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question_answer1)
The degree of the differential equation \[3\frac{{{d}^{2}}y}{d{{x}^{2}}}={{\left\{ 1+{{\left( \frac{dy}{dx} \right)}^{2}} \right\}}^{3/2}}\] is [MP PET 1994, 95]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
6 done
clear
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question_answer2)
The differential equation representing the family of curves \[{{y}^{2}}=2c(x+\sqrt{c}),\]where c is a positive parameter, is of [IIT 1999; AIEEE 2005; MP PET 2002]
A)
Order 1 done
clear
B)
Order 2 done
clear
C)
Degree 3 done
clear
D)
Degree 4 done
clear
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question_answer3)
The order of the differential equation whose general solution is given by \[y={{C}_{1}}{{e}^{2x+{{C}_{2}}}}+\] \[{{C}_{3}}{{e}^{x}}+{{C}_{4}}\sin (x+{{C}_{5}})\] is [AMU 2000]
A)
5 done
clear
B)
4 done
clear
C)
3 done
clear
D)
2 done
clear
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question_answer4)
The order and degree of the differential equation \[{{\left( 1+3\frac{dy}{dx} \right)}^{\frac{2}{3}}}=4\frac{{{d}^{3}}y}{d{{x}^{3}}}\] are [AIEEE 2002]
A)
\[1,\,\frac{2}{3}\] done
clear
B)
3, 1 done
clear
C)
3, 3 done
clear
D)
1, 2 done
clear
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question_answer5)
The degree of the differential equation \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+3{{\left[ \frac{dy}{dx} \right]}^{2}}={{x}^{2}}\log \left[ \frac{{{d}^{2}}y}{d{{x}^{2}}} \right]\] is [Pb. CET 2004]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
None of these done
clear
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question_answer6)
The differential equation of the family of curves \[y=A{{e}^{3x}}+B{{e}^{5x}},\]where A and B are arbitrary constants, is [MNR 1988]
A)
\[\frac{{{d}^{2}}y}{d{{x}^{2}}}+8\frac{dy}{dx}+15y=0\] done
clear
B)
\[\frac{{{d}^{2}}y}{d{{x}^{2}}}-8\frac{dy}{dx}+15y=0\] done
clear
C)
\[\frac{{{d}^{2}}y}{d{{x}^{2}}}-\frac{dy}{dx}+y=0\] done
clear
D)
None of these done
clear
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question_answer7)
The differential equation of the family of parabolas with focus at the origin and the x-axis as axis is [EAMCET 2003]
A)
\[y\,{{\left( \frac{dy}{dx} \right)}^{2}}+4x\frac{dy}{dx}=4y\] done
clear
B)
\[-y\,{{\left( \frac{dy}{dx} \right)}^{2}}=2x\frac{dy}{dx}-y\] done
clear
C)
\[y\,{{\left( \frac{dy}{dx} \right)}^{2}}+y=2xy\frac{dy}{dx}\] done
clear
D)
\[y\,{{\left( \frac{dy}{dx} \right)}^{2}}+2xy\frac{dy}{dx}+y=0\] done
clear
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question_answer8)
The differential equation of the family of curves for which the length of the normal is equal to a constant k, is given by [Pb. CET 2004]
A)
\[{{y}^{2}}\frac{dy}{dx}={{k}^{2}}-{{y}^{2}}\] done
clear
B)
\[{{\left( y\frac{dy}{dx} \right)}^{2}}={{k}^{2}}-{{y}^{2}}\] done
clear
C)
\[y{{\left( \frac{dy}{dx} \right)}^{2}}={{k}^{2}}+{{y}^{2}}\] done
clear
D)
\[{{\left( y\frac{dy}{dx} \right)}^{2}}={{k}^{2}}+{{y}^{2}}\] done
clear
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question_answer9)
The solution of the differential equation \[y-x\frac{dy}{dx}=a\left( {{y}^{2}}+\frac{dy}{dx} \right)\] is [AISSE 1989, 90]
A)
\[y=c(x+a)(1+ay)\] done
clear
B)
\[y=c(x+a)(1-ay)\] done
clear
C)
\[y=c(x-a)(1+ay)\] done
clear
D)
None of these done
clear
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question_answer10)
The solution of the differential equation \[\sqrt{a+x}\frac{dy}{dx}+xy=0\]is [MP PET 1998]
A)
\[y=A{{e}^{2/3(2a-x)\sqrt{x+a}}}\] done
clear
B)
\[y=A{{e}^{-2/3(a-x)\sqrt{x+a}}}\] done
clear
C)
\[y=A{{e}^{2/3(2a+x)\sqrt{x+a}}}\] done
clear
D)
\[y=A{{e}^{-2/3(2a-x)\sqrt{x+a}}}\] (Where A is an arbitrary constant.) done
clear
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question_answer11)
The solution of the differential equation \[x\frac{dy}{dx}=y(\log y-\log x+1)\]is [IIT 1986; AIEEE 2005]
A)
\[y=x{{e}^{cx}}\] done
clear
B)
\[y+x{{e}^{cx}}=0\] done
clear
C)
\[y+{{e}^{x}}=0\] done
clear
D)
None of these done
clear
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question_answer12)
The solution of the differential equation \[\frac{dy}{dx}=\frac{y}{x}+\frac{\varphi \,\left( \frac{y}{x} \right)}{{\varphi }'\,\left( \frac{y}{x} \right)}\] is [DCE 2002]
A)
\[\varphi \,\left( \frac{y}{x} \right)=kx\] done
clear
B)
\[x\,\varphi \,\left( \frac{y}{x} \right)=k\] done
