JEE Main & Advanced Mathematics Differential Equations Question Bank

done Variable Separable type differential equations Total Questions - 81

• question_answer1)
  The solution of the differential equation \[3{{e}^{x}}\tan ydx+(1-{{e}^{x}}){{\sec }^{2}}ydy=0\] is [MP PET 1993; AISSE 1985]
  A)
                  \[\tan y=c{{(1-{{e}^{x}})}^{3}}\] done clear
  B)
                  \[{{(1-{{e}^{x}})}^{3}}\tan y=c\] done clear
  C)
                  \[\tan y=c(1-{{e}^{x}})\]   done clear
  D)
                  \[(1-{{e}^{x}})\tan y=c\] done clear
  View Solution play_arrow

• question_answer2)
  The solution of the differential equation \[\frac{dy}{dx}=\frac{1+{{y}^{2}}}{1+{{x}^{2}}}\] is [SCRA 1986]
  A)
                  \[1+xy+c(y+x)=0\]             done clear
  B)
                  \[x+y=c(1-xy)\] done clear
  C)
                  \[y-x=c(1+xy)\] done clear
  D)
                  \[1+xy=c(x+y)\] done clear
  View Solution play_arrow

• question_answer3)
  The solution of the differential equation \[x\cos ydy=(x{{e}^{x}}\log x+{{e}^{x}})dx\] is  [DSSE 1988]
  A)
                  \[\sin y=\frac{1}{x}{{e}^{x}}+c\]   done clear
  B)
                  \[\sin y+{{e}^{x}}\log x+c=0\] done clear
  C)
                  \[\sin y={{e}^{x}}\log x+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer4)
  The solution of the equation \[\frac{dy}{dx}={{e}^{x-y}}+{{x}^{2}}{{e}^{-y}}\] is       [MP PET 2004]
  A)
                  \[{{e}^{y}}={{e}^{x}}+\frac{{{x}^{3}}}{3}+c\]  done clear
  B)
                  \[{{e}^{y}}={{e}^{x}}+2x+c\] done clear
  C)
                  \[{{e}^{y}}={{e}^{x}}+{{x}^{3}}+c\]   done clear
  D)
                  \[y={{e}^{x}}+c\] done clear
  View Solution play_arrow

• question_answer5)
  The solution of the differential equation \[\frac{dy}{dx}+\frac{1+{{x}^{2}}}{x}=0\] is 
  A)
                  \[y=-\frac{1}{2}{{\tan }^{-1}}x+c\]               done clear
  B)
                  \[y+\log x+\frac{{{x}^{2}}}{2}+c=0\] done clear
  C)
                  \[y=\frac{1}{2}{{\tan }^{-1}}x+c\] done clear
  D)
                  \[y-\log x-\frac{{{x}^{2}}}{2}=c\] done clear
  View Solution play_arrow

• question_answer6)
  The solution of the differential equation \[\frac{dy}{dx}=\sec x(\sec x+\tan x)\]is
  A)
                  \[y=\sec x+\tan x+c\]           done clear
  B)
                  \[y=\sec x+\cot x+c\] done clear
  C)
                  \[y=\sec x-\tan x+c\]             done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer7)
  The solution of the differential equation \[(1+{{x}^{2}})\frac{dy}{dx}=x\] is
  A)
                  \[y={{\tan }^{-1}}x+c\]    done clear
  B)
                  \[y=-{{\tan }^{-1}}x+c\] done clear
  C)
                  \[y=\frac{1}{2}{{\log }_{e}}(1+{{x}^{2}})+c\]         done clear
  D)
                  \[y=-\frac{1}{2}{{\log }_{e}}(1+{{x}^{2}})+c\] done clear
  View Solution play_arrow

• question_answer8)
  The solution of the differential equation \[\frac{dy}{dx}={{e}^{x}}+\cos x+x+\tan x\]is
  A)
                  \[y={{e}^{x}}+\sin x+\frac{{{x}^{2}}}{2}+\log \cos x+c\] done clear
  B)
                  \[y={{e}^{x}}+\sin x+\frac{{{x}^{2}}}{2}+\log \sec x+c\] done clear
  C)
                  \[y={{e}^{x}}-\sin x+\frac{{{x}^{2}}}{2}+\log \cos x+c\] done clear
  D)
                  \[y={{e}^{x}}-\sin x+\frac{{{x}^{2}}}{2}+\log \sec x+c\] done clear
  View Solution play_arrow

• question_answer9)
  The solution of differential equation \[\frac{dy}{dx}+{{\sin }^{2}}y=0\] is [MP PET 1994]
  A)
                  \[y+2\cos y=c\]     done clear
  B)
                  \[y-2\sin y=c\] done clear
  C)
                  \[x=\cot y+c\]        done clear
  D)
                  \[y=\cot x+c\] done clear
  View Solution play_arrow

