JEE Main & Advanced Mathematics Differential Equations Question Bank

done Self Evaluation Test - Differential Equations Total Questions - 80

• question_answer1)
  The differential equation which represents the three parameter family of circles\[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] is
  A)
  \[y'''=\frac{3y'y'{{'}^{2}}}{1+y{{'}^{2}}}\]    done clear
  B)
  \[y'''=\frac{3y'{{'}^{2}}}{1+y{{'}^{2}}}\] done clear
  C)
  \[y'''=\frac{3y'}{1+y{{'}^{2}}}\] done clear
  D)
  \[y'''=\frac{3y'}{1-y{{'}^{2}}}\] done clear
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• question_answer2)
  A curve passing through (2, 3) and satisfying the differential equation \[\int_{0}^{x}{ty(t)dt={{x}^{2}}y(x),(x>0)}\] is
  A)
  \[{{x}^{2}}+{{y}^{2}}=13\] done clear
  B)
  \[{{y}^{2}}=\frac{9}{2}x\] done clear
  C)
  \[\frac{{{x}^{2}}}{8}+\frac{{{y}^{2}}}{18}=1\] done clear
  D)
  \[xy=c\] done clear
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• question_answer3)
  The solution of the differential equation \[3{{e}^{x\tan }}y\,\,dx+(1-{{e}^{x}}){{\sec }^{2}}y\,\,dy=0\] is
  A)
  \[{{e}^{x}}\tan y=C\] done clear
  B)
  \[C{{e}^{x}}={{(1-\tan \,\,y)}^{3}}\] done clear
  C)
  \[C\tan \,\,y={{(1-{{e}^{x}})}^{2}}\] done clear
  D)
  \[\tan \,\,y=C{{(1-{{e}^{x}})}^{3}}\] done clear
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• question_answer4)
  The differential equation of family of curves whose tangent form an angle of \[\pi /4\] with the hyperbola \[xy={{C}^{2}}\] is
  A)
  \[\frac{dy}{dx}=\frac{{{x}^{2}}+{{C}^{2}}}{{{x}^{2}}-{{C}^{2}}}\] done clear
  B)
  \[\frac{dy}{dx}=\frac{{{x}^{2}}-{{C}^{2}}}{{{x}^{2}}+{{C}^{2}}}\] done clear
  C)
  \[\frac{dy}{dx}=-\frac{{{C}^{2}}}{{{x}^{2}}}\] done clear
  D)
  None of these done clear
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• question_answer5)
  The solution of the differential equation\[\frac{dy}{dx}+\frac{y}{x}\log y=\frac{y}{{{x}^{2}}}{{(\log \,\,y)}^{2}}\] is
  A)
  \[y=\log ({{x}^{2}}+cx)\] done clear
  B)
  \[\log \,\,y=x\left( c{{x}^{2}}+\frac{1}{2} \right)\] done clear
  C)
  \[x=\log \,\,y\left( c{{x}^{2}}+\frac{1}{2} \right)\] done clear
  D)
  None of these done clear
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• question_answer6)
  If \[x\,dy=y\,dx+{{y}^{2}}dy,y>0\] and\[y\text{(1})=1\], then what is \[y(-3)\] equal to?
  A)
  3 only done clear
  B)
  -1 only done clear
  C)
  Both -1 and 3 done clear
  D)
  Neither -1 nor 3 done clear
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• question_answer7)
  The degree of the differential equation\[\frac{dy}{dx}-x={{\left( y-x\frac{dy}{dx} \right)}^{-4}}\] is
  A)
  2 done clear
  B)
  3 done clear
  C)
  4 done clear
  D)
  5 done clear
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• question_answer8)
  The solution of \[\frac{dy}{dx}=\left| x \right|\] is:
  A)
  \[y=\frac{x\left| x \right|}{2}+c\] done clear
  B)
  \[y=\frac{\left| x \right|}{2}+c\] done clear
  C)
  \[y=\frac{{{x}^{2}}}{2}+c\] done clear
  D)
  \[y=\frac{{{x}^{3}}}{2}+c\] Where c is an arbitrary constant done clear
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• question_answer9)
  The solution to the differential equation\[\frac{dy}{dx}=\frac{yf'(x)-{{y}^{2}}}{f(x)}\] Where f(x) is a given function is
  A)
  \[f(x)=y(x+c)\] done clear
  B)
  \[f(x)=cxy\] done clear
  C)
  \[f(x)=c(x+y)\] done clear
  D)
  \[yf(x)=cx\] done clear
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• question_answer10)
  Solution of differential equation \[{{x}^{2}}=1+{{\left( \frac{x}{y} \right)}^{-1}}\frac{dy}{dx}+\frac{{{\left( \frac{x}{y} \right)}^{-2}}{{\left( \frac{dy}{dx} \right)}^{2}}}{2!}\]\[+\frac{{{\left( \frac{x}{y} \right)}^{-3}}{{\left( \frac{dy}{dx} \right)}^{3}}}{3!}+.........\] is
  A)
  \[{{y}^{2}}={{x}^{2}}(ln\,\,{{x}^{2}}-1)+C\] done clear
  B)
  \[y={{x}^{2}}(ln\,\,x-1)+C\] done clear
  C)
  \[{{y}^{2}}=x(ln\,\,x-1)+C\] done clear
  D)
  \[y={{x}^{2}}{{e}^{{{x}^{2}}}}+C\]   done clear
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• question_answer11)
