JEE Main & Advanced Mathematics Differential Equations Question Bank

done Mock Test - Differential Equations Total Questions - 25

• question_answer1)
  The solution to the differential equation\[y\log y+xy'=0\], where \[y(1)=e\], is
  A)
  \[x(log\,y)=1\]    done clear
  B)
  \[xy(log\,y)=1\] done clear
  C)
  \[{{(log\,y)}^{2}}=2\]  done clear
  D)
  \[\log y+\left( \frac{{{x}^{2}}}{2} \right)y=1\] done clear
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• question_answer2)
  If \[y=\frac{x}{\log \left| cx \right|}\] (where c is an arbitrary constant) I the general solution of the differential equation \[dy/dx=y/x+\phi (x/y)\] then the function \[\phi (x/y)\]is
  A)
  \[{{x}^{2}}/{{y}^{2}}\]         done clear
  B)
  \[-{{x}^{2}}/{{y}^{2}}\] done clear
  C)
  \[{{y}^{2}}/{{x}^{2}}\]         done clear
  D)
  \[-{{y}^{2}}/{{x}^{2}}\] done clear
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• question_answer3)
  The differential equation of all parabolas whose axis are parallel to the y-axis is
  A)
  \[\frac{{{d}^{3}}y}{d{{x}^{3}}}=0\] done clear
  B)
  \[\frac{{{d}^{2}}x}{d{{y}^{2}}}=C\] done clear
  C)
  \[\frac{{{d}^{3}}y}{d{{x}^{3}}}+\frac{{{d}^{2}}x}{d{{y}^{2}}}=0\] done clear
  D)
  \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+2\frac{dy}{dx}=C\] done clear
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• question_answer4)
  Solution of differential equation \[dy-sin\text{ }x\text{ }sin\text{ }y\text{ }dx=0\] is
  A)
  \[{{e}^{\cos x}}\tan \frac{y}{2}=c\] done clear
  B)
  \[{{e}^{\cos x}}\tan \,y=c\] done clear
  C)
  \[\cos x\tan y=c\] done clear
  D)
  \[\cos x\sin y=]c\] done clear
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• question_answer5)
  Solution of \[\frac{dy}{dx}+2xy=y\] is
  A)
  \[y=c{{e}^{x-{{x}^{2}}}}\] done clear
  B)
  \[y=c\,{{e}^{{{x}^{2}}-x}}\] done clear
  C)
  \[y=c\,{{e}^{x}}\]       done clear
  D)
  \[y=c\,{{e}^{-{{x}^{2}}}}\] done clear
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• question_answer6)
  The solution of (y + x + 5) dy = (y - x + 1) dx is
  A)
  \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})+ta{{n}^{-1}}\frac{y+3}{y+2}+C\] done clear
  B)
  \[\log ({{(y+3)}^{2}}+{{(x-2)}^{2}})+ta{{n}^{-1}}\frac{y-3}{x-2}=C\] done clear
  C)
  \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})+2ta{{n}^{-1}}\frac{y+3}{x+2}=C\] done clear
  D)
  \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})-2ta{{n}^{-1}}\frac{y+3}{x+2}=C\] done clear
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• question_answer7)
  The slope of the tangent at (x, y) to a curve passing through \[\left( 1,\frac{\pi }{4} \right)\] is given by \[\frac{y}{x}-{{\cos }^{2}}\left( \frac{y}{x} \right)\] , then the equation of
  A)
  \[y={{\tan }^{-1}}\left( \log \left( \frac{e}{x} \right) \right)\] done clear
  B)
  \[y=x{{\tan }^{-1}}\left( \log \left( \frac{x}{e} \right) \right)\] done clear
  C)
  \[y=x{{\tan }^{-1}}\left( \log \left( \frac{e}{x} \right) \right)\] done clear
  D)
  None of thee done clear
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• question_answer8)
  Solution of the differential equation \[(y+x\sqrt{xy}(x+y))\,dx+(y\sqrt{xy}(x+y)-x)dy=0\] is
  A)
  \[\frac{{{x}^{2}}+{{y}^{2}}}{2}+{{\tan }^{-1}}\sqrt{\frac{y}{x}=c}\] done clear
