CBSE Questions for Class 12 Commerce Maths Differential Equations Quiz 5 - MCQExams.com

The solution of $$\frac{dy}{dx}=|x|$$ is :
  • $$y=\frac{x|x|}{2}+c$$
  • $$y=\frac{|x|}{2}+c$$
  • $$y=\frac{x^2}{2}+c$$
  • $$y=\frac{x^3}{2}+c$$
The solution of the differential equation $$\dfrac { dy }{ dx } ={ e }^{ x-y }\left( { e }^{ x }-{ e }^{ y } \right) $$ is
  • $${ e }^{ y }=\left( { e }^{ x }+1 \right) +C{ e }^{ -{ e }^{ x } }$$
  • $${ e }^{ y }=\left( { e }^{ x }-1 \right) +C$$
  • $${ e }^{ y }=\left( { e }^{ x }-1 \right) +C{ e }^{ -{ e }^{ x } }$$
  • None of the above
The solution of the differential equation $$\dfrac {dy}{dx} = \tan \left (\dfrac {y}{x}\right ) + \dfrac {y}{x}$$ is:
  • $$\cos \left (\dfrac {y}{x}\right ) = cx$$
  • $$\sin \left (\dfrac {y}{x}\right ) = cx$$
  • $$\cos \left (\dfrac {y}{x}\right ) = cy$$
  • $$\sin \left (\dfrac {y}{x}\right ) = cy$$
The solution of the differential equation $$\displaystyle (x^2-yx^2)\frac{dy}{dx}+y^2+xy^2=0$$ is?
  • $$\displaystyle \log\left(\frac{x}{y}\right)=\frac{1}{x}+\frac{1}{y}+C$$
  • $$\displaystyle \log\left(\frac{y}{x}\right)=\frac{1}{x}+\frac{1}{y}+C$$
  • $$\displaystyle \log(xy)=\frac{1}{x}+\frac{1}{y}+C$$
  • $$\displaystyle \log(xy)+\frac{1}{x}+\frac{1}{y}=C$$
The solution of $$\dfrac{dy}{dx} = 1+y+y^2+x+xy+xy^2$$ is
  • $$\tan^{-1}\left(\dfrac{2y+1}{\sqrt{3}}\right) = x+x^2+C$$
  • $$4\tan^{-1}\left(\dfrac{2y+1}{\sqrt{3}}\right) = \sqrt{3}(2x+x^2)+C$$
  • $$\sqrt{3}\tan^{-1}\left(\dfrac{3y+1}{3}\right)=4(1+x+x^2)+C$$
  • $$\tan^{-1}\left(\dfrac{2y+1}{3}\right)=4(2x+x^2)+C$$
The solution of the differential equation $$\sec^{2} x \cdot \tan y \,dx + \sec^{2} y \cdot \tan x\ dy = 0$$ is 
  • $$\tan x \cdot \cot y = C$$
  • $$\cot x \cdot \tan y = C $$
  • $$\tan x \cdot \tan y = C$$
  • $$\sin x \cdot \cos y = C$$
The solution of the differential equation $$\dfrac {dy}{dx} = (x + y)^{2}$$ is:
  • $$\dfrac {1}{x + y} = c$$
  • $$\sin^{-1} (x + y) = x + c$$
  • $$\tan^{-1} (x + y) = c$$
  • $$\tan^{-1} (x + y) = x + c$$
The solution of the differential equation $$(1+{ x }^{ 2 }{ y }^{ 2 })ydx+({ x }^{ 2 }{ y }^{ 2 }-1)xdy=0$$ is
  • $$xy=\log { \dfrac { x }{ y } } +C$$
  • $$xy=2\log { \dfrac { y }{ x } } +C$$
  • $${x}^{2}{y}^{2}=\log { \dfrac { y }{ x } } +C$$
  • $${x}^{2}{y}^{2}=2\log { \dfrac { y }{ x } } +C$$
What is the curve which passes through the point (1, 1) and whose slope is $$\dfrac{2y}{x} $$?
  • Circle
  • Parabola
  • Ellipse
  • Hyperbola
If $$x  dy = y  dx + y^2 dy, y > 0$$ and y(1) = 1, then what is y(-3) equal to?
