CBSE Questions for Class 12 Commerce Maths Differential Equations Quiz 2 - MCQExams.com

$$\dfrac{dy}{dx}=\dfrac{4x+2y+1}{x-2y+3}$$ is a differential equation of the type:
  • variable separable
  • exact
  • Homogeneous
  • non homogeneous
The solution of $$\displaystyle \frac{dy}{dx}=\displaystyle \frac{3(y+1)}{x-2}$$ is:
  • $$y -1= c (x - 2)^{3}$$
  • $$y +1= c (x - 2)^{3}$$
  • $$y +1= c (x - 2)^{2}$$
  • $$y -1= c (x - 2)^{2}$$
The solution of $$\dfrac{dy}{dx}=\dfrac{x+x^{2}}{y-y^{2}}$$ is:
  • $$3(y^{2}-x^{2}{})=2(x^{3}+y^{3})+c$$
  • $$3(x^{2}+y^{2}{})=2(x^{3}+y^{3})+c$$
  • $$x^{2}-y^{2}{}=x^{3}+y^{3}+c$$
  • $$x^{2}+y^{2}{}=x^{3}+y^{3}+c$$
The solution of the differential equation
$$\displaystyle \frac{dy}{dx}=\displaystyle \frac{xy+y}{xy+x}$$ is
  • $$x + y = log \left (\displaystyle \frac{cy}{cx} \right )$$
  • $$x + y = \log (cxy)$$
  • $$x - y = \log \left ( \displaystyle \frac{cx}{y} \right )$$
  • $$y - x = \log \left ( \displaystyle \frac{cx}{y} \right )$$
The solution of $$\dfrac{dy}{dx}= \tan\  y$$ is
  • $$\log \sin x = 2x + c$$
  • $$\log \sin y = x + c$$
  • $$\log \sin y = 2x + c$$
  • $$\log \sin x = 2 y + c$$
The solution of the differential equation $$(1+x^{2})\displaystyle \frac{dy}{dx}=2 x \cot y$$, is:
  • $$\sec y = c (1+x^{2})$$
  • $$\cos y = c (1+x^{2})$$
  • $$\tan y = c (1+x^{2})$$
  • $$\cot y = c (1+x^{2})$$
The solution of $$\frac{dy}{dx}=e^{x-y}$$ is:
  • $$e^{x}-e^{y}=c$$
  • $$e^{x}+e^{y}=c$$
  • $$e^{-x}-e^{y}=c$$
  • $$e^{-x}-e^{-y}=c$$
The solution of $$(1+e^{x})ydy=e^{x}dx$$ is:
  • $$y^{2}=\log c(e^{x}+1)$$
  • $$\dfrac{y^{2}}{2}=\log ce^{x}$$
  • $$\dfrac{y^{2}}{2}=\log c(e^{x}+1)$$
  • $$2y = \log c (e^{x}+1)$$
The solution of $$\displaystyle \frac{dy}{dx}+y \tan x = 0$$ is:
  • $$y = a \cos x$$
  • $$y = a \sin x$$
  • $$y = \log \cos x + c$$
  • $$y = a \tan x + c$$
The solution of $$ \displaystyle \frac{dy}{dx}=(1+y^{2})(1+x^{2})^{-1}$$ is
  • $$y - x = c (1+ xy)$$
  • $$y + x = c (1+ xy)$$
  • $$y = (1+ 2x) c$$
  • $$xy = x^2 + x + c$$
The solution of $$\dfrac{dy}{dx}=\dfrac{1+y^{2}}{\sec x}$$ is
  • $$\tan^{-1}y=\cos x+c$$
  • $$\tan^{-1}y=\sin x+c$$
  • $$y^{3}=\csc x +c$$
  • $$\log\left ( 1+y^{2} \right )=\cos x+c$$
The solution of $$x^{3}dy-y^{3}dx=0$$ is:
  • $$x^{4}-y^{4}=c$$
  • $$x^{2}-y^{2}=c$$
  • $$\dfrac{1}{x}-\dfrac{1}{y}=c$$