  B)
  \[\frac{{{x}^{2}}+{{y}^{2}}}{2}+2{{\tan }^{-1}}\sqrt{\frac{x}{y}=c}\] done clear
  C)
  \[\frac{{{x}^{2}}+{{y}^{2}}}{2}+2{{\cot }^{-1}}\sqrt{\frac{x}{y}=c}\] done clear
  D)
  None of these done clear
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• question_answer9)
  The solution of \[\frac{dy}{dx}=\frac{{{x}^{2}}+{{y}^{2}}+1}{2xy}\] satisfying y(1)=1 is given by
  A)
  a system of parabolas done clear
  B)
  a system of circles done clear
  C)
  \[{{y}^{2}}=x(1+x)-1\] done clear
  D)
  \[{{(x-2)}^{2}}+{{(y-3)}^{2}}=5\] done clear
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• question_answer10)
  If integrating factor of \[x(1-{{x}^{2}})dy+(2{{x}^{2}}y-y-a{{x}^{3}})dx=0\] is \[_{e}\int pdx\], then P is equal to
  A)
  \[\frac{2{{x}^{2}}-a{{x}^{3}}}{x(1-{{x}^{2}})}\]    done clear
  B)
  \[2{{x}^{3}}-1\] done clear
  C)
  \[\frac{2{{x}^{2}}-a}{a{{x}^{3}}}\] done clear
  D)
  \[\frac{2{{x}^{2}}-1}{x(1-{{x}^{2}})}\] done clear
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• question_answer11)
  The solution of the differential equation \[{{x}^{2}}\frac{dy}{dx}\cos \frac{1}{x}-y\sin \frac{1}{x}=-1\], where \[y\to -1\,\,as\,\,x\to \infty \]is
  A)
  \[y=\sin \frac{1}{x}-\cos \frac{1}{x}\] done clear
  B)
  \[y=\frac{x+1}{x\sin \frac{1}{x}}\] done clear
  C)
  \[y=\cos \frac{1}{x}+sin\frac{1}{x}\] done clear
  D)
  \[y=\frac{x+1}{x\cos 1/x}\] done clear
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• question_answer12)
  The solution of differential equation \[(2y+x{{y}^{3}})dx+(x+{{x}^{2}}{{y}^{2}})dy=0\] is
  A)
  \[{{x}^{2}}y+\frac{{{x}^{3}}{{y}^{3}}}{3}=c\] done clear
  B)
  \[x{{y}^{2}}+\frac{{{x}^{3}}{{y}^{3}}}{3}=c\] done clear
  C)
  \[{{x}^{2}}y+\frac{{{x}^{4}}{{y}^{4}}}{4}=c\] done clear
  D)
  none of these done clear
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• question_answer13)
  The general solution of the equation \[\frac{dy}{dx}=1+xy\] is
  A)
  \[y=c{{e}^{-{{x}^{2}}/2}}\] done clear
  B)
  \[y=c{{e}^{{{x}^{2}}/2}}\] done clear
  C)
  \[y=(x+c){{e}^{-{{x}^{2}}/2}}\] done clear
  D)
  None of these done clear
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• question_answer14)
  The solution of differential equation \[\frac{x+y\frac{dy}{dx}}{y-x\frac{dy}{dx}}=\frac{x{{\cos }^{2}}({{x}^{2}}+{{y}^{2}})}{{{y}^{3}}}\] is
  A)
  \[\tan ({{x}^{2}}+{{y}^{2}})=\frac{{{x}^{2}}}{{{y}^{2}}}+c\] done clear
  B)
  \[cot({{x}^{2}}+{{y}^{2}})=\frac{{{x}^{2}}}{{{y}^{2}}}+c\] done clear
  C)
  \[tan({{x}^{2}}+{{y}^{2}})=\frac{{{y}^{2}}}{{{x}^{2}}}+c\] done clear
  D)
  \[cot({{x}^{2}}+{{y}^{2}})=\frac{{{y}^{2}}}{{{x}^{2}}}+c\] done clear
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• question_answer15)
    The solution of the differential equation \[\frac{dy}{dx}=\frac{3{{x}^{2}}{{y}^{4}}+2xy}{{{x}^{2}}-2{{x}^{3}}{{y}^{3}}}\] is
  A)
  \[\frac{{{y}^{2}}}{x}-{{x}^{3}}{{y}^{2}}=c\] done clear
  B)
  \[\frac{{{x}^{2}}}{{{y}^{2}}}+{{x}^{3}}{{y}^{3}}=c\] done clear
  C)
  \[\frac{{{x}^{2}}}{y}+{{x}^{3}}{{y}^{2}}=c\] done clear
  D)
  \[\frac{{{x}^{2}}}{3y}-{{x}^{3}}{{y}^{2}}=c\] done clear
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• question_answer16)