  The solution of the differential equation\[\frac{dy}{dx}=\frac{1-3y-3x}{1+x+y}\] is
  A)
  \[x+y-\ell n\left| x+y \right|=c\] done clear
  B)
  \[3x+y+2\ell n\left| 1-x-y \right|=c\] done clear
  C)
  \[x+3y-2\ell n\left| 1-x-y \right|=c\] done clear
  D)
  None of these done clear
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• question_answer12)
  A function \[y=f(x)\] satisfies the condition \[f'(x)\sin x+f(x)\cos x=1\] being bounded when\[x\to 0\]. If\[l=\int_{0}^{\pi /2}{f(x)dx}\], then
  A)
  \[\frac{\pi }{2}<l<\frac{{{\pi }^{2}}}{4}\] done clear
  B)
  \[\frac{\pi }{4}<l<\frac{{{\pi }^{2}}}{2}\] done clear
  C)
  \[1<l<\frac{\pi }{2}\] done clear
  D)
  \[0<l<1\] done clear
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• question_answer13)
  The solution of differential equation\[yy'=x\left( \frac{{{y}^{2}}}{{{x}^{2}}}+\frac{f({{y}^{2}}/{{x}^{2}})}{f'({{y}^{2}}/{{x}^{2}})} \right)\] is
  A)
  \[f({{y}^{2}}/{{x}^{2}})=c{{x}^{2}}\] done clear
  B)
  \[{{x}^{2}}f({{y}^{2}}/{{x}^{2}})={{c}^{2}}{{y}^{2}}\] done clear
  C)
  \[{{x}^{2}}f({{y}^{2}}/{{x}^{2}})=c\] done clear
  D)
  \[f({{y}^{2}}/{{x}^{2}})=cy/x\] done clear
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• question_answer14)
  The expression which is the general solution of the differential equation \[\frac{dy}{dx}+\frac{x}{1-{{x}^{2}}}y=x\sqrt{y}\] is
  A)
  \[\sqrt{y}+\frac{1}{3}(1-{{x}^{2}})=c{{(1-{{x}^{2}})}^{\frac{1}{4}}}\] done clear
  B)
  \[y{{(1-{{x}^{2}})}^{\frac{1}{4}}}=c(1-{{x}^{2}})\] done clear
  C)
  \[\sqrt{y}{{(1-{{x}^{2}})}^{\frac{1}{4}}}=\frac{1}{3}(1-{{x}^{2}})+c\] done clear
  D)
  None of these done clear
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• question_answer15)
  The solutions of \[(x+y+1)\text{ }dy=dx\] are
  A)
  \[x+y+2=C{{e}^{y}}\] done clear
  B)
  \[x+y+4=C\,log\,y\] done clear
  C)
  \[\log (x+y+2)=Cy\] done clear
  D)
  \[\log (x+y+2)=C-y\] done clear
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• question_answer16)
  The solution to of the differential equation \[(x+1)\frac{dy}{dx}-y={{e}^{3x}}{{(x+1)}^{2}}\] is
  A)
  \[y=(x+1){{e}^{3x}}+c\] done clear
  B)
  \[3y=(x+1)+{{e}^{3x}}+c\] done clear
  C)
  \[\frac{3y}{x+1}={{e}^{3x}}+c\] done clear
  D)
  \[y{{e}^{-3x}}=3(x+1)+c\] done clear
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• question_answer17)
  The curve satisfying the equation \[\frac{dy}{dx}=\frac{y(x+{{y}^{3}})}{x({{y}^{3}}-x)}\]and passing through the point (4, -2) is
  A)
  \[{{y}^{2}}=-2x\] done clear
  B)
  \[y=-2x\] done clear
  C)
  \[{{y}^{3}}=-2x\] done clear
  D)
  None of these done clear
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• question_answer18)
  The general solution of the differential equation \[\frac{dy}{dx}-\frac{\tan \,\,y}{1+x}=(1+x){{e}^{x}}\sec \,\,y\] is
  A)
  \[\sin (1+x)=y({{e}^{x}}+c)\] done clear
  B)
  \[y\sin (1+x)=c{{e}^{x}}\] done clear
  C)
  \[(1+x)sin\,\,y={{e}^{x}}+c\] done clear
  D)
  \[\sin \,\,y=(1+x)({{e}^{x}}+c)\] done clear
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• question_answer19)
  Which one of the following differential equations represents the family of straight lines which are at unit distance from the origin?
  A)
  \[{{\left( y-x\frac{dy}{dx} \right)}^{2}}=1-{{\left( \frac{dy}{dx} \right)}^{2}}\] done clear
  B)
  \[{{\left( y+x\frac{dy}{dx} \right)}^{2}}=1+{{\left( \frac{dy}{dx} \right)}^{2}}\] done clear
  C)
  \[{{\left( y-x\frac{dy}{dx} \right)}^{2}}=1+{{\left( \frac{dy}{dx} \right)}^{2}}\] done clear
  D)
  \[{{\left( y+x\frac{dy}{dx} \right)}^{2}}=1-{{\left( \frac{dy}{dx} \right)}^{2}}\] done clear
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• question_answer20)
  The solution of \[\frac{dy}{dx}=\sqrt{1-{{x}^{2}}-{{y}^{2}}+{{x}^{2}}{{y}^{2}}}\] is
  A)
  \[si{{n}^{-1}}y=si{{n}^{-1}}x+c\] done clear
  B)
  \[2si{{n}^{-1}}y=\sqrt{1-{{x}^{2}}}+si{{n}^{-1}}x+c\] done clear
  C)
  \[2si{{n}^{-1}}y=x\sqrt{1-{{x}^{2}}}+si{{n}^{-1}}x+c\] done clear
  D)