• question_answer10)
  The solution of the differential equation \[(\sin x+\cos x)dy+(\cos x-\sin x)dx=0\]is
  A)
                  \[{{e}^{x}}(\sin x+\cos x)+c=0\]    done clear
  B)
                  \[{{e}^{y}}(\sin x+\cos x)=c\] done clear
  C)
                  \[{{e}^{y}}(\cos x-\sin x)=c\]            done clear
  D)
                  \[{{e}^{x}}(\sin x-\cos x)=c\] done clear
  View Solution play_arrow

• question_answer11)
  The solution of the differential equation \[\frac{dy}{dx}=(1+x)(1+{{y}^{2}})\] is
  A)
                  \[y=\tan ({{x}^{2}}+x+c)\]             done clear
  B)
                  \[y=\tan (2{{x}^{2}}+x+c)\] done clear
  C)
                  \[y=\tan ({{x}^{2}}-x+c)\]               done clear
  D)
                  \[y=\tan \left( \frac{{{x}^{2}}}{2}+x+c \right)\] done clear
  View Solution play_arrow

• question_answer12)
  The solution of \[\frac{dy}{dx}={{e}^{x}}(\sin x+\cos x)\]is
  A)
                  \[y={{e}^{x}}(\sin x-\cos x)+c\]      done clear
  B)
                  \[y={{e}^{x}}(\cos x-\sin x)+c\] done clear
  C)
                  \[y={{e}^{x}}\sin x+c\]    done clear
  D)
                  \[y={{e}^{x}}\cos x+c\] done clear
  View Solution play_arrow

• question_answer13)
  The general solution of \[{{x}^{2}}\frac{dy}{dx}=2\] is             [AISSE 1984]
  A)
                  \[y=c+\frac{2}{x}\]              done clear
  B)
                  \[y=c-\frac{2}{x}\] done clear
  C)
                  \[y=2cx\]                done clear
  D)
                  \[y=c-\frac{3}{{{x}^{2}}}\] done clear
  View Solution play_arrow

• question_answer14)
  The solution of \[\frac{dy}{dx}=x\log x\]is       [MP PET 2003]
  A)
                  \[y={{x}^{2}}\log x-\frac{{{x}^{2}}}{2}+c\]            done clear
  B)
                  \[y=\frac{{{x}^{2}}}{2}\log x-{{x}^{2}}+c\] done clear
  C)
                  \[y=\frac{1}{2}{{x}^{2}}+\frac{1}{2}{{x}^{2}}\log x+c\]    done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer15)
  The solution of the differential equation \[\frac{dy}{dx}=1+x+y+xy\] is    [AISSE 1985; AI CBSE 1990; MP PET 2003]
  A)
                  \[\log (1+y)=x+\frac{{{x}^{2}}}{2}+c\]      done clear
  B)
                  \[{{(1+y)}^{2}}=x+\frac{{{x}^{2}}}{2}+c\] done clear
  C)
                  \[\log (1+y)=\log (1+x)+c\]                done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer16)
  The solution of the differential equation \[({{x}^{2}}-y{{x}^{2}})\frac{dy}{dx}+{{y}^{2}}+x{{y}^{2}}=0\] is      [Pb. CET 2003]
  A)
                  \[\log \left( \frac{x}{y} \right)=\frac{1}{x}+\frac{1}{y}+c\]           done clear
  B)
                  \[\log \left( \frac{y}{x} \right)=\frac{1}{x}+\frac{1}{y}+c\] done clear
  C)
                  \[\log \left( xy \right)=\frac{1}{x}+\frac{1}{y}+c\]          done clear
  D)
                  \[\log \left( xy \right)+\frac{1}{x}+\frac{1}{y}=c\] done clear
  View Solution play_arrow

• question_answer17)
  The solution of the differential equation\[x\sec y\frac{dy}{dx}=1\]is
  A)
                  \[x\sec y\tan y=c\]  done clear
  B)
                  \[cx=\sec y+\tan y\] done clear
  C)
                  \[cy=\sec x\tan x\]  done clear
  D)
                  \[cy=\sec x+\tan x\] done clear
  View Solution play_arrow

• question_answer18)
  The solution of differential equation \[x\frac{dy}{dx}+y={{y}^{2}}\] is
  A)
                  \[y=1+cxy\]           done clear
  B)
                  \[y=\log \{cxy\}\] done clear
  C)
                  \[y+1=cxy\]           done clear
  D)
                  \[y=c+xy\] done clear
  View Solution play_arrow

• question_answer19)
  If \[\frac{dy}{dx}+\frac{1}{\sqrt{1-{{x}^{2}}}}=0\], then         [MNR 1983]
  A)
                  \[y+{{\sin }^{-1}}x=c\]    done clear
  B)
                  \[{{y}^{2}}+2{{\sin }^{-1}}x+c=0\] done clear
  C)
                  \[x+{{\sin }^{-1}}y=0\]    done clear
  D)
                  \[{{x}^{2}}+2{{\sin }^{-1}}y=1\] done clear
  View Solution play_arrow

