-
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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