  \[2si{{n}^{-1}}y=x\sqrt{1-{{x}^{2}}}+{{\cos }^{-1}}x+c\] done clear
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• question_answer21)
  What is the solution of \[\frac{dy}{dx}+2y=1\] satisfying\[y(0)=0\]?
  A)
  \[y=\frac{1-{{e}^{-2x}}}{2}\] done clear
  B)
  \[y=\frac{1+{{e}^{-2x}}}{2}\] done clear
  C)
  \[y=1+{{e}^{x}}\] done clear
  D)
  \[y=\frac{1+{{e}^{x}}}{2}\] done clear
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• question_answer22)
  What is the solution of the differential equation\[\frac{dy}{dx}=\frac{y}{(x+2{{y}^{3}})}\]?
  A)
  \[y(1-xy)=cx\] done clear
  B)
  \[{{y}^{3}}-x=cy\] done clear
  C)
  \[x(1-xy)=cy\] done clear
  D)
  \[x(1+xy)=cy\] done clear
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• question_answer23)
  What is the solution of the differential equation\[a\left( x\frac{dy}{dx}+2y \right)=xy\frac{dy}{dx}\]?
  A)
  \[{{x}^{2}}=ky{{e}^{\frac{y}{a}}}\] done clear
  B)
  \[y{{x}^{2}}=ky{{e}^{\frac{y}{a}}}\] done clear
  C)
  \[{{y}^{2}}{{x}^{2}}=ky{{e}^{\frac{{{y}^{2}}}{a}}}\]    done clear
  D)
  None of the above done clear
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• question_answer24)
  A function \[y=f(x)\] satisfies the differential equation \[\frac{dy}{dx}-y=\cos x-\sin x\] with initial condition that y is bounded when \[x\to \infty \]. The area enclosed by \[y=f(x),y=cos\,\,x\] and the y-axis is
  A)
  \[\sqrt{2}-1\] done clear
  B)
  \[\sqrt{2}\] done clear
  C)
  1 done clear
  D)
  \[\frac{1}{\sqrt{2}}\] done clear
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• question_answer25)
  The differential equation of the family of circles with fixed radius 5 units and centre on the line \[y=2\] is
  A)
  \[(y-2)y{{'}^{2}}=25-{{(y-2)}^{2}}\] done clear
  B)
  \[{{(y-2)}^{2}}y{{'}^{2}}=25-{{(y-2)}^{2}}\] done clear
  C)
  \[{{(x-2)}^{2}}y{{'}^{2}}=25-{{(y-2)}^{2}}\] done clear
  D)
  \[(x-2)y{{'}^{2}}=25-{{(y-2)}^{2}}\] done clear
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• question_answer26)
  The general solution of \[(x+1)\frac{dy}{dx}+1=2{{e}^{-y}}\] is
  A)
  \[{{e}^{y}}(x+1)=x+C\] done clear
  B)
  \[{{e}^{-y}}=2x+C\] done clear
  C)
  \[{{e}^{y}}(x+1)=2x+C\] done clear
  D)
  \[{{e}^{y}}(x+1)=C\] done clear
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• question_answer27)
  Which of the following does not represent the orthogonal trajectory of the system of curves \[{{\left( \frac{dy}{dx} \right)}^{2}}=\frac{a}{x}\]
  A)
  \[9a{{(y+c)}^{2}}=4{{x}^{3}}\] done clear
  B)
  \[y+c=\frac{-2}{3\sqrt{a}}{{x}^{3/2}}\] done clear
  C)
  \[y+c=\frac{2}{3\sqrt{a}}{{x}^{3/2}}\] done clear
  D)
  All are orthogonal trajectories done clear
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• question_answer28)
  The differential equation\[(1+{{y}^{2}})xdx-(1+{{x}^{2}})ydy=0\] Represents a family of:
  A)
  Ellipses of constant eccentricity done clear
  B)
  Ellipses of variable eccentricity- done clear
  C)
  Hyperbolas of constant eccentricity done clear
  D)
  Hyperbolas of variable eccentricity done clear
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• question_answer29)
  The degree of differential equation satisfying the relation\[\sqrt{1+{{x}^{2}}}+\sqrt{1+{{y}^{2}}}=\lambda (x\sqrt{1+{{y}^{2}}}-y\sqrt{1+{{x}^{2}}})\] is
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
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• question_answer30)
  The particular solution of the differential equation \[{{\sin }^{-1}}\left( \frac{{{d}^{2}}y}{d{{x}^{2}}}-1 \right)=x\], where\[y=\frac{dy}{dx}=0\] when\[x=0\], is
  A)
  \[y={{x}^{2}}+x-\sin x\] done clear
  B)
  \[y=\frac{{{x}^{2}}}{2}+x-\sin x\] done clear
  C)
  \[y=\frac{{{x}^{2}}}{2}+\frac{x}{2}-\sin x\] done clear
  D)
  \[2y={{x}^{2}}+x-\sin x\] done clear
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• question_answer31)
  The solution of the differential equation\[\left\{ 1+x\sqrt{\left( {{x}^{2}}+{{y}^{2}} \right)} \right\}dx+\left\{ \sqrt{\left( {{x}^{2}}+{{y}^{2}} \right)}-1 \right\}ydy=0\] is
  A)