• question_answer20)
  If \[\frac{dy}{dx}=\frac{xy+y}{xy+x}\], then the solution of the differential equation is           [SCRA 1980]
  A)
                  \[y=x{{e}^{x}}+c\]           done clear
  B)
                  \[y={{e}^{x}}+c\] done clear
  C)
                  \[y=Ax{{e}^{x-y}}\]          done clear
  D)
                  \[y=x+A\] done clear
  View Solution play_arrow

• question_answer21)
  The general solution of the equation \[({{e}^{y}}+1)\cos xdx+{{e}^{y}}\sin xdy=0\] is    [SCRA 1986]
  A)
                  \[({{e}^{y}}+1)\cos x=c\] done clear
  B)
                  \[({{e}^{y}}-1)\sin x=c\] done clear
  C)
                  \[({{e}^{y}}+1)\sin x=c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer22)
  The solution of the differential equation\[{{x}^{2}}dy=-2xydx\] is [SCRA 1990]
  A)
                  \[x{{y}^{2}}=c\]                done clear
  B)
                  \[{{x}^{2}}{{y}^{2}}=c\] done clear
  C)
                  \[{{x}^{2}}y=c\]                done clear
  D)
                  \[xy=c\] done clear
  View Solution play_arrow

• question_answer23)
  The solution of the differential equation \[\frac{dy}{dx}=(a{{e}^{bx}}+c\cos mx)\] is
  A)
                  \[y=\frac{a{{e}^{x}}}{b}+\frac{c}{m}\sin mx+k\]      done clear
  B)
                  \[y=a{{e}^{x}}+c\sin mx+k\] done clear
  C)
                  \[y=\frac{a{{e}^{bx}}}{b}+\frac{c}{m}\sin mx+k\]   done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer24)
  The solution of the differential equation \[(1+\cos x)dy=(1-\cos x)dx\]is
  A)
                  \[y=2\tan \frac{x}{2}-x+c\] done clear
  B)
                  \[y=2\tan x+x+c\] done clear
  C)
                  \[y=2\tan \frac{x}{2}+x+c\]               done clear
  D)
                  \[y=x-2\tan \frac{x}{2}+c\] done clear
  View Solution play_arrow

• question_answer25)
  The solution of the differential equation \[\frac{dy}{dx}=\frac{(1+x)y}{(y-1)x}\]is  [AISSE 1986; AI CBSE 1982; MP PET 2004]
  A)
                  \[\log xy+x+y=c\] done clear
  B)
                  \[\log \left( \frac{x}{y} \right)+x-y=c\] done clear
  C)
                  \[\log xy+x-y=c\]  done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer26)
  The solution of the equation \[{{\sin }^{-1}}\left( \frac{dy}{dx} \right)=x+y\]is
  A)
                  \[\tan (x+y)+\sec (x+y)=x+c\]           done clear
  B)
                  \[\tan (x+y)-\sec (x+y)=x+c\] done clear
  C)
                  \[\tan (x+y)+\sec (x+y)+x+c=0\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer27)
  The solution of the differential equation \[\frac{dy}{dx}={{x}^{2}}+\sin 3x\] is [DSSE 1981]
  A)
                  \[y=\frac{{{x}^{3}}}{3}+\frac{\cos 3x}{3}+c\]           done clear
  B)
                  \[y=\frac{{{x}^{3}}}{3}-\frac{\cos 3x}{3}+c\] done clear
  C)
                  \[y=\frac{{{x}^{3}}}{3}+\sin 3x+c\]             done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer28)
  The solution of the equation \[(1+{{x}^{2}})\frac{dy}{dx}=1\]is
  A)
                  \[y=\log (1+{{x}^{2}})+c\]             done clear
  B)
                  \[y+\log (1+{{x}^{2}})+c=0\] done clear
  C)
                  \[y-\log (1+x)=c\] done clear
  D)
                  \[y={{\tan }^{-1}}x+c\] done clear
  View Solution play_arrow

