CBSE Questions for Class 12 Commerce Maths Differential Equations Quiz 4 - MCQExams.com

Solution of the given differential equation $$\displaystyle \left ( 1-x^{2} \right )\frac{dy}{dx}+xy=xy^{2}$$ is
  • $$y(1-y)=c\displaystyle \sqrt{1-x^2}$$
  • $$y(y-1)=c\displaystyle \sqrt{1-x^2}$$
  • $$x(1-x)=c\displaystyle \sqrt{1-y^2}$$
  • $$x(x-1)=c\displaystyle \sqrt{1-y^2}$$
Which of the following functions are homogeneous?
  • $$x \sin y + y \sin x$$
  • $$x\, e^{y/x}\, +\, y\, e^{x/y}$$
  • $$x^2\, -\, xy$$
  • $$arc \sin (xy)$$
Find general solution of  $$\sqrt{1 + 4x^{2}} dy = y^{3} xdx$$.
  • $$\displaystyle - \frac{1}{2y^{2}} = \frac{1}{4}\sqrt{1 + 4x^{2}} + k$$
  • $$\displaystyle \frac{1}{2y^{2}} = \frac{1}{4}\sqrt{1 + 4x^{2}} + k$$
  • $$\displaystyle \frac{1}{2y^{2}} = -\frac{1}{4}\sqrt{1 + 4x^{2}} + k$$
  • $$\displaystyle \frac{1}{2y^{2}} = -\frac{1}{4}\sqrt{1 - 4x^{2}} + k$$
The solution of the differential equation $$\displaystyle \left( 1+\cos { x }  \right) \frac { dy }{ dx } =1-\cos { x } $$ is
  • $$\displaystyle y=\tan { \frac { x }{ 2 }  } +x+c$$
  • $$\displaystyle y=2\tan { \frac { x }{ 2 }  } -x+c$$
  • $$\displaystyle y=\tan { \frac { x }{ 2 }  } -x+c$$
  • $$\displaystyle y=x-2\tan { \frac { x }{ 2 }  } +c$$
The equation of the curve in which sub-normal varies as the square of the ordinate is ($$k$$ is constant of proportionality)
  • $$ y=Ae^{2kx} $$
  • $$ y=e^{kx} $$
  • $$ y^{2}/2+kx=A $$
  • $$  y^{2}+kx^{2}=A $$
Solve : $$(\tan y)\displaystyle \frac{dy}{dx} = \sin(x + y) + \sin(x - y)$$
  • $$\sec y = -2 \cos x + C$$
  • $$\sec y = 2 \cos x + C$$
  • $$\sec y = - \cos x + C$$
  • $$\sec y = \cos x + C$$
Find general solution of $$\displaystyle \frac{2dy}{dx} = \frac{y(x + 1)}{x}$$.
  • $$\log y^{2} = x + \log |x| + k$$
  • $$\log y^{2} = x + \log (x) + k$$
  • $$\log y = x + \log (x) + k$$
  • $$2\log y = x - \log |x| + k$$
The solution to the differential equation $$y\ln y \, +\, xy'\, =\, 0\,$$ where$$\, y(1)\, =\, e$$, is:
  • $$x(\ln y)\, =\, 1$$
  • $$xy(\ln y)\, =\, 1$$
  • $$(\ln y )^2\, =\, 2$$
  • $$\ln y\, +\, \left ( \displaystyle \frac {x^2}{2} \right )\, y\, =\, 1$$
A solution of the differential equation, $$\left ( \displaystyle \dfrac {dy}{dx} \right ) ^2\, -\,x\, \displaystyle \dfrac {dy}{dx}\, +\, y\, =\, 0$$ is
  • $$y\, =\, 2$$
  • $$y\, =\, 2x$$
  • $$y\, =\, 2x\, \, -4$$
  • $$y\, =\, 2x^2\, \,- 4$$
The solution to the differential equation $$(x + 1) \displaystyle \frac{dy}{dx} - y = e^{3x} (x + 1)^2 $$ is
  • $$y = (x + 1)e^{3x} + c$$
  • $$3y = (x + 1) + e^{3x} + c$$
  • $$\displaystyle \frac{3y}{x+ 1} = e^{3x} + c$$
  • $$ye^{-3x} = 3 (x + 1) + c$$
The solution of $$\displaystyle \frac {dy}{dx}\, =\, \displaystyle \frac {ax\, +\, h}{by\, +\, k}$$ represent a parabola if:
  • $$a\, =\, 2,\, b\, =\, 0$$
  • $$a\, =\, 2,\, b\, =\, 2$$
  • $$a\, =\, 0,\, b\, =\, 2$$
  • $$a\, =\, 0,\, b\, =\, 0$$
