the differential equation which represents the three

The differential equation which represents the three parameter family of circles $${x^2} + {y^2} + 2gx + 2fy + c = 0{\text{ is :}}$$

A. $$y''' = \frac{{3y'y'{'^2}}}{{1 + y{'^2}}}$$

B. $$y''' = \frac{{3y'{'^2}}}{{1 + y{'^2}}}$$

C. $$y''' = \frac{{3y'}}{{1 + y{'^2}}}$$

D. $$y''' = \frac{{3y'}}{{1 - y{'^2}}}$$

Answer : $$y''' = \frac{{3y'y'{'^2}}}{{1 + y{'^2}}}$$

Solution :
To eliminate the parameters $$g,\,f$$ and $$c$$ differentiate thrice w.r.t. $