• question_answer29)
  The solution of the equation \[\frac{dy}{dx}=y({{e}^{x}}+1)\]is [AISSE 1986; AI CBSE 1984]
  A)
                  \[y+{{e}^{({{e}^{x}}+x+c)}}=0\]             done clear
  B)
                  \[\log y={{e}^{x}}+x+c\] done clear
  C)
                  \[\log y+{{e}^{x}}=x+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer30)
  The solution of the equation \[\frac{dy}{dx}+\sqrt{\frac{1-{{y}^{2}}}{1-{{x}^{2}}}}=0\] is [Orissa JEE 2003]
  A)
                  \[x\sqrt{1-{{y}^{2}}}-y\sqrt{1-{{x}^{2}}}=c\]          done clear
  B)
                  \[x\sqrt{1-{{y}^{2}}}+y\sqrt{1-{{x}^{2}}}=c\] done clear
  C)
                  \[x\sqrt{1+{{y}^{2}}}+y\sqrt{1+{{x}^{2}}}=c\]    done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer31)
  The solution of the differential equation \[\frac{dy}{dx}+\frac{1+\cos 2y}{1-\cos 2x}=0\]      [AISSE 1982; Karnataka CET 2004]
  A)
                  \[\tan y+\cot x=c\] done clear
  B)
                  \[\tan y\cot x=c\] done clear
  C)
                  \[\tan y-\cot x=c\]   done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer32)
  The solution of the differential equation \[(1+{{x}^{2}})\frac{dy}{dx}=x(1+{{y}^{2}})\]is              [AISSE 1983]
  A)
                  \[2{{\tan }^{-1}}y=\log (1+{{x}^{2}})+c\] done clear
  B)
                  \[{{\tan }^{-1}}y=\log (1+{{x}^{2}})+c\] done clear
  C)
                  \[2{{\tan }^{-1}}y+\log (1+{{x}^{2}})+c=0\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer33)
  Solution of the equation \[({{e}^{x}}+1)ydy=(y+1){{e}^{x}}dx\] is [AISSE 1988]
  A)
                  \[c(y+1)({{e}^{x}}+1)+{{e}^{y}}=0\]      done clear
  B)
                  \[c(y+1)({{e}^{x}}-1)+{{e}^{y}}=0\] done clear
  C)
                  \[c(y+1)({{e}^{x}}-1)-{{e}^{y}}=0\]          done clear
  D)
                  \[c(y+1)({{e}^{x}}+1)={{e}^{y}}\] done clear
  View Solution play_arrow

• question_answer34)
  Solution of the equation \[(1-{{x}^{2}})dy+xydx=x{{y}^{2}}dx\] is [DSSE 1989]
  A)
                  \[{{(y-1)}^{2}}(1-{{x}^{2}})=0\] done clear
  B)
                  \[{{(y-1)}^{2}}{{(1-x)}^{2}}={{c}^{2}}{{y}^{2}}\] done clear
  C)
                  \[{{(y-1)}^{2}}(1+{{x}^{2}})={{c}^{2}}{{y}^{2}}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer35)
  The solution of the equation \[\sqrt{a+x}\frac{dy}{dx}+x=0\] is [DSSE 1988]
  A)
                  \[3y+2\sqrt{a+x}.(x-2a)=3c\] done clear
  B)
                  \[3y+2\sqrt{x+a}.(x+2a)=3c\] done clear
  C)
                  \[3y+\sqrt{x+a}.(x+2a)=3c\]            done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer36)
  Solution of the equation \[\cos x\cos y\frac{dy}{dx}=-\sin x\sin y\]is [DSSE 1987]
  A)
                  \[\sin y+\cos x=c\] done clear
  B)
                  \[\sin y-\cos x=c\] done clear
  C)
                  \[\sin y.\cos x=c\]   done clear
  D)
                  \[\sin y=c\cos x\] done clear
  View Solution play_arrow

• question_answer37)
  The solution of the differential equation \[x({{e}^{2y}}-1)dy+({{x}^{2}}-1){{e}^{y}}dx=0\]is     [AISSE 1990]
  A)
                  \[{{e}^{y}}+{{e}^{-y}}=\log x-\frac{{{x}^{2}}}{2}+c\] done clear
  B)
                  \[{{e}^{y}}-{{e}^{-y}}=\log x-\frac{{{x}^{2}}}{2}+c\]\[\] done clear
  C)
                  \[{{e}^{y}}+{{e}^{-y}}=\log x+\frac{{{x}^{2}}}{2}+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer38)
  The solution of \[\frac{dy}{dx}=\sin (x+y)+\cos (x+y)\]is
  A)
                  \[\log \left[ 1+\tan \left( \frac{x+y}{2} \right) \right]+c=0\] done clear
  B)
                  \[\log \left[ 1+\tan \left( \frac{x+y}{2} \right) \right]=x+c\] done clear
  C)
                  \[\log \left[ 1-\tan \left( \frac{x+y}{2} \right) \right]=x+c\]               done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer39)
  The solution of the differential equation \[\frac{dy}{dx}=\frac{x-y+3}{2(x-y)+5}\] is
  A)
                  \[2(x-y)+\log (x-y)=x+c\] done clear
  B)
                  \[2(x-y)-\log (x-y+2)=x+c\] done clear
  C)
                  \[2(x-y)+\log (x-y+2)=x+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer40)
  The solution of the differential equation \[(1-{{x}^{2}})(1-y)dx=xy(1+y)dy\] is
  A)
                  \[\log [x\,{{(1-y)}^{2}}]=\frac{{{x}^{2}}}{2}+\frac{{{y}^{2}}}{2}-2y+c\] done clear
  B)
                  \[\log [x{{(1-y)}^{2}}]=\frac{{{x}^{2}}}{2}-\frac{{{y}^{2}}}{2}+2y+c\] done clear
  C)
                  \[\log [x{{(1+y)}^{2}}]=\frac{{{x}^{2}}}{2}+\frac{{{y}^{2}}}{2}+2y+c\] done clear
  D)
                  \[\log [x{{(1-y)}^{2}}]=\frac{{{x}^{2}}}{2}-\frac{{{y}^{2}}}{2}-2y+c\] done clear
  View Solution play_arrow