  \[{{x}^{2}}+\frac{{{y}^{2}}}{2}+\frac{1}{3}{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{3/2}}=C\] done clear
  B)
  \[x-\frac{{{y}^{2}}}{3}+\frac{1}{2}{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{1/2}}=C\] done clear
  C)
  \[x-\frac{{{y}^{2}}}{2}+\frac{1}{3}{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{3/2}}=C\] done clear
  D)
  None of these done clear
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• question_answer32)
  The gradient of the curve passing through (4, 0) is given by \[\frac{dy}{dx}-\frac{y}{x}+\frac{5x}{(x+2)(x-3)}=0\] if the point (5, a) lies on the curve, then the value of a is
  A)
  \[\frac{67}{12}\] done clear
  B)
  \[5\sin \frac{7}{12}\] done clear
  C)
  \[5\log \frac{7}{12}\] done clear
  D)
  None of these done clear
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• question_answer33)
  The solution of \[\frac{dy}{dx}=\frac{{{e}^{x}}({{\sin }^{2}}x+\sin 2x)}{y(2\,\,\log \,\,y+1)}\] is
  A)
  \[{{y}^{2}}(\log \,y)-{{e}^{x}}{{\sin }^{2}}x+c=0\] done clear
  B)
  \[{{y}^{2}}(\log \,y)-{{e}^{x}}{{\cos }^{2}}x+c=0\] done clear
  C)
  \[{{y}^{2}}(\log \,y)+{{e}^{x}}{{\cos }^{2}}x+c=0\] done clear
  D)
  None of these done clear
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• question_answer34)
  A differential equation associated with the primitive \[y=a+b{{e}^{5x}}+c{{e}^{-~7x}}\] is
  A)
  \[{{y}_{3}}+2{{y}_{2}}-{{y}_{1}}=0\] done clear
  B)
  \[{{y}_{3}}+2{{y}_{2}}-35{{y}_{1}}=0\] done clear
  C)
  \[4{{y}_{3}}+5{{y}_{2}}-20{{y}_{1}}=0\] done clear
  D)
  None of these done clear
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• question_answer35)
  The differential equations of all conies whose axes coincide with the co-ordinate axis
  A)
  \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}+y\frac{dy}{dx}=0\] done clear
  B)
  \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}+x\frac{dy}{dx}=0\] done clear
  C)
  \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}-y\frac{dy}{dx}=0\] done clear
  D)
  \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}-x{{\left( \frac{dy}{dx} \right)}^{2}}+y\frac{dy}{dx}=0\] done clear
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• question_answer36)
  What is the order of the differential equation\[\frac{dx}{dy}+\int{y\,dx={{x}^{3}}}\]?
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  Cannot be determined done clear
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• question_answer37)
  The order and degree of the differential equation of parabolas having vertex at the origin and focus at (a, 0) where a > 0, are respectively
  A)
  1, 1 done clear
  B)
  2, 1 done clear
  C)
  1, 2 done clear
  D)
  2, 2 done clear
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• question_answer38)
  If \[{{y}^{2}}=p(x)\] is a polynomial of degree 3, then what is \[2\frac{d}{dx}\left[ {{y}^{3}}\frac{{{d}^{2}}y}{d{{x}^{2}}} \right]\] equal to?
  A)
  p'(x)p"'(x) done clear
  B)
  p"(x)p'"(x) done clear
  C)
  p(x)p"'(x) done clear
  D)
  A constant done clear
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• question_answer39)
  The solution of the equation \[\frac{dy}{dx}=\sqrt{\frac{1-{{y}^{2}}}{1-{{x}^{2}}}}\] is
  A)
  \[{{\sin }^{-1}}y-{{\sin }^{-1}}x=c\] done clear
  B)
  \[{{\sin }^{-1}}y{{\sin }^{-1}}x=c\] done clear
  C)
  \[{{\sin }^{-1}}(xy)=2\] done clear
  D)
  None of these done clear
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• question_answer40)
  A continuously differentiable function \[\phi \,(x)\],\[x\in [0,\pi ]-\left\{ \frac{\pi }{2} \right\}\] satisfying \[y'=1+{{y}^{2}},y(0)=0=y(\pi )\] is
  A)
  \[\tan x\] done clear
  B)
  \[x(x-\pi )\] done clear
  C)
  \[(x-\pi )(1-{{e}^{x}})\] done clear
  D)
  \[{{\sec }^{2}}x\]   done clear
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• question_answer41)
  The solution of the differential equation\[x\sin x\frac{dy}{dx}+(x\cos x+\sin x)y=\sin x\]. When \[y(0)=0\] is
  A)
  \[xy\sin x=1-\cos x\] done clear
  B)
  \[xy\sin x+\cos x=0\] done clear
  C)