  • $$\dfrac{1}{x^{2}}-\dfrac{1}{y^{2}}=c$$
The solution of $$\dfrac{dy}{dx}+2x=e^{3x}$$ is:
  • $$y=3e^{3x}-x^{2}+c$$
  • $$y=\dfrac{e^{3x}}{3}-x^{2}+c$$
  • $$y=e^{3x}-x^{2x}+c$$
  • $$y=e^{3x}-x^{2}+c$$
The solution of $$\dfrac{d^{2}y}{dx^{2}}=xe^{x}+1$$ is:
  • $$\displaystyle y=(x-1)e^{x}+\frac{1}{2}x^{2}+C_{1}x+C_{2}$$
  • $$\displaystyle y=(x-2)e^{x}+\frac{1}{2}x^{2}+C_{1}x+C_{2}$$
  • $$\displaystyle y=(x+2)e^{x}+\frac{1}{2}x^{2}+C_{1}x+C_{2}$$
  • $$\displaystyle y=(x+2)e^{x}+C_{1}$$
The solution of $$x^{2}dy-y^{2}dx=0$$ is:
  • $$\dfrac{1}{x}-\dfrac{1}{y}=c$$
  • $$\dfrac{1}{x}+\dfrac{1}{y}=c$$
  • $$x^{3}-y^{3}=c$$
  • $$x^{2}-y^{2}=c$$
The solution of $$\dfrac{dy}{dx}=\text{cosech }y$$ is:
  • $$x=\cosh y+c$$
  • $$x=\text{sech}\ y+c$$
  • $$x=\sinh y+c$$
  • $$x=\coth y+c$$
The solution of $$\dfrac{dy}{dx}+\dfrac{x^{2}}{y^{2}}=0$$ is:
  • $$x^{2}+y^{2}=c$$
  • $$x^{2}-y^{2}=c$$
  • $$x^{3}-y^{3}=c$$
  • $$x^{3}+y^{3}=c$$
The solution of $$\dfrac{dy}{dx}=\dfrac{1+y}{1-x}$$ is:
  • $$1+y=c(1-x)$$
  • $$(1-x)(1+y)=c$$
  • $$(1-y)(1-x)=c$$
  • $$(1+x)(1+y)=c$$
The solution of $$\dfrac{dy}{dx}+y=1$$ is:
  • $$x+\log\left | 1-y \right |=c$$
  • $$y+\log\left | 1-x \right |=c$$
  • $$x(1-y)=c$$
  • $$y(1-x)=c$$
The solution of $$\dfrac{dy}{dx}=\dfrac{x+x^{2}}{y+y^{2}}$$ is:
  • $$x^{3}-y^{3}-y^{2}-x^{2}=c$$
  • $$2(x^{3}-y^{3})+3(x^{2}-y^{2})=c$$
  • $$x^{2}+y^{2}+x+y=c$$
  • $$x^{2}y+xy^{2}=c$$
The solution of $$\dfrac{dy}{dx}+\dfrac{\sqrt{1-y^{2}}}{\sqrt{1-x^{2}}}=0$$ is
  • $$sin^{-1}x+sin^{-1}y=c$$
  • $$cot^{-1}x+cot^{-1}y=c$$
  • $$Tan^{-1}x+Tan^{-1}y=c$$
  • $$sinh^{-1}x+sinh^{-1}y=c$$
Solution of $$\dfrac{dy}{dx}=\dfrac{y+2}{x-1}$$ is:
  • $$y+2=c(x-1)$$
  • $$(y+2)(x-1)=c$$
  • $$log(y+2)=c(x-y)$$
  • $$log(x-1)=c(y+2)$$
The solution of $$\dfrac{dy}{dx}=\dfrac{x}{y}$$ is:
  • $$x^{2}-y^{2}=c$$
  • $$x^{2}+y^{2}=c$$
  • $$x^{2}-y=c$$
  • $$x^{2}+y=c$$
The solution of $$y\dfrac{dy}{dx}=1+y^{2}$$ is:
  • $$2x=log\left [ c\left ( 1+y^{2} \right ) \right ]$$
  • $$x=cy^{2}$$
  • $$c(1+y^{2})=x$$
  • $$2y=log\left \{ c\left ( 1+x^{2} \right ) \right \}$$
The solution of $$\dfrac{dy}{dx}=\tan x $$ is
  • $$e^{y}= c \sin x$$
  • $$e^{y}= c \cos x$$
  • $$e^{y}= c \csc x$$
  • $$e^{y}= c \sec x$$
The solution of $$x+y\dfrac{dy}{dx}=5$$ is:
  • $$x^{2}-10x+y^{2}=c$$
  • $$x^{2}-5x+y^{2}=c$$