• question_answer41)
  The solution of the differential equation \[(x-{{y}^{2}}x)dx=(y-{{x}^{2}}y)dy\] is              [DSSE 1984]
  A)
                  \[(1-{{y}^{2}})={{c}^{2}}(1-{{x}^{2}})\]               done clear
  B)
                  \[(1+{{y}^{2}})={{c}^{2}}(1-{{x}^{2}})\] done clear
  C)
                  \[(1+{{y}^{2}})={{c}^{2}}(1+{{x}^{2}})\]           done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer42)
  The solution of \[(\text{cosec}\,x\log y)dy+({{x}^{2}}y)dx=0\] is [AISSE 1986]
  A)
                  \[\frac{\log y}{2}+(2-{{x}^{2}})\cos x+2\sin x=c\] done clear
  B)
                  \[{{\left( \frac{\log y}{2} \right)}^{2}}+(2-{{x}^{2}})\cos x+2x\sin x=c\] done clear
  C)
                  \[{{\frac{(\log y)}{2}}^{2}}+(2-{{x}^{2}})\cos x+2x\sin x=c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer43)
  The solution of \[\frac{dy}{dx}=\frac{{{e}^{x}}({{\sin }^{2}}x+\sin 2x)}{y(2\log y+1)}\] is            [AISSE 1990]
  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
  View Solution play_arrow

• question_answer44)
  The solution of the differential equation \[xy\frac{dy}{dx}=\frac{(1+{{y}^{2}})(1+x+{{x}^{2}})}{(1+{{x}^{2}})}\] is                 [AISSE 1983]
  A)
                  \[\frac{1}{2}\log (1+{{y}^{2}})=\log x-{{\tan }^{-1}}x+c\] done clear
  B)
                  \[\frac{1}{2}\log (1+{{y}^{2}})=\log x+{{\tan }^{-1}}x+c\] done clear
  C)
                  \[\log (1+{{y}^{2}})=\log x-{{\tan }^{-1}}x+c\] done clear
  D)
                  \[\log (1+{{y}^{2}})=\log x+{{\tan }^{-1}}x+c\] done clear
  View Solution play_arrow

• question_answer45)
  The solution of \[(x\sqrt{1+{{y}^{2}}})dx+(y\sqrt{1+{{x}^{2}}})dy=0\] is
  A)
                  \[\sqrt{1+{{x}^{2}}}+\sqrt{1+{{y}^{2}}}=c\]        done clear
  B)
                  \[\sqrt{1+{{x}^{2}}}-\sqrt{1+{{y}^{2}}}=c\] done clear
  C)
                  \[{{(1+{{x}^{2}})}^{3/2}}+{{(1+{{y}^{2}})}^{3/2}}=c\]              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer46)
  The solution of \[{{e}^{2x-3y}}dx+{{e}^{2y-3x}}dy=0\] is
  A)
                  \[{{e}^{5x}}+{{e}^{5y}}=c\]      done clear
  B)
                  \[{{e}^{5x}}-{{e}^{5y}}=c\] done clear
  C)
                  \[{{e}^{5x+5y}}=c\]        done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer47)
  The solution of the differential equation \[(1+{{x}^{2}})(1+y)dy+(1+x)(1+{{y}^{2}})dx=0\] is   [DSSE 1986]
  A)
                  \[{{\tan }^{-1}}x+\log (1+{{x}^{2}})+{{\tan }^{-1}}y+\log (1+{{y}^{2}})=c\] done clear
  B)
                  \[{{\tan }^{-1}}x-\frac{1}{2}\log (1+{{x}^{2}})+{{\tan }^{-1}}y-\frac{1}{2}\log (1+{{y}^{2}})=c\] done clear
  C)
                  \[{{\tan }^{-1}}x+\frac{1}{2}\log (1+{{x}^{2}})+{{\tan }^{-1}}y+\frac{1}{2}\log (1+{{y}^{2}})=c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer48)
  The solution of the equation \[\frac{dy}{dx}={{(x+y)}^{2}}\] is
  A)
                  \[x+y+\tan (x+c)=0\]          done clear
  B)
                  \[x-y+\tan (x+c)=0\] done clear
  C)
                  \[x+y-\tan (x+c)=0\]            done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer49)
  The solution of the differential equation \[\cos y\log (\sec x+\tan x)dx=\cos x\log (\sec y+\tan y)dy\] is [AI CBSE 1990]
  A)
                  \[{{\sec }^{2}}x+{{\sec }^{2}}y=c\]            done clear
  B)
                  \[\sec x+\sec y=c\] done clear
  C)
                  \[\sec x-\sec y=c\]  done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer50)
  The solution of \[\frac{dy}{dx}=\frac{1}{x}\] is
  A)
                  \[y+\log x+c=0\]  done clear
  B)
                  \[y=\log x+c\] done clear
  C)
                  \[{{y}^{\log x}}+c=0\]     done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer51)
  The general solution of the differential equation \[\log \left( \frac{dy}{dx} \right)=x+y\] is        [MP PET 1994; 1995; DSSE 1984]
  A)
                  \[{{e}^{x}}+{{e}^{y}}=c\]           done clear
  B)
                  \[{{e}^{x}}+{{e}^{-y}}=c\] done clear
  C)
                  \[{{e}^{-x}}+{{e}^{y}}=c\]                          done clear
  D)
                  \[{{e}^{-x}}+{{e}^{-y}}=c\] done clear
  View Solution play_arrow