  • 3 only
  • -1 only
  • Both -1 and 3
  • Neither -1 nor 3
Solution of the differential equation $$\dfrac { dx }{ x } +\dfrac { dy }{ y } =0$$ is
  • $$\dfrac { 1 }{ x } + \dfrac { 1 }{ y } =c$$
  • $$\log { x } \log { y } =c$$
  • $$xy=c$$
  • $$x+y=c$$
A solution of the differential equation $${ \left( \dfrac { dy }{ dx }  \right)  }^{ 2 }-x\dfrac { dy }{ dx } +y=0$$ is
  • $$y=2$$
  • $$y=2x$$
  • $$y=2x-4$$
  • $$y=2{ x }^{ 2 }-4$$
The solution of the differential equation $${ y }^{ ' }\left( { y }^{ 2 }-x \right) =y$$ is
  • $${ y }^{ 3 }-3xy=C$$
  • $${ y }^{ 3 }+3xy=C$$
  • $${ x }^{ 3 }-3xy=C$$
  • $${ y }^{ 3 }-xy=C$$
  • $${ x }^{ 3 }-xy=C$$
The solution of differential equation
$$4xy\cfrac { dy }{ dx } =\cfrac { 3{ \left( 1+x \right)  }^{ 2 }\left( 1+{ y }^{ 2 } \right)  }{ \left( 1+{ x }^{ 2 } \right)  } $$ is 
  • $$\log { (1+y) } =\log { x } +2\tan { x } +constant$$
  • $$\log { \left( 1+{ y }^{ 2 } \right) } =3\log { \left( \cfrac { 1 }{ x } \right) } +6\tan ^{ -1 }{ x } +constant$$
  • $$2\log { \left( 1+{ y }^{ 2 } \right) } =3\log { x } +6\tan ^{ -1 }{ x } +constant$$
  • None of the above
The solution of the differential equation $$\dfrac { dy }{ dx } +\sin { \left( \dfrac { y+x }{ 2 }  \right)  } +\sin { \left( \dfrac { y-x }{ 2 }  \right)  } =0$$ is
  • $$\log { \tan { \left( \dfrac { y }{ 2 } \right) } } =C-2\sin { x } $$
  • $$\log { \tan { \left( \dfrac { y }{ 4 } \right) } } =C-2\sin { \left( \dfrac { x }{ 2 } \right) } $$
  • $$\log { \tan { \left( \dfrac { y }{ 2 } +\dfrac { \pi }{ 4 } \right) } } =C-2\sin { x } $$
  • $$\log { \tan { \left( \dfrac { y }{ 2 } +\dfrac { \pi }{ 4 } \right) } } =C-2\sin { \left( \dfrac { x }{ 2 } \right) } $$
The solution of $$ \dfrac {dy}{dx} + \sqrt{ \left( \dfrac {1-y^2}{1-x^2} \right) } = 0 $$ is :
  • $$ \tan^{-1} x + \cos^{-1} x = C $$
  • $$ \sin^{-1} x + \sin^{-1} y = C $$
  • $$ \sec^{-1} x + \text{cosec} ^{-1} x = C $$
  • None of the above.
An integrating factor of the differential equation $$\sin x \dfrac{dy}{dx} + 2 y \cos  x = 1$$ is
  • $$\sin^2 x$$
  • $$\dfrac{2}{\sin x}$$
  • $$\log |\sin x|$$
  • $$\dfrac{1}{\sin^2 x}$$
  • $$2 \sin x$$
The general solution of the differential equation $$\left( x+y \right) dx+xdy=0$$ is
  • $${ x }^{ 2 }+{ y }^{ 2 }=C$$
  • $$2{ x }^{ 2 }-{ y }^{ 2 }=C$$
  • $${ x }^{ 2 }+2xy=C$$
  • $${ y }^{ 2 }+2xy=C$$
The solution of the differential equation $$\left( { x }^{ 2 }-y{ x }^{ 2 } \right) \dfrac { dy }{ dx } +{ y }^{ 2 }+x{ y }^{ 2 }=0$$ is
  • $$\log { \left( \dfrac { x }{ y } \right) } =\dfrac { 1 }{ x } +\dfrac { 1 }{ y } +C$$