  • $$\dfrac{y^{2}}{2}=10x+x^{2}+c$$
  • $$y^2=10x+x^{2}+c$$
The solution of $$\displaystyle \frac{dy}{dx}=e^{\displaystyle (y-x)}$$ is
  • $$e^{\displaystyle y}+e^{\displaystyle x}=c$$
  • $$e^{\displaystyle -x}=e^{\displaystyle -y}+c$$
  • $$e^{\displaystyle y-x}=c$$
  • $$e^{\displaystyle y/x}=c$$
The solution of $$\displaystyle \dfrac{dy}{dx}=2y\tanh x $$ is
  • $$cy=\sinh^{2}x$$
  • $$cy=\sec h^{2}x$$
  • $$cy=\cosh^{2}x$$
  • $$cy=\coth^{2}x$$
The solution of $$\left ( 1+y^{2} \right )dx=xydy$$ is:
  • $$1+y^{2}=cx^{2}$$
  • $$\left ( 1+y^{2} \right )x^{2}=c$$
  • $$1+y^{2}=cx$$
  • $$\left ( 1+y^{2} \right )x=c$$
The solution of $$\dfrac{dy}{dx}=2^{x-y}$$ is:
  • $$2^{x}+2^{y}=c$$
  • $$2^{x}-2^{y}=c$$
  • $$2^{x-y}=c$$
  • $$2^{x+y}=c$$
The solution of $$(x^{2}+x)\frac{dy}{dx}=1+2x$$ is:
  • $$e^{y}=c(x^{2}+x)$$
  • $$y=x(x+1)+c$$
  • $$y=(1+2x)+c$$
  • $$xy=x^{2}+x+c$$
The solution of

$$(x^{2}-yx^{2})\dfrac{dy}{dx}+(y^{2}+x^{2}y^{2})=0$$
  • $$log(xy)=\dfrac{1}{x}+\dfrac{1}{y}+c$$
  • $$log y+\dfrac{1}{y}=x-\dfrac{1}{x}+c$$
  • $$y-\dfrac{1}{y}=x-\dfrac{1}{x}+c$$
  • $$log x+\dfrac{1}{x}=y-\dfrac{1}{y}+c$$
The solution of $$\tan x\ dy + \tan y\ dx$$ $$=$$ 0
  • $$\tan x.\tan y= c$$
  • $$\sec^{2}x+\sec^{2}y=c$$
  • $$\sin x.\sin y$$ $$= c$$ 
  • $$\cot x.\cot y$$ $$= c$$ 
The solution of $$e^{ x }\tan ydx+(1-e^{ x })\sec^{ 2 }ydy=0$$ is
  • $$\tan y=c(1-e^{x})$$
  • $$\sec y=c(1-e^{x})$$
  • $$\tan y(1-e^{x})=c$$
  • $$\sec y=1-e^{x}$$
The general solution of the differential equation $$\dfrac{dy}{dx}-\dfrac{2xy}{1+x^{2}}=0$$ is
  • $$y=A(1+x^{2})$$
  • $$y=A\sqrt{1+x^{2}}$$
  • $$y=\dfrac{A}{1+x^{2}}$$
  • $$y=\dfrac{a}{\sqrt{1+x^{2}}}$$
Solution of  $$\tan y.\sec^{2}xdx + \tan x.\sec^{2} ydy=0$$ is:
  • $$\sec x \sec y=c$$
  • $$\tan x. \tan y=c$$
  • $$\sin x . \sin y=c$$
  • $$\cos x .\cos y=c$$
The solution of $$x\sqrt{1+x^{2}}dx+y\sqrt{1+y^{2}}dy=0$$ is:
  • $$\sqrt{1+x^{2}}+\sqrt{1+y^{2}}=c$$
  • $$(\sqrt{1+x^{2}}$$x$$\sqrt{1+y^{2}}))=c$$
  • $$(1+x^{2})^{\frac{3}{2}}+(1+y^{2})^{\frac{3}{2}}=c$$
  • $$\frac{x}{\sqrt{1+x^{2}}}+\frac{y}{\sqrt{1+y^{2}}}=c$$
The solution of $$x^{2}+y^{2}\dfrac{dy}{dx}=4$$ is
  • $$x^{2}+y^{2}=12x+c$$
  • $$x^{2}+y^{2}=3x+c$$
  • $$x^{3}+y^{3}=3x+c$$
  • $$x^{3}+y^{3}=12x+c$$
The solution of $$\dfrac{dy}{dx}=e^{2x-y}+x^{3}e^{-y}$$ is:
  • $$e^{y}=e^{2x}+3x^{2}+c$$