• question_answer52)
  The general solution of the differential equation \[\frac{dy}{dx}=\cot x\cot y\] is        [AISSE 1983; MP PET 1994]
  A)
                  \[\cos x=c\,\cos \text{ec}y\]  done clear
  B)
                  \[\sin x=c\sec y\] done clear
  C)
                  \[\sin x=c\cos y\]    done clear
  D)
                  \[\cos x=c\sin y\] done clear
  View Solution play_arrow

• question_answer53)
  The solution of the equation \[\frac{dy}{dx}=\frac{{{y}^{2}}-y-2}{{{x}^{2}}+2x-3}\] is
  A)
                  \[\frac{1}{3}\log \left| \frac{y-2}{y+1} \right|=\frac{1}{4}\log \left| \frac{x+3}{x-1} \right|+c\] done clear
  B)
                  \[\frac{1}{3}\log \left| \frac{y+1}{y-2} \right|=\frac{1}{4}\log \left| \frac{x-1}{x+3} \right|+c\] done clear
  C)
                  \[4\log \left| \frac{y-2}{y+1} \right|=3\log \left| \frac{x-1}{x+3} \right|+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer54)
  The general solution of the differential equation \[ydx\,+(1+{{x}^{2}}){{\tan }^{-1}}xdy=0,\] is     [MP PET 1995]
  A)
                  \[y{{\tan }^{-1}}x=c\]       done clear
  B)
                  \[x{{\tan }^{-1}}y=c\] done clear
  C)
                  \[y+{{\tan }^{-1}}x=c\]    done clear
  D)
                  \[x+{{\tan }^{-1}}y=c\] done clear
  View Solution play_arrow

• question_answer55)
  The general solution of the differential equation \[\frac{dy}{dx}=\frac{{{x}^{2}}}{{{y}^{2}}}\] is
  A)
                  \[{{x}^{3}}-{{y}^{3}}=c\]            done clear
  B)
                  \[{{x}^{3}}+{{y}^{3}}=c\] done clear
  C)
                  \[{{x}^{2}}+{{y}^{2}}=c\]          done clear
  D)
                  \[{{x}^{2}}-{{y}^{2}}=c\] done clear
  View Solution play_arrow

• question_answer56)
  If \[\frac{dy}{dx}={{e}^{-2y}}\] and \[y=0\] when \[x=5,\] the value of x for \[y=3\] is         [MP PET 2001]
  A)
                  \[{{e}^{5}}\]       done clear
  B)
                  \[{{e}^{6}}+1\] done clear
  C)
                  \[\frac{{{e}^{6}}+9}{2}\]                done clear
  D)
                  \[{{\log }_{e}}6\] done clear
  View Solution play_arrow

• question_answer57)
  The solution of differential equation \[dy-\sin x\sin ydx=0\] is [MP PET 1996]
  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
  View Solution play_arrow

• question_answer58)
  The general solution of the differential equation \[{{e}^{y}}\frac{dy}{dx}+({{e}^{y}}+1)\cot x=0\] is
  A)
                  \[({{e}^{y}}+1)\cos x=K\]               done clear
  B)
                  \[({{e}^{y}}+1)\text{cosec}\,x=K\] done clear
  C)
                  \[({{e}^{y}}+1)\sin x=K\]                done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer59)
  Solution of differential equation \[\frac{dy}{dx}=\sin x+2x\], is [MP PET 1997]
  A)
                  \[y={{x}^{2}}-\cos x+c\]  done clear
  B)
                  \[y=\cos x+{{x}^{2}}+c\] done clear
  C)
                  \[y=\cos x+2\]       done clear
  D)
                  \[y=\cos x+2+c\] done clear
  View Solution play_arrow

• question_answer60)
  Solution of differential equation \[\frac{dy}{dx}=2xy\]is [MP PET 1997]
  A)
                  \[y=c{{e}^{{{x}^{2}}}}\]              done clear
  B)
                  \[{{y}^{2}}=2{{x}^{2}}+c\] done clear
  C)
                  \[y={{e}^{-{{x}^{2}}}}+c\]         done clear
  D)
                  \[y={{x}^{2}}+c\] done clear
  View Solution play_arrow

• question_answer61)
  Solution of \[ydx-xdy={{x}^{2}}ydx\]is         [MP PET 1999]
  A)
                  \[y{{e}^{{{x}^{2}}}}=c{{x}^{2}}\]          done clear
  B)
                  \[y{{e}^{-{{x}^{2}}}}=c{{x}^{2}}\] done clear
  C)
                  \[{{y}^{2}}{{e}^{{{x}^{2}}}}=c{{x}^{2}}\]       done clear
  D)
                  \[{{y}^{2}}{{e}^{-{{x}^{2}}}}=c{{x}^{2}}\] done clear
  View Solution play_arrow