The solution of differential equation 
$$\displaystyle \frac { dy }{ dx } =-\left( \frac { y+\sin { x }  }{ x }  \right) $$ satisfying condition $$\displaystyle y(0) = 1$$, is
  • $$\displaystyle \cos x = xy - 1$$
  • $$\displaystyle \cos x = xy + 1$$
  • $$\displaystyle \cos x = xy $$
  • $$\displaystyle \cos x = x + 1$$
The solution of the equation $$\displaystyle \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ e }^{ x }+{ e }^{ -x }$$ is-
Note : (where c & d are arbitrary constants in the given options)
  • $$\displaystyle y={ e }^{ x }-{ e }^{ -x }+cx+d$$
  • $$\displaystyle y={ e }^{ x }+{ e }^{ -x }+cx+d$$
  • $$\displaystyle y=-{ e }^{ x }+{ e }^{ -x }+cx+d$$
  • None of these
Solution of the differential equation 
$$\displaystyle \left( 2x-y+2 \right) dx+\left( 4x-2y-1 \right) dy=0$$ is 
  • $$\displaystyle 2x-y={ ce }^{ -\left( x+2y \right) }$$
  • $$\displaystyle 2x+y={ ce }^{ \left( 2x+y \right) }$$
  • $$\displaystyle x-2y={ ce }^{ -\left( x+2y \right) }$$
  • $$\displaystyle 2x+y={ ce }^{ \left( x+2y \right) }$$
Solution of differential equation $$\displaystyle \frac { dx }{ dy } =\tan { x } \left( 1+ysinx \right) $$ is given by -
  • $$\displaystyle cosec x= -y+1+{ Ce }^{ -y }$$
  • $$\displaystyle y=\tan { x } +{ Ce }^{ X }$$
  • $$\displaystyle \sin { x } { e }^{ y }=1+y+C$$
  • $$\displaystyle cosecx=y+{ Ce }^{ y }$$
Find the equation of the curve for which the normal at any point (x, y) passes through the origin. The curve represents a :
  • ellipse
  • rectange
  • circle
  • hyperbola
Find general solution of $$y-x\, \cfrac {dy}{dx}\, =\, b(1+x^2\cfrac {dy}{dx})$$ is:
  • $$\quad b+kx=y(1+bx)$$
  • $$\quad b+ky=x(1+bx)$$
  • $$\quad b+ky=x(1+by)$$
  • $$\quad b+kx=x(1+by)$$
Which of the following differential equations has y = x as one of its particular solution?
  • $$\cfrac{d^2y}{dx^2}-x^2\cfrac{dy}{dx}+xy=x$$
  • $$\cfrac{d^2y}{dx^2}+x\cfrac{dy}{dx}+xy=x$$
  • $$\cfrac{d^2y}{dx^2} -x^2\cfrac{dy}{dx}+xy=0$$
  • $$\cfrac{d^2y}{dx^2}+x\cfrac{dy}{dx}+xy=0$$
Which of the following differential equations has $$y = c_1 e^x + c_2 e^{x}$$ as the general solution?
  • $$\dfrac{d^2y}{dx^2}+y = 0$$
  • $$\dfrac{d^2y}{dx^2}-y = 0$$
  • $$\dfrac{d^2y}{dx^2}+1 = 0$$
  • $$\dfrac{d^2y}{dx^2}-1 = 0$$
The solution of the differential equation $$\dfrac { dy }{ dx } =\dfrac { x-y+3 }{ 2\left( x-y \right) +5 } $$ is
  • $$2\left( x-y \right) +\log { \left( x-y \right) } =x+c$$
  • $$2\left( x-y \right) -\log { \left( x-y+2 \right) } =x+c$$
  • $$2\left( x-y \right) +\log { \left( x-y+2 \right) } =x+c$$
  • None of the above
The general solution of the differential equation $$\log _{ e }{ \left( \cfrac { dy }{ dx }  \right)  } =x+y$$ is:
  • $${ e }^{ x }+{ e }^{ -y }=C$$
  • $${ e }^{ x }+{ e }^{ y }=C$$
  • $${ e }^{ y }+{ e }^{ x }=C$$
  • $${ e }^{ -x }+{ e }^{ -y }=C$$
Find the general solution of $$dy=y \sec x dx$$.