  \[x\sin x+y\cos x=0\] done clear
  D)
  \[x\sin x+y\cos x=1\] done clear
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• question_answer42)
  The solution of the equation\[x\int\limits_{0}^{x}{y(t)dt=(x+1)\int_{0}^{x}{ty(t)dt,x>0}}\] is
  A)
  \[y=\frac{c}{{{x}^{3}}}{{e}^{{{x}^{3}}}}\] done clear
  B)
  \[y=c{{x}^{3}}{{e}^{-{{x}^{3}}}}\] done clear
  C)
  \[\frac{c}{{{x}^{3}}}{{e}^{-x}}\] done clear
  D)
  None of these done clear
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• question_answer43)
  If \[y={{(x+\sqrt{1+{{x}^{2}}})}^{n}},\] then \[(1+{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}+x\frac{dy}{dx}\] is
  A)
  \[{{n}^{2}}y\] done clear
  B)
  \[-{{n}^{2}}y\] done clear
  C)
  \[-y\] done clear
  D)
  \[2{{x}^{2}}y\] done clear
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• question_answer44)
  The general solution of the differential equation \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=\cos \,\,nx\] is
  A)
  \[{{n}^{2}}y+\cos \,\,nx={{n}^{2}}(Cx+D)\] done clear
  B)
  \[{{n}^{2}}y-sin\,\,nx={{n}^{2}}(-Cx+D)\] done clear
  C)
  \[{{n}^{2}}y+\cos \,\,nx=\frac{Cx+D}{{{n}^{2}}}\] done clear
  D)
  None of these. [Where C and D are arbitrary constants] done clear
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• question_answer45)
  The solution of \[(y+x+5)dy=(y-x+1)dx\] is
  A)
  \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})+{{\tan }^{-1}}\frac{y+3}{y+2}+C\] done clear
  B)
  \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})+{{\tan }^{-1}}\frac{y-3}{y-2}=C\] done clear
  C)
  \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})+2{{\tan }^{-1}}\frac{y+3}{y+2}=C\] done clear
  D)
  \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})-2{{\tan }^{-1}}\frac{y+3}{y+2}=C\] done clear
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• question_answer46)
  The solution to of the differential equation\[(x+1)\frac{dy}{dx}-y={{e}^{3x}}{{(x+1)}^{2}}\] is
  A)
  \[y=(x+1){{e}^{3x}}+c\] done clear
  B)
  \[3y=(x+1)+{{e}^{3x}}+c\] done clear
  C)
  \[\frac{3y}{x+1}={{e}^{3x}}+c\] done clear
  D)
  \[y{{e}^{-3x}}=3(x+1)+c\] done clear
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• question_answer47)
  The solution of the differential equation\[\frac{dy}{dx}+\frac{2yx}{1+{{x}^{2}}}=\frac{1}{{{(1+{{x}^{2}})}^{2}}}\] is:
  A)
  \[y(1+{{x}^{2}})=c+{{\tan }^{-1}}x\] done clear
  B)
  \[\frac{y}{1+{{x}^{2}}}=c+{{\tan }^{-1}}x\] done clear
  C)
  \[y\log (1+{{x}^{2}})=c+{{\tan }^{-1}}x\] done clear
  D)
  \[y(1+{{x}^{2}})=c+{{\sin }^{-1}}x\] done clear
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• question_answer48)
  If \[y={{e}^{4x}}+2{{e}^{-x}}\] satisfies the relation \[\frac{{{d}^{3}}y}{d{{x}^{3}}}+A\frac{dy}{dx}+By=0,\] then values of A and B respectively are:
  A)
  -13, 14 done clear
  B)
  -13, -12 done clear
  C)
  -13, 12 done clear
  D)
  12, -13 done clear
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• question_answer49)
  Solution of the differential equation\[x=1+xy\frac{dy}{dx}+\frac{{{x}^{2}}{{y}^{2}}}{2!}{{\left( \frac{dy}{dx} \right)}^{2}}+\]\[\frac{{{x}^{3}}{{y}^{3}}}{3!}{{\left( \frac{dy}{dx} \right)}^{3}}+............\]
  A)
  \[y=ln\,(x)+c\] done clear
  B)
  \[y=\,{{(ln\,x)}^{2}}+c\] done clear
  C)
  \[y=\pm \,ln\,(x)+c\] done clear
  D)
  \[xy={{x}^{y}}+c\] done clear
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• question_answer50)
  What is the solution of the differential equation\[\sin \left( \frac{dy}{dx} \right)-a=0\]?
  A)
  \[y=x{{\sin }^{-1}}a+c\] done clear
  B)
  \[x=y{{\sin }^{-1}}a+c\] done clear
  C)
  \[y=x+x{{\sin }^{-1}}a+c\] done clear
  D)
  \[y=\,{{\sin }^{-1}}a+c\] where c is an arbitrary constant. done clear
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• question_answer51)
  What is the solution of the equation\[\ln \,\left( \frac{dy}{dx} \right)+x=0\]?
  A)
  \[y+{{e}^{x}}=c\] done clear
  B)
  \[y-{{e}^{-x}}=c\] done clear
  C)
  \[y+{{e}^{-x}}=c\] done clear
  D)
  \[y-{{e}^{x}}=c\] done clear
  View Solution play_arrow