  • $$\log { \left( \dfrac { y }{ x } \right) } =\dfrac { 1 }{ x } +\dfrac { 1 }{ y } +C$$
  • $$\log { \left( xy \right) } =\dfrac { 1 }{ x } +\dfrac { 1 }{ y } +C$$
  • $$\log { \left( xy \right) } +\dfrac { 1 }{ x } +\dfrac { 1 }{ y } =C$$
The general solution of the differential equation $$xdy-ydx={y}^{2}dx$$ is
  • $$y=\cfrac { x }{ C-x } $$
  • $$x=\cfrac { 2y }{ C+x } $$
  • $$(C+x)(2x)$$
  • $$y=\cfrac { 2x }{ C+x } $$
  • $$x=\cfrac { y }{ C-x } $$
The degree of the differential equation $$\left (\dfrac {d^{2}y}{dx^{2}}\right )^{3} + \left (\dfrac {dy}{dx}\right )^{2} + \sin\left (\dfrac {dy}{dx}\right )^{2} + \sin \left (\dfrac {dy}{dx}\right ) + 1 = 0$$ is
  • $$3$$
  • $$2$$
  • $$1$$
  • None of the above
Solution of $$\cfrac { dx }{ dy } +mx=0$$, where $$m< 0$$ is:
  • $$x=C{ e }^{ my }$$
  • $$x=C{ e }^{ -my }$$
  • $$x=my+C$$
  • $$x=C$$
The solution of the differential equation $$\dfrac {dy}{dx} = e^{x - y} (e^{x} - e^{y})$$ is
  • $$e^{y} = (e^{x} + 1) + Ce^{-x}$$
  • $$e^{y} = (e^{x} - 1) + C$$
  • $$e^{y} = (e^{x} - 1) + Ce^{-x}$$
  • None of the above
The solution of $$\left( y-3{ x }^{ 2 } \right) dx+xdy=0$$ is
  • $$y(x)=\sin { x } +\cfrac { 1 }{ { x }^{ 2 } } +C$$
  • $$y(x)=\cos { x } -\cfrac { 1 }{ { x }^{ 2 } } +C\quad $$
  • $$y(x)={ x }^{ 2 }+\cfrac { C }{ x } $$
  • $$y(x)=\sqrt { x } +\cfrac { C }{ x } $$
Let $$f(x)$$ be differentiable on the interval $$(0,\infty)$$ and $$\lim _{ t\rightarrow x }{ \cfrac { { t }^{ 3 }f(x)+{ x }^{ 3 }f(t) }{ t-x }  } =2$$ gives a linear differential equation whose integrating factor is
  • $${ x }^{ 3 }$$
  • $$1/{ x }^{ 3 }$$
  • $${ x }^{ 2 }$$
  • $$1/{ x }^{ 2 }$$
The solution of $$x \log x \displaystyle\frac{dy}{dx}+y=1$$ is?
  • $$\log x=\displaystyle\frac{c}{(y-1)}$$
  • $$y\log x\displaystyle\frac{dy}{dx}+y=1$$
  • $$xy=\log (\log x)+c$$
  • $$\displaystyle\frac{x}{y}\log y=c$$
Solution of the differential equation $$(x^{2} + y^{3}) (2x^{2}dx + 3ydy) = 12x\ dx + 18y^{2}dy$$ is
  • $$\dfrac {2}{3}x^{3} + \dfrac {3}{2}y^{2} = 6ln (x^{2} + y^{3}) + c$$
  • $$x^{2} + y^{3} = 9ln (x^{2} + y^{3}) + c$$
  • $$\dfrac {2}{3}x^{3} + \dfrac {3}{2}y^{2} = 6ln (x^{3} + y^{2}) + c$$
  • $$x^{3} + y^{2} = 6ln (x^{2} + y^{3}) + c$$
The solution of $$y' = e^{x - y} + x^{2} e^{-y}$$ is
  • $$3(e^{y} - e^{x}) - x^{3} = c$$
  • $$e^{y} - e^{x} - x^{3} = c$$
  • $$e^{y} - e^{x} + x^{3} = c$$
  • $$3(e^{y} - e^{x}) + x^{3} = c$$
If $$y=\sqrt{(a-x)(x-b)}-(a-b)\tan^{-1}\sqrt{\displaystyle\frac{a-x}{x-b}}(a > b)$$ then $$\displaystyle\frac{dy}{dx}=$$.