  • $$y=C(\sec x -2\tan x)$$
  • $$y=C(\sec x +\tan x)$$
  • $$y=C(2\sec x +\tan x)$$
  • None of these
The solution of the differential equation $$\dfrac{dx}{x}+\dfrac{dy}{y}=0$$ is
  • $$xy=c$$
  • $$x+y=c$$
  • $$log\, x\, log\, y = c$$
  • $$x^2+y^2=c$$
The general solution of $$\cfrac{dy}{dx}=\cfrac{2x-y}{x+2y}$$ is
  • $${x}^{2}-xy+{y}^{2}=c$$
  • $${x}^{2}-xy-{y}^{2}=c$$
  • $${x}^{2}+xy-{y}^{2}=c$$
  • $${x}^{2}-x{y}^{2}=c$$
The order and degree of differential equation $${ \left( 1+3\cfrac { dy }{ dx }  \right)  }^{ 2/3 }=4\cfrac { { d }^{ 3 }y }{ d{ x }^{ 3 } } $$, are
  • $$1,\cfrac{2}{3}$$
  • $$3,1$$
  • $$3,3$$
  • $$1,2$$
The general solution of the differential equation $$\dfrac{y\, dx-x\, dy}{y}=0$$ is:
  • $$xy = C$$
  • $$x = Cy^2$$
  • $$y=Cx$$
  • $$y=Cx^2$$
The general solution of the differential equation $$\displaystyle \frac { dy }{ dx } +\frac { 1+\cos { 2y }  }{ 1-\cos { 2x }  } =0$$ is given by:
  • $$\tan { y } +\cot { x } =c$$
  • $$\displaystyle \tan { y } -\cot { x } =c$$
  • $$\displaystyle \tan { x } -\cot { y } =c$$
  • $$\displaystyle \tan { x } +\cot { x } =c$$
Which of the following are solutions of the differential equation $$y^{\prime\prime}-y=0$$ ?
  • $$y=Ce^x$$
  • $$y=C \dfrac {e^x}{2}$$
  • $$y=Ce^{-x}$$
  • None of these
Find a particular solution for the following differential equation.
$$y'-4y'-12y=te^{4t}$$
  • $$y(t)=\dfrac{1}{32}(3t+1)e^{4t}$$
  • $$y(t)=-\dfrac{1}{18}(3t+1)e^{4t}$$
  • $$y(t)=-\dfrac{1}{36}(3t+1)e^{4t}$$
  • None of these
If the general solutions of a differential equation is $$(y+c)^2=cx$$, where $$c$$ is an arbitrary constant, then the order and degree of differential equation are:
  • $$1,2$$
  • $$2,1$$
  • $$1,1$$
  • None of these
Which of the following are true?
  • Particular solution is a solution of a differential equation containing no arbitrary constants.
  • Particular Solution is a solution to a differential equation that contains arbitrary, unevaluated constants.
  • General solution is a solution of a differential equation containing no arbitrary constants.
  • General Solution is a solution to a differential equation that contains arbitrary, unevaluated constants.
Verify that $$y=Cx^3$$ is a solution of the differential equation $$xy'-3y=0$$ for any value of C. Then

find the particular solution determined by the initial condition $$y=2$$ when $$x=-3$$.