• question_answer52)
  \[y=2\,cos\text{ }x+3\,sin\text{ }x\] satisfies which of the following differential equations?

  1. \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+y=0\]

  2. \[{{\left( \frac{dy}{dx} \right)}^{2}}+\frac{dy}{dx}=0\]

  Select the correct answer using the code given below.

  A)
  1 only done clear
  B)
  2 only done clear
  C)
  Both 1 and 2 done clear
  D)
  Neither 1 nor 2 done clear
  View Solution play_arrow

• question_answer53)
  If \[y=y(x)\] and \[\frac{2+\sin x}{1+y}\left( \frac{dy}{dx} \right)=-\cos \,x,y(0)=1,\]then \[y\left( \frac{\pi }{2} \right)\] equals
  A)
  1/3 done clear
  B)
  2/3 done clear
  C)
  -1/3 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer54)
  An integrating factor of the differential equation \[\sin x\frac{dy}{dx}+2y\cos x=1\] is
  A)
  \[{{\sin }^{2}}x\] done clear
  B)
  \[\frac{2}{\sin x}\] done clear
  C)
  \[\log \left| \sin \,\,x \right|\] done clear
  D)
  \[\frac{1}{{{\sin }^{2}}x}\] done clear
  View Solution play_arrow

• question_answer55)
  If for the differential equation \[y'=\frac{y}{x}+\phi \left( \frac{x}{y} \right),\] the general solution is \[y=\frac{x}{\log \left| Cx \right|},\] then \[\phi (x/y)\] is given by
  A)
  \[-{{x}^{2}}/{{y}^{2}}\] done clear
  B)
  \[-{{y}^{2}}/{{x}^{2}}\] done clear
  C)
  \[{{x}^{2}}/{{y}^{2}}\] done clear
  D)
  \[-{{y}^{2}}/{{x}^{2}}\] done clear
  View Solution play_arrow

• question_answer56)
  A curve is such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2). The equation of the curve is
  A)
  \[xy=1\] done clear
  B)
  \[xy=2\] done clear
  C)
  \[xy=3\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer57)
  What is the differential equation for\[{{y}^{2}}=4a(x-a)\]?
  A)
  \[yy'-2xyy'+{{y}^{2}}=0\] done clear
  B)
  \[yy'(yy'+2x)+{{y}^{2}}=0\] done clear
  C)
  \[yy'(yy'-2x)+{{y}^{2}}=0\] done clear
  D)
  \[yy'-2xyy'+y=0\] done clear
  View Solution play_arrow

• question_answer58)
  The degree and order respectively of the differential equation are \[\frac{dy}{dx}=\frac{1}{x+y+1}\].
  A)
  1, 1 done clear
  B)
  1, 2 done clear
  C)
  2, 1 done clear
  D)
  2, 2 done clear
  View Solution play_arrow

• question_answer59)
  What is the degree of the differential equation\[y=x\frac{dy}{dx}+{{\left( \frac{dy}{dx} \right)}^{-1}}\]?
  A)
  1 done clear
  B)
  2 done clear
  C)
  -1 done clear
  D)
  Degree does not exist. done clear
  View Solution play_arrow