  • $$\sqrt{\displaystyle \frac{a-x}{x-b}}$$
  • $$\sqrt{(a-x)(x-b)}$$
  • $$0$$
  • $$1$$
The solution of the differential equation $$3xy'-3y+{ \left( { x }^{ 2 }-{ y }^{ 2 } \right)  }^{ 1/2 }=0$$, satisfying the condition $$y(1)=1$$ is
  • $$3\cos ^{ -1 }{ \left( \cfrac { y }{ x } \right) } =\ln { \left| x \right| } $$
  • $$3\cos { \left( \cfrac { y }{ x } \right) } =\ln { \left| x \right| } $$
  • $$3\cos ^{ -1 }{ \left( \cfrac { y }{ x } \right) } =2\ln { \left| x \right| } $$
  • $$3\sin ^{ -1 }{ \left( \cfrac { y }{ x } \right) } =\ln { \left| x \right| } $$
The solution of equation $$\dfrac{dy}{dx}=\dfrac{ax+b}{cy+d}$$ represents : 
  • A straight line if $$a=c=0$$ and $$b,d \neq 0$$
  • A parabola if $$a=2, c=0, d \neq 0$$
  • A huberbola if $$b=d=0, a$$ and $$c\neq 0$$
  • A rectangular hyberbola of $$b=d=0, a=c=2$$
The solution of differential equation $$x \dfrac {dy}{dx} + y=y^2$$ is:
  • $$y=1+ cxy$$
  • $$y=\ln(cxy)$$
  • $$y+1=cxy$$
  • $$y=c+xy$$
The solution of the differential equation $$\left( x+3{ y }^{ 2 } \right) \dfrac { dy }{ dx } =y,\ y>0$$ is
  • $$\dfrac { x }{ y } =3y+c$$
  • $$x=2{ y }^{ 3 }+3{ y }^{ 2 }+c$$
  • $$y=3{ x }^{ 2 }+c$$
  • $$y=3x+c$$
The solution of the differential equation $$y'=\cfrac { 1 }{ { e }^{ -y }-x } $$, is
  • $$x={ e }^{ -y }\left( y+c \right) $$
  • $$y+{ e }^{ -y }=x+c$$
  • $$x={ e }^{ y }(y+c)$$
  • $$x+y={ e }^{ -y }+c$$
The general solution of the differential equation $$\cfrac { dy }{ dx } +\sin { \cfrac { x+y }{ 2 }  } =\sin { \cfrac { x-y }{ 2 }  } $$ is
  • $$\log _{ e }{ \left| \tan { \cfrac { y }{ 2 } } \right| } =-2\cos { \cfrac { x }{ 2 } } +C$$
  • $$\log _{ e }{ \left| \tan { \cfrac { y }{ 2 } } \right| } =2\cos { \cfrac { x }{ 2 } } +C$$
  • $$\log _{ e }\left|{ \tan { \cfrac { y }{ 2 } } }\right| =2 \sin { \cfrac { x }{ 2 } } +C$$
  • $$\log _{ e }\left|{ \tan { \cfrac { y }{ 2 } } }\right| =-2\sin { \cfrac { x }{ 2 } } +C\quad $$
The differential equation $$\dfrac{dy}{dx} = e^x.e^y$$ has solution ____________
  • $$e^x + e^y = C$$
  • $$e^{-x} + e^y = C$$
  • $$e^x + e^{-y} = C$$
  • $$e^{-x} + e^{-y} = C$$
The particular solution of differential equation $$\cfrac { dy }{ dx } =-4x{ y }^{ 2 },y(0)=1$$ is ______
  • $$y=\left( 2{ x }^{ 2 }+1 \right) =1$$
  • $${ x }^{ 2 }=\cfrac { 1 }{ { y }^{ 2 } } $$
  • $$y={ x }^{ 2 }+\log { x } $$
  • $$4{ e }^{ x }+\cfrac { 1 }{ { y }^{ 2 } } =8\quad $$
The order of the differential equation whose general solution is $$y \, = \, c, \, cos \, 2x \, + \, c_2 \, cos^2x \, + \, c_3 \, sin^2x \, + \, c_4$$
  • $$2$$
  • $$4$$
  • $$3$$
  • None of these
The solution of the differential equation $$2x \dfrac{dy}{dx} = y; y(1) = 2$$ represents $$=$$ ____.
  • parabola
  • ellipse
  • circle
  • line
What are the order and degree, respectively, of the differential equation
$${ \left( \cfrac { { d }^{ 3 }y }{ d{ x }^{ 3 } }  \right)  }^{ 2 }={ y }^{ 4 }+{ \left( \cfrac { dy }{ dx }  \right)  }^{ 5 }\quad $$?