  • $$y=\dfrac{2}{27}x^2$$
  • $$y=-\dfrac{2}{27}x^3$$
  • $$y=-\dfrac{2}{25}x^3$$
  • None of these
The solution for the differential equation $$\cfrac { dy }{ y } +\cfrac { dx }{ x } =0$$ is:
  • $$\cfrac { 1 }{ y } +\cfrac { 1 }{ x } =c$$
  • $$\log { x } .\log { y } =c$$
  • $$xy=C$$
  • $$x+y=c$$
The solution of differential equation $$x \dfrac {dy}{dx} + 2y= x^{2}$$ is ____
  • $$y = \dfrac {x^{2} + C}{4x^{2}}$$
  • $$y = \dfrac {x^{2}}{4} + C$$
  • $$y = \dfrac {x^{4} + C}{x^{2}}$$
  • $$y = \dfrac {x^{4} + C}{4x^{2}}$$
The solution of the differential equation $$y\sin\left(\dfrac{x}{y}\right)dx=\left(x \sin\left(\dfrac{x}{y} \right)-y \right) dy$$ satisfying $$y(\dfrac{\pi}{4})=1$$ is
  • $$\cos\dfrac{x}{y}=-\log_ey+\dfrac{1}{\sqrt{2}}$$
  • $$\sin\dfrac{x}{y}=\log_ey+\dfrac{1}{\sqrt{2}}$$
  • $$\sin\dfrac{x}{y}=\log_ex-\dfrac{1}{\sqrt{2}}$$
  • $$\cos\dfrac{x}{y}=-\log_ex-\dfrac{1}{\sqrt{2}}$$
The integrating factor of linear differential equation $$\cfrac { dy }{ dx } +y\sec { x } =\tan { x } $$ is:
  • $$\sec { x } -\tan { x } $$
  • $$\sec { x } .\tan { x } $$
  • $$\sec { x } +\tan { x } $$
  • $$\sec { x } .\cot { x } $$
The integrating factor of the differential equation $$\cfrac { dy }{ dx } -y\tan { x } =\cos { x } $$ is:
  • $$\sec{x}$$
  • $$\cos{x}$$
  • $${e}^{\tan{x}}$$
  • $$\cot{x}$$
The solution of $$\dfrac {d^{2}x}{dy^{2}} - x = k$$, where $$k$$ is a non-zero constant, vanishes when $$y = 0$$ and tends of finite limit as $$y$$ tends to infinity, is
  • $$x = k(1 + e^{-y})$$
  • $$x = k(e^{y} + e^{-y} - 2)$$
  • $$x = k(e^{-y} - 1)$$
  • $$x = k(e^{y} - 1)$$
For the differential equation $${ \left( \cfrac { dy }{ dx }  \right)  }^{ 2 }-x\left( \cfrac { dy }{ dx }  \right) +y=0$$, which one of the following is not its solution?
  • $$y=x-1$$
  • $$4y={x}^{2}$$
  • $$y=x$$
  • $$y=-x-1$$
The solution of the differential equation $$\dfrac {dy}{dx} = \dfrac {yf'(x) - y^{2}}{f(x)}$$ is:
  • $$f(x) = y + C$$
  • $$f(x) = y(x + C)$$
  • $$f(x) = x + C$$
  • None of the above
Solution of $$\cfrac { dx }{ dy } +mx=0$$, $$m< 0$$ is
  • $$x=c{ e }^{ my }$$
  • $$x=c{ e }^{ -my }$$
  • $$x=my+c$$
  • $$x=c$$
What is the general solution of the differential equation $${ e }^{ x }\tan { y } dx+\left( 1-{ e }^{ x } \right) \sec ^{ 2 }{ y } dy=0$$?
  • $$\sin { y } =c\left( 1-{ e }^{ x } \right) $$ where $$c$$ is the constant of integration
  • $$\cos { y } =c\left( 1-{ e }^{ x } \right) $$ where $$c$$ is the constant of integration
  • $$\cot { y } =c\left( 1-{ e }^{ x } \right) $$ where $$c$$ is the constant of integration
  • None of the above
The solution of $$\dfrac{dy}{dx}=\sqrt{1-x^2-y^2+x^2y^2}$$  is:
where c is an arbitrary constant
  • $${\sin}^{-1}y={\sin}^{-1}x+c$$
  • $$2{\sin}^{-1}y=\sqrt{1-x^2}+{\sin}^{-1}x+c$$
  • $$2{\sin}^{-1}y=x \sqrt{1-x^2}+{\sin}^{-1}x+c$$
  • $$2{\sin}^{-1}y=x \sqrt{1-x^2}+{\cos}^{-1}x+c$$
What is the general solution of the differential equation $${x}^{2}dy+{y}^{2}dx=0$$?
  • $$x+y=c$$ where $$c$$ is the constant of integration
  • $$xy=c$$ where $$c$$ is the constant of integration
  • $$c(x+y)=xy$$ where $$c$$ is the constant of integration
  • None of the above
What is the solution of $$\frac{dy}{dx}=2y-1$$ is :
  • $$y=\frac{1-e^{-2x}}{2}$$
  • $$y=\frac{1+e^{2x}}{2}$$
  • $$y=1+e^x$$
  • $$y=\frac{1+e^{x}}{2}$$
What is the number of arbitrary constants in the particular solution of differential equation of third order ? 
  • 0
  • 1
  • 2
  • 3
What is D equal to ?
  • -1
  • 1
  • -2
  • None of the above
What is B equal to ?
  • -1
  • 1
  • 2
  • None of the above
What is the equation of a curve passing through (0, 1) and whose differential equation is given by dy = y tan x dx ? 
  • y = cos x
  • y = sin x
  • y = sec x
  • y = cosec x
What is C equal to ? 
  • 1
  • -1
  • 2
  • None of the above
0:0:1


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