• question_answer60)
  The general solution the differential equation\[\frac{dy}{dx}-\frac{\tan \,\,y}{1+x}={{(1+x\,\,e)}^{x}}\sec \,\,y\] is
  A)
  \[\sin (1+x)=y({{e}^{x}}+c)\] done clear
  B)
  \[y\sin (1+x)=c{{e}^{x}}\] done clear
  C)
  \[(1+x)\sin \,\,y={{e}^{x}}+c\] done clear
  D)
  \[\sin \,\,y=(1+x)({{e}^{x}}+c)\] done clear
  View Solution play_arrow

• question_answer61)
  The solution of the differential equation \[\frac{dy}{dx}+\frac{y}{x}\log \,\,y=\frac{y}{{{x}^{2}}}(\log \,\,{{y}^{2}})\] is
  A)
  \[y=\log ({{x}^{2}}+cx)\] done clear
  B)
  \[\log \,\,y=x\left( c{{x}^{2}}+\frac{1}{2} \right)\] done clear
  C)
  \[x=\log \,\,y\left( c{{x}^{2}}+\frac{1}{2} \right)\] done clear
  D)
  None of these. done clear
  View Solution play_arrow

• question_answer62)
  The differential equation\[\frac{{{d}^{2}}y}{d{{x}^{2}}}+x\frac{dy}{dx}+\sin y+{{x}^{2}}=0\] is of the following type
  A)
  Linear done clear
  B)
  Homogeneous done clear
  C)
  Order two done clear
  D)
  Degree two done clear
  View Solution play_arrow

• question_answer63)
  The function \[f(\theta )=\frac{d}{d\theta }\int\limits_{0}^{\theta }{\frac{dx}{1-\cos \theta \,\,\cos x}}\] satisfies the differential equation
  A)
  \[\frac{df}{d\theta }+2f(\theta )cot\theta =0\] done clear
  B)
  \[\frac{df}{d\theta }-2f(\theta )cot\theta =0\] done clear
  C)
  \[\frac{df}{d\theta }+2f(\theta )=0\] done clear
  D)
  \[\frac{df}{d\theta }-2f(\theta )=0\] done clear
  View Solution play_arrow

• question_answer64)
  The equation of the curve passing through the point \[\left( 0,\frac{\pi }{4} \right)\] whose differential equation is\[sin\text{ }x\text{ }cos\text{ }y\text{ }dx+cos\text{ }x\text{ }sin\text{ }y\text{ }dy=0\], is
  A)
  \[sec\,\,x\,\,sec\,\,y=\sqrt{2}\] done clear
  B)
  \[cos\,\,x\,\,cos\,\,y=\sqrt{2}\] done clear
  C)
  \[\sec \,\,x=\sqrt{2}\,\,\cos \,\,y\] done clear
  D)
  \[cos\,\,y=\sqrt{2}\,\,\sec \,\,y\] done clear
  View Solution play_arrow

• question_answer65)
  Solution of the differential equation\[\frac{dx}{dy}-\frac{x\,\,\log \,\,x}{1+\log \,\,x}=\frac{{{e}^{y}}}{1+\log \,\,x'}\] if \[y(1)=0\], is
  A)
  \[{{x}^{x}}={{e}^{y{{e}^{y}}}}\] done clear
  B)
  \[{{e}^{y}}={{x}^{{{e}^{y}}}}\] done clear
  C)
  \[{{x}^{x}}=y{{e}^{^{y}}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer66)
  What is the degree of the differential equation\[k\frac{{{d}^{2}}y}{d{{x}^{2}}}={{\left[ 1+{{\left( \frac{dy}{dx} \right)}^{3}} \right]}^{3/2}}\], where k is a constant?
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer67)
  What is the degree of the differential equation\[{{\left( \frac{{{d}^{3}}y}{d{{x}^{3}}} \right)}^{2/3}}+4-3\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)+5\left( \frac{dy}{dx} \right)=0\]?
  A)
  3 done clear
  B)
  2 done clear
  C)
  2/3 done clear
  D)
  Not defined done clear
  View Solution play_arrow

• question_answer68)
  The marginal cost of manufacturing a certain item is given by\[c'(x)=\frac{dc}{dx}=2+0.15x\]. The total cost function c (x), is
  A)
  \[0.075{{x}^{2}}+2x+100\] done clear
  B)
  \[0.15{{x}^{2}}+3x+30\] done clear
  C)
  \[{{x}^{2}}+100.075x+100\] done clear
  D)
  None of these It is given that c (0) = 100 done clear
  View Solution play_arrow

• question_answer69)
  The population of a country doubles in 40 years. Assuming that the rate of increase is proportional to the number of inhabitants, the number of years in which it would treble itself is
  A)
  80 years done clear
  B)
  \[80\frac{\log 2}{\log 3}years\] done clear
  C)
  \[40\frac{\log 3}{\log 2}years\] done clear
  D)
  \[40\log 2\log 3\,years\] done clear
  View Solution play_arrow