• question_answer62)
  For solving \[\frac{dy}{dx}=(4x+y+1)\], suitable substitution is [MP PET 1999]
  A)
                  \[y=vx\]  done clear
  B)
                  \[y=4x+v\] done clear
  C)
                  \[y=4x\]  done clear
  D)
                  \[y+4x+1=v\] done clear
  View Solution play_arrow

• question_answer63)
  Solution of \[(x+y-1)dx+(2x+2y-3)dy=0\] is [MP PET 1999]
  A)
                  \[y+x+\log (x+y-2)=c\] done clear
  B)
                  \[y+2x+\log (x+y-2)=c\] done clear
  C)
                  \[2y+x+\log (x+y-2)=c\] done clear
  D)
                  \[2y+2x+\log (x+y-2)=c\] done clear
  View Solution play_arrow

• question_answer64)
  Solution of the differential equation \[\sin \frac{dy}{dx}=a\] with \[y(0)=1\] is           [Kurukshetra CEE 1998]
  A)
                  \[{{\sin }^{-1}}\frac{(y-1)}{x}=a\]  done clear
  B)
                  \[\sin \frac{(y-1)}{x}=a\] done clear
  C)
                  \[\sin \frac{(1-y)}{(1+x)}=a\]             done clear
  D)
                  \[\sin \frac{y}{(x+1)}=a\] done clear
  View Solution play_arrow

• question_answer65)
  The solution of \[\cos (x+y)\,dy=\,\,dx\] is         [DCE 1999]
  A)
                  \[y=\tan \,\left( \frac{x+y}{2} \right)+c\]            done clear
  B)
                  \[y+{{\cos }^{-1}}\left( \frac{y}{x} \right)=c\] done clear
  C)
                  \[y=x\,\,\sec \left( \frac{y}{x} \right)+c\]             done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer66)
  The solution of  \[\frac{dy}{dx}+\sqrt{\,\left( \frac{1-{{y}^{2}}}{1-{{x}^{2}}} \right)}\,=\,0\]is        [DCE 1999]
  A)
                  \[{{\tan }^{-1}}x+{{\cot }^{-1}}x=c\]         done clear
  B)
                  \[{{\sin }^{-1}}x+{{\sin }^{-1}}y=c\] done clear
  C)
                  \[{{\sec }^{-1}}x+\text{cose}{{\text{c}}^{-1}}x=c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer67)
  The solution of \[\frac{dy}{dx}={{2}^{y-x}}\] is          [Karnataka CET 2000]
  A)
                  \[{{2}^{x}}+{{2}^{y}}=c\]          done clear
  B)
                  \[{{2}^{x}}-{{2}^{y}}=c\] done clear
  C)
                  \[\frac{1}{{{2}^{x}}}-\frac{1}{{{2}^{y}}}=c\]          done clear
  D)
                  \[x+y=c\] done clear
  View Solution play_arrow

• question_answer68)
  The solution of \[{y}'-y=1,\ y(0)=-1\] is given by \[y(x)=\] [MP PET 2000]
  A)
                  \[-\exp (x)\]             done clear
  B)
                  \[-\exp (-x)\] done clear
  C)
                  ? 1           done clear
  D)
                  \[\exp (x)-2\] done clear
  View Solution play_arrow

• question_answer69)
  The number of solutions of  \[{y}'=\frac{y+1}{x-1},\,y(1)=2\] is [MP PET 2000]
  A)
                  None      done clear
  B)
                  One done clear
  C)
                  Two        done clear
  D)
                  Infinite done clear
  View Solution play_arrow

• question_answer70)
  The general solution of the differential equation \[\frac{dy}{dx}+\sin \left( \frac{x+y}{2} \right)=\sin \left( \frac{x-y}{2} \right)\] is   [MP PET 2001]
  A)
                  \[\log \tan \left( \frac{y}{2} \right)=c-2\sin x\]    done clear
  B)
                  \[\log \tan \,\left( \frac{y}{4} \right)=c-2\sin \left( \frac{x}{2} \right)\] done clear
  C)
                  \[\log \tan \,\left( \frac{y}{2}+\frac{\pi }{4} \right)=c-2\sin x\] done clear
  D)
   \[\log \tan \left( \frac{y}{4}+\frac{\pi }{4} \right)=c-2\sin \left( \frac{x}{2} \right)\] done clear
  View Solution play_arrow

