) ^ { y } = c$$
  • $$( \sin x ) ^ { x } ( \cos y ) ^ { y } = c$$
  • none of these
The solution of the differential equation $$y ^ { 2 } d y = x ( y d x - x d y ) \text { is } y = y ( x )$$. If $$y ( \sqrt { 3 } e ) = e$$ and $$y \left( x _ { 0 } \right) = 1$$ then $$x_0$$ is
  • $$0$$
  • $$1$$
  • $$\frac { 1 } { e }$$
  • $$-1$$
The differential equation whose solution is $$y = Ax^{5} + Bx^{4}$$ is
  • $$x^{2} \dfrac {d^{2}y}{dx^{2}} + 8x \dfrac {dy}{dx} + 20y = 0$$
  • $$x^{2} \dfrac {d^{2}y}{dx^{2}} + 8x \dfrac {dy}{dx} - 20y = 0$$
  • $$x^{2} \dfrac {d^{2}y}{dx^{2}} - 8x \dfrac {dy}{dx} + 20y = 0$$
  • $$x^{2} \dfrac {d^{2}y}{dx^{2}} - 8x \dfrac {dy}{dx} - 20y = 0$$
Solution of the equation $$\dfrac{{dy}}{{dx}} = 1 + xy + x + y$$ is
  • $$\left( {1 + y} \right)\left( {1 + x} \right) = c$$
  • $$\log \left( {1 + y} \right) = 1 + x + c$$
  • $$\ln |1 + y| = x + \dfrac{{{x^2}}}{2} + c$$
  • $$\ln |1 + y| + x + \dfrac{{{x^2}}}{2} + c = 0$$
The general solution of the differential equation $$\dfrac { dy }{ dx } +y={ x }^{ 3 }$$ is ______.
  • $${ ye }^{ x }={ e }^{ x }\left( { x }^{ 3 }+{ 3x }^{ 2 }-6x-6 \right) +c$$
  • $${ ye }^{ x }={ e }^{ x }\left( { x }^{ 3 }-{ 3x }^{ 2 }-6x+6 \right) +c$$
  • $$y=\left( { x }^{ 3 }-{ 3x }^{ 2 }+6x-6 \right) +c{ e }^{ -x }$$
  • $$y={ e }^{ x }$$
If $$\dfrac{dy}{dx}=y+3>0$$ and $$y(0)=2$$ then $$y(\ln{2})$$ is equal to :
  • $$7$$
  • $$5$$
  • $$13$$
  • $$-2$$
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