• question_answer70)
  The differential equation of the curve \[\frac{x}{c-1}+\frac{y}{c+1}=1\] is given by
  A)
  \[\left( \frac{dy}{dx}-1 \right)\left( y+x\frac{dy}{dx} \right)=2\frac{dy}{dx}\] done clear
  B)
  \[\left( \frac{dy}{dx}+1 \right)\left( y-x\frac{dy}{dx} \right)=\frac{dy}{dx}\] done clear
  C)
  \[\left( \frac{dy}{dx}+1 \right)\left( y-x\frac{dy}{dx} \right)=2\frac{dy}{dx}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer71)
  The equation of the curve satisfying \[xdy-ydx=\sqrt{{{x}^{2}}-{{y}^{2}}}\] and \[y(1)=0\] is:
  A)
  \[y={{x}^{2}}\log (\sin \,x)\] done clear
  B)
  \[y=x\sin (log\,x)\] done clear
  C)
  \[{{y}^{2}}=x{{(x-1)}^{2}}\] done clear
  D)
  \[y=2{{x}^{2}}(x-1)\] done clear
  View Solution play_arrow

• question_answer72)
  If \[y+x\frac{dy}{dx}=x\frac{\phi (xy)}{\phi '(xy)}\] then \[\phi (xy)\] is equation to
  A)
  \[k{{e}^{{{x}^{2}}/2}}\] done clear
  B)
  \[k{{e}^{{{y}^{2}}/2}}\] done clear
  C)
  \[k{{e}^{xy/2}}\] done clear
  D)
  \[k{{e}^{xy}}\] done clear
  View Solution play_arrow

• question_answer73)
  What are the order and degree respectively of the differential equation\[y=x\frac{dy}{dx}+\frac{dx}{dy}\]?
  A)
  1, 1 done clear
  B)
  1, 2 done clear
  C)
  2, 1 done clear
  D)
  2, 2 done clear
  View Solution play_arrow

• question_answer74)
  The solution to the differential equation\[\frac{dy}{dx}=\frac{yf'(x)-{{y}^{2}}}{f(x)}\]Where \[f(x)\] is a given function is
  A)
  \[f(x)=y(x+c)\] done clear
  B)
  \[f(x)=cxy\] done clear
  C)
  \[f(x)=c(x+y)\] done clear
  D)
  \[yf(x)=cx\] done clear
  View Solution play_arrow

• question_answer75)
  If \[\phi (x)\] is a differentiable function, then the solution of the differential equation\[dy+\{y\phi '(x)-\phi (x)\phi '(x)\}dx=0\] is
  A)
  \[y=\{\phi (x)-1\}+c{{e}^{-\phi (x)}}\] done clear
  B)
  \[y\phi (x)={{\{\phi (x)\}}^{2}}+c\] done clear
  C)
  \[y{{e}^{\phi (x)}}=\phi (x){{e}^{\phi (x)}}+c\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer76)
  What is the solution of the differential equation\[\frac{dx}{dy}+\frac{x}{y}-{{y}^{2}}=0\]?
  A)
  \[xy={{x}^{4}}+c\] done clear
  B)
  \[xy={{y}^{4}}+c\] done clear
  C)
  \[4xy={{y}^{4}}+c\] done clear
  D)
  \[3xy={{y}^{3}}+c\] where c is an arbitrary constant. done clear
  View Solution play_arrow

• question_answer77)
  What is the solution of the differential equation\[(x+y)(dx-dy)=dx+dy\]?
  A)
  \[x+y+ln\,\,(x+y)=c\] done clear
  B)
  \[x-y+ln\,\,(x+y)=c\] done clear
  C)
  \[y-x+ln\,\,(x+y)=c\] done clear
  D)
  \[y-x-ln\,\,(x-y)=c\] done clear
  View Solution play_arrow

• question_answer78)
  Consider a differential equation of order m and degree n. Which one of the following pairs is not feasible?
  A)
  (3, 2) done clear
  B)
  (2, 3/2) done clear
  C)
  (2, 4) done clear
  D)
  (2, 2) done clear
  View Solution play_arrow

• question_answer79)
  Under which one of the following conditions does the solution of \[\frac{dy}{dx}=\frac{ax+b}{cy+d}\] represent a parabola?
  A)
  \[a=0,\text{ }c=0\] done clear
  B)
  \[a=1,\text{ }b=2,\text{ }c\ne 0\] done clear
  C)
  \[a=0,c\ne 0,b\ne 0\] done clear
  D)
  \[a=1,c=1\] done clear
  View Solution play_arrow

• question_answer80)
  Consider the following statements in respect of the differential equation\[\frac{{{d}^{2}}y}{d{{x}^{2}}}+\cos \left( \frac{dy}{dx} \right)=0\]

  1. The degree of the differential equation is not defined.

  2. The order of the differential equation is 2.

  Which of the above statements is/are correct?

  A)
  1 only done clear
  B)
  2 only done clear
  C)
  Both 1 and 2 done clear
  D)
  Neither 1 nor 2 done clear
  View Solution play_arrow