clear
C)
\[\varphi \,\left( \frac{y}{x} \right)=ky\] done
clear
D)
\[y\,\varphi \left( \frac{y}{x} \right)=k\] done
clear
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question_answer13)
The general solution of \[{{y}^{2}}\,dx+({{x}^{2}}-xy+{{y}^{2}})\,\,dy=0\] is [EAMCET 2003]
A)
\[{{\tan }^{-1}}\left( \frac{x}{y} \right)+\log y+c=0\] done
clear
B)
\[2{{\tan }^{-1}}\left( \frac{x}{y} \right)+\log x+c=0\] done
clear
C)
\[\log (y+\sqrt{{{x}^{2}}+{{y}^{2}}})+\log y+c=0\] done
clear
D)
\[{{\sinh }^{-1}}\left( \frac{x}{y} \right)+\log y+c=0\] done
clear
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question_answer14)
The solution of the equation \[\frac{dy}{dx}=\frac{1}{x+y+1}\] is
A)
\[x=c{{e}^{y}}-y-2\] done
clear
B)
\[y=x+c{{e}^{y}}-2\] done
clear
C)
\[x+c{{e}^{y}}-y-2=0\] done
clear
D)
None of these done
clear
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question_answer15)
The solution of the given differential equation \[\frac{dy}{dx}+2xy=y\] is [Roorkee 1995]
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_answer16)
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 [MP PET 1999]
A)
\[\frac{2{{x}^{2}}-a{{x}^{3}}}{x(1-{{x}^{2}})}\] done
clear
B)
\[(2{{x}^{2}}-1)\] done
clear
C)
\[\frac{2{{x}^{2}}-1}{a{{x}^{3}}}\] done
clear
D)
\[\frac{(2{{x}^{2}}-1)}{x(1-{{x}^{2}})}\] done
clear
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question_answer17)
A solution of the differential equation \[{{\left( \frac{dy}{dx} \right)}^{2}}-x\frac{dy}{dx}+y=0\] is [IIT 1999; Karnataka CET 2002]
A)
\[y=2\] done
clear
B)
\[y=2x\] done
clear
C)
\[y=2x-4\] done
clear
D)
\[y=2{{x}^{2}}-4\] done
clear
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question_answer18)
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 the curve is [Kurukshetra CEE 2002]
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 these done
clear
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question_answer19)
The equation of family of curves for which the length of the normal is equal to the radius vector is
A)
\[{{y}^{2}}\pm {{x}^{2}}=k\] done
clear
B)
\[y\pm x=k\] done
clear
C)
\[{{y}^{2}}=kx\] done
clear
D)
None of these done
clear
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question_answer20)
A continuously differentiable function \[\varphi (x)\,\text{in}\,(0,\,\pi )\] satisfying \[{y}'=1+{{y}^{2}},\,\,y(0)=0=y(\pi )\] is [MP PET 2000]
A)
\[\tan x\] done
clear
B)
\[x(x-\pi )\] done
clear
C)
\[(x-\pi )\] \[(1-{{e}^{x}})\] done
clear
D)
Not possible done
clear
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question_answer21)
The rate of increase of bacteria in a certain culture is proportional to the number present. If it double in 5 hours then in 25 hours, its number would be [Pb. CET 2004]
A)
8 times the original done
clear
B)
16 times the original done
clear
C)
32 times the original done
clear
D)
64 times the original done
clear
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question_answer22)
The solution of \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=\cos x-\sin x\]is
A)
\[y=-\cos x+\sin x+{{c}_{1}}x+{{c}_{2}}\] done
clear
B)
\[y=-\cos x-\sin x+{{c}_{1}}x+{{c}_{2}}\] done
clear
C)
\[y=\cos x-\sin x+{{c}_{1}}{{x}^{2}}+{{c}_{2}}x\] done
clear
D)
\[y=\cos x+\sin x+{{c}_{1}}{{x}^{2}}+{{c}_{2}}x\] done
clear
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question_answer23)
The solution of the differential equation \[{{x}^{4}}\frac{dy}{dx}+{{x}^{3}}y+\text{cosec}\,(xy)=0\] is equal to [Pb. CET 2004]
A)
\[2\cos \,(xy)+{{x}^{-2}}=c\] done
clear
B)
\[2\cos \,(xy)+{{y}^{-2}}=c\] done
clear
C)
\[2\sin \,(xy)+{{x}^{-2}}=c\] done
clear
D)
\[2\sin \,(xy)+{{y}^{-2}}=c\] done
clear
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question_answer24)
The solution of the equation \[\frac{{{x}^{2}}{{d}^{2}}y}{d{{x}^{2}}}=\ln x,\] when \[x=1\], \[y=0\] and \[\frac{dy}{dx}=-1\] is [Orissa JEE 2003]
A)
\[\frac{1}{2}{{(\ln x)}^{2}}+\ln x\] done
clear
B)
\[\frac{1}{2}{{(\ln x)}^{2}}-\ln x\] done
clear
C)
\[-\frac{1}{2}{{(\ln x)}^{2}}+\ln x\] done
clear
D)
\[-\frac{1}{2}{{(\ln x)}^{2}}-\ln x\] done
clear
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question_answer25)
If \[y\cos x+x\cos y=\pi \], then \[{{y}'}'(0)\] is [IIT Screening 2005]
A)
1 done
clear
B)
\[\pi \] done
clear
C)
0 done
clear
D)
\[-\pi \] done
clear
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