  Tangent to a curve intercepts the y-axis at a point P. A line perpendicular to this tangent through P passes through another point (1, 0). The differential equation of the curve
  A)
  \[y\frac{dy}{dx}-x{{\left( \frac{dy}{dx} \right)}^{2}}=1\] done clear
  B)
  \[\frac{x{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}}=0\] done clear
  C)
  \[y\frac{dx}{dy}+x=1\] done clear
  D)
  None of these done clear
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• question_answer17)
  The normal to a curve at P(x, y) meets the x-axis at G. If the distance of G from the origin is twice the abscissa of P, then the curve is a
  A)
  parabola            done clear
  B)
  circle done clear
  C)
  hyperbola          done clear
  D)
  ellipse done clear
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• question_answer18)
  Orthogonal trajectories of family of the curve \[{{x}^{2/3}}+{{y}^{2/3}}={{a}^{2/3}}\], where a is any arbitrary constant, is
  A)
  \[{{x}^{2/3}}-{{y}^{2/3}}=c\] done clear
  B)
  \[{{x}^{4/3}}-{{y}^{4/3}}=c\] done clear
  C)
  \[{{x}^{4/3}}+{{y}^{4/3}}=c\] done clear
  D)
  \[{{x}^{1/3}}-{{y}^{1/3}}=c\] done clear
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• question_answer19)
  The solution of the differential equation\[y(2{{x}^{4}}+y)\frac{dy}{dx}=(1-4x{{y}^{2}}){{x}^{2}}\] is given by
  A)
  \[3{{({{x}^{2}}y)}^{2}}+{{y}^{3}}-{{x}^{3}}=c\] done clear
  B)
  \[x{{y}^{2}}+\frac{{{y}^{3}}}{3}-\frac{{{x}^{3}}}{3}+c=0\] done clear
  C)
  \[\frac{2}{3}y{{x}^{5}}+\frac{{{y}^{3}}}{3}=\frac{{{x}^{3}}}{3}-\frac{4x{{y}^{3}}}{3}+c\] done clear
  D)
  None of these done clear
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• question_answer20)
  The solution to the differential equation \[\frac{dy}{dx}=\frac{x+y}{x}\] satisfying the condition y(1)=1 is
  A)
  \[y=\ln \,x+x\] done clear
  B)
  \[y=x\,\ln \,x+{{x}^{2}}\] done clear
  C)
  \[y=x{{e}^{(x-1)}}\]    done clear
  D)
  \[y=x\,\,\ln \,x+x\] done clear
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• question_answer21)
  The differential equation of all parabolas each of which has a latus rectum 4a and whose axes are parallel to the x-axis is of order 2 degree _____.
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• question_answer22)
  Differential equation of the family of curves v = A/r + B, where A and B are arbitrary constants, is \[\frac{{{d}^{2}}v}{d{{r}^{2}}}+\frac{k}{r}\frac{dv}{dr}=0\]. Then the value of k is _____.
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• question_answer23)
  The population of a country increases at a rate proportional to the number of inhabitants. f is the population which doubles in 30 years, then the population will triple in approximately ________ years.
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• question_answer24)
  The solution of the differential equation\[\frac{{{d}^{2}}v}{d{{r}^{2}}}=\sin 3x+{{e}^{x}}+{{x}^{2}}\] when \[{{y}_{1}}(0)=1\] and \[y(0)=0\] is k\[\sin 3x+{{e}^{x}}+\frac{{{x}^{4}}}{12}+\frac{1}{3}x-1\] the value of k is ________.
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• question_answer25)
  The solution to the equation \[\frac{{{d}^{2}}x}{d{{x}^{2}}}={{e}^{-2x}}\] is \[k{{e}^{-2e}}+cx+d\]. Then the value of k is _____.
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