  • $$4,5$$
  • $$2,3$$
  • $$3,2$$
  • $$5,4$$
What is the degree of the differential eqaution : $$\dfrac{d^3y}{dx^3}-6(\dfrac{dy}{dx})^2-4y=0$$
  • 1
  • 2
  • 3
  • None of these
the general solution of differential equation $$x^4 \, \frac{dy}{dx} \, + \, x^3 y \, + cosec \, xy \, =0$$,   is 
  • 2 cos ( x y ) + $$\dfrac{1}{x^2}$$ = C
  • 2 cos ( x y ) + $$\dfrac{1}{y^2}$$ = C
  • 2 sin y + $$\dfrac{1}{x^2}$$ = C
  • 2 sin ( x y ) + $$\dfrac{1}{y^2}$$ = C
Solution of the differential equation
$$\tan { y } .\sec ^{ 2 }{ x } dx+\tan { x } .\sec ^{ 2 }{ y } dy=0$$ is
  • $$\tan { x }+\tan { y }=k$$
  • $$\tan { x }-\tan { y }=k$$
  • $$\cfrac { \tan { x } }{ \tan { y } } =k$$
  • $$\tan { x }.\tan { y }=k$$
Let the function f satisfies $$f(x) .f'(-x) =f(-x) .f'(x)$$ for all $$x$$ and $$f(0) =3$$ 
The value of $$f(x) .f(-x)$$ for all $$x$$, is
  • $$4$$
  • $$9$$
  • $$12$$
  • $$6$$
The order of differential equation of family of all concentric circles centered at $$(h,k)$$ is
  • 1
  • 2
  • 3
  • 4
The solution of $$\dfrac { dx }{ dy } -\dfrac { 2 }{ 3 } xy={ x }^{ 4 }{ y }^{ 3 }$$ is
  • $$\dfrac { 1 }{ { x }^{ 3 } } =\dfrac { 3 }{ 2 } \left( 1-{ y }^{ 2 } \right) +c{ e }^{ -{ y }^{ 2 } }$$
  • $$\dfrac { 1 }{ { x }^{ 3 } } =\dfrac { 3 }{ 2 } \left( 1+{ y }^{ 2 } \right) +c{ e }^{ -{ y }^{ 2 } }$$
  • $$\dfrac { 1 }{ { x }^{ 4 } } =\dfrac { 3 }{ 2 } \left( 1-{ y }^{ 2 } \right) +c{ e }^{ { y }^{ 2 } }$$
  • $$\dfrac { 1 }{ { x }^{ 4 } } =\dfrac { 3 }{ 2 } \left( 1+{ y }^{ 2 } \right) +c{ e }^{ { y }^{ 2 } }$$
The solution of the equation $$\dfrac{dy }{dx} = e^{2x}$$ is
  • $$\dfrac{e^{-2x}}{4} = y$$
  • $$y = \dfrac{e^{2x}}{2} + c$$
  • $$\dfrac{1}{4} e^{-2x} + cx^2 + d = y$$
  • $$\dfrac{1}{4} e^{-2x} + cx + d = y$$
The solution of $$x\dfrac{dy}{dx}+y\log{y}=xy{e}^{x}$$ is
  • $$x\log{y}=(x+1){e}^{x}+c$$
  • $$\log{y}=(x-1){e}^{x}+c$$
  • $$(x-1)\log{y}=x{e}^{x}+c$$
  • $$x\log{y}=(x-1){e}^{x}+c$$
The solution of $$\sec ^{2}y\dfrac {dy}{dx}+2x\tan y=x^{3}$$ is
  • $$\tan { y } =\frac { { x }^{ 2 } }{ 2 } +\frac { 1 }{ 2 } +c{ e }^{ -x^{ 2 } }$$
  • $$\tan { y } =\frac { { x }^{ 2 } }{ 2 } -\frac { 1 }{ 2 } +c{ e }^{ -x^{ 2 } }$$
  • $$\tan{ y}$$ $$=\dfrac {x^{2}}{2}-\dfrac {1}{2}+c ({e}^{x})^{2}$$
  • $$\tan y=\dfrac {x^{2}}{2}+\dfrac{1}{2}$$+$$c({e^{x}})^{2}$$
The solution of $$\dfrac{dx}{dy} + \dfrac{x}{y} = x^2$$  is 
  • $$\dfrac{1}{y} = cx - xlogx$$
  • $$\dfrac{1}{x} = cy - ylogy$$
  • $$\dfrac{1}{x} = cx + xlogy$$
  • $$\dfrac{1}{y} = cx - ylogx$$
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