• question_answer71)
  The solution of the differential equation \[{{(x+y)}^{2}}\frac{dy}{dx}={{a}^{2}}\] is [AMU 2001]
  A)
                  \[{{(x+y)}^{2}}=\frac{{{a}^{2}}}{2}x+c\]               done clear
  B)
                  \[{{(x+y)}^{2}}={{a}^{2}}x+c\] done clear
  C)
                  \[{{(x+y)}^{2}}=2{{a}^{2}}x+c\]              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer72)
  Solution of the differential equation \[\frac{dx}{x}+\frac{dy}{y}=0\] is [Karnataka CET 2002]
  A)
                  \[xy=c\]  done clear
  B)
                  \[x+y=c\] done clear
  C)
                  \[\log x\,\,\log y=c\]                done clear
  D)
                  \[{{x}^{2}}+{{y}^{2}}=c\] done clear
  View Solution play_arrow

• question_answer73)
  The solution of differential equation \[y-x\frac{dy}{dx}=a\left( {{y}^{2}}+\frac{dy}{dx} \right)\] is    [MP PET 2002]
  A)
                  \[(x+a)(x+ay)=cy\]              done clear
  B)
                  \[(x+a)(1-ay)=cy\] done clear
  C)
                  \[(x+a)(1-ay)=c\]  done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer74)
  The solution of \[\log \,\left( \frac{dy}{dx} \right)=ax+by\] is         [AMU 2002]
  A)
                  \[\frac{{{e}^{by}}}{b}=\frac{{{e}^{ax}}}{a}+c\]    done clear
  B)
                  \[\frac{{{e}^{-by}}}{-b}=\frac{{{e}^{ax}}}{a}+c\] done clear
  C)
                  \[\frac{{{e}^{-by}}}{a}=\frac{{{e}^{ax}}}{b}+c\]   done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer75)
  The solution of \[\frac{dy}{dx}={{\left( \frac{y}{x} \right)}^{1/3}}\] is        [EAMCET 2002]
  A)
                  \[{{x}^{2/3}}+{{y}^{2/3}}=c\]    done clear
  B)
                  \[{{x}^{1/3}}+{{y}^{1/3}}=c\] done clear
  C)
                  \[{{y}^{2/3}}-{{x}^{2/3}}=c\]      done clear
  D)
                  \[{{y}^{1/3}}-{{x}^{1/3}}=c\] done clear
  View Solution play_arrow

• question_answer76)
  The solution of the equation \[(2y-1)\,\,dx-(2x+3)\,dy=0\] is [Kerala (Engg.)  2002]
  A)
                  \[\frac{2x-1}{2y+3}=c\]     done clear
  B)
                  \[\frac{2y+1}{2x-3}=c\] done clear
  C)
                  \[\frac{2x+3}{2y-1}=c\]     done clear
  D)
  s                 \[\frac{2x-1}{2y-1}=c\] done clear
  View Solution play_arrow

• question_answer77)
  The differential equation \[\cot y\,\,dx=x\,\,dy\] has a solution of the form     [Orissa JEE  2002]
  A)
                  \[y=\cos x\]            done clear
  B)
                  \[x=c\sec y\] done clear
  C)
                  \[x=\sin y\]             done clear
  D)
                  \[y=\sin x\] done clear
  View Solution play_arrow

• question_answer78)
  Solution of \[\frac{dy}{dx}=\frac{x\log {{x}^{2}}+x}{\sin y+y\,\,\cos y}\] is           [EAMCET 2003]
  A)
                  \[y\sin y={{x}^{2}}\log x+c\]          done clear
  B)
                  \[y\sin y={{x}^{2}}+c\] done clear
  C)
                  \[y\sin y={{x}^{2}}+\log x+c\]       done clear
  D)
                  \[y\sin y=x\log x+c\] done clear
  View Solution play_arrow

• question_answer79)
  If \[\left( \frac{2+\sin x}{1+y} \right)\frac{dy}{dx}=-\cos x,\ y(0)=1,\] then \[y\text{ }\left( \frac{\pi }{2} \right)\]= [IIT Screening 2004]
  A)
                  1              done clear
  B)
                  \[\frac{1}{2}\] done clear
  C)
                  \[\frac{1}{3}\]        done clear
  D)
                  \[\frac{1}{4}\] done clear
  View Solution play_arrow

• question_answer80)
  The solution of \[{{e}^{dy/dx}}=(x+1)\], \[y(0)=3\] is [Kerala (Engg.) 2005]
  A)
                  \[y=x\log x-x+2\] done clear
  B)
              \[y=(x+1)\log |x+1|-x+3\] done clear
  C)
                  \[y=(x+1)\log |x+1|+x+3\] done clear
  D)
                  \[y=x\log x+x+3\] done clear
  E)
                  \[y=-(x+1)\log |x+1|+x+3\] done clear
  View Solution play_arrow

• question_answer81)
  Solution of the differential equation \[\frac{dy}{dx}\tan y=\sin (x+y)+\sin (x-y)\] is  [Kerala (Engg.) 2005]
  A)
                  \[\sec y+2\cos x=c\]              done clear
  B)
                  \[\sec y-2\cos x=c\] done clear
  C)
                  \[\cos y-2\sin x=c\]                done clear
  D)
                  \[\tan y-2\sec y=c\] done clear
  E)
                  \[\sec y+2\sin x=c\] done clear
  View Solution play_arrow