  • $$e^{y}=e^{2x}+\dfrac{x^{4}}{4}+c$$
  • $$4e^{y}=2e^{2x}+x^{4}+c$$
  • $$2e^{y}=e^{2x}+3x^{4}+c$$
Solve the differential equation:
$$\sqrt{1+x^{2}}dx+\sqrt{1+y^{2}}dy=0$$
  • $$x\sqrt{1+x^{2}}+y\sqrt{1+y^{2}}+log\left [ \left ( x+\sqrt{1+x^{2}} \right )\left ( y+\sqrt{1+y^{2}} \right ) \right ]=c$$
  • $$\sqrt{1+x^{2}}+\sqrt{1+y^{2}}=c$$
  • $$\dfrac{1}{\sqrt{1+x^{2}}}+\dfrac{1}{\sqrt{1+y^{2}}}=c$$
  • $$log\left \{ \left ( \sqrt{1+x^{2}} \right )+\left ( \sqrt{1+y^{2}}\right ) \right \}=x+c$$
The solution of $$\frac{dy}{dx}=e^{x-y}+e^{2logx-y}$$
  • $$e^{y}=e^{x}+\frac{x^{2}}{3}+c$$
  • $$e^{y}=e^{x}+\frac{x^{3}}{3}+c$$
  • $$e^{y}=e^{x}+ log x + c$$
  • $$y=\dfrac{e^{3x+y}}{3}$$
The solution of $$x\sqrt{1-y^{2}}dx+y\sqrt{1-x^{2}}dy=0$$ is:
  • $$sin^{-1}x+sin^{-1}y=c$$
  • $$\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=c$$
  • $$\sqrt{1-x^{2}}\sqrt{1-y^{2}}=c$$
  • $$\sqrt{\dfrac{1-x^{2}}{1-y^{2}}}=c$$
The solution of $$xdx+ydy=x^{2}ydy-xy^{2}dx$$ is
  • $$x^{2}-1=c(1+y^{2})$$
  • $$x^{2}+1=c(1-y^{2})$$
  • $$x^{3}-1=c(1+y^{3})$$
  • $$x^{3}+1=c(1-y^{3})$$
The solution of $$2xy\dfrac{dy}{dx}=1+y^{2}$$ is:
  • $$1+y^{2}=cx$$
  • $$1-y^{2}=cx$$
  • $$1+x^{2}=cy$$
  • $$1-x^{2}=cy$$
The solution of $$xcos^{2} y(dx) + tan y(dy)=0$$ is:
  • $$x^{2}+sec^{2}y=c$$
  • $$x^{2}+cot^{2}y=c$$
  • $$x^{2}+sin^{2}y=c$$
  • $$x^{2}+cos^{2}y=c$$
The solution of $$log\left (\displaystyle \frac{dy}{dx} \right )=ax+by$$
  • $$be^{\displaystyle ax}+ae^{\displaystyle -by}=k$$
  • $$e^{\displaystyle ax}+e^{\displaystyle -by}=c$$
  • $$e^{\displaystyle ax+by}=c$$
  • $$(ax+by)=cxy$$
The solution of $$\dfrac{dy}{dx}=x log x$$
  • $$2y=x^{2}\left [ logx+\dfrac{1}{2} \right ]+c$$
  • $$2y=x^{2}\left [ logx-\dfrac{1}{2} \right ]+c$$
  • $$y=\dfrac{x^{2}}{2}(log2-x)+c$$
  • $$y^{2}=x^{2} log x + x + c$$
Solution of $$(xy^{2}+x)dx+(yx^{2}+y)dy=0$$
  • $$(x^{2}+1)(y^{2}+1)=c$$
  • $$(xy+1)(xy-1)=c$$
  • $$(x^{3}+1)(y^{3}+1)=c$$
  • $$(1-x^{2})(1-y^{2})=c$$
The solution of $$\sqrt{1-x^{2}}sin^{-1}xdy+ydx=0$$ is:
  • $$ytan^{-1}x=c$$
  • $$ysin^{-1}x=c$$
  • $$ycos^{-1}x=c$$
  • $$xsin^{-1}x=c$$
The solution of $$\frac{dy}{dx}=e^{3x+y}$$ given $$y=0$$ when $$x=0$$ is:
  • $$e^{3x}+3e^{-y}=4$$
  • $$e^{-y}=e^{3x}+4$$
  • $$3e^{-y}=e^{3x}+12$$
  • $$y=\dfrac{e^{3x+y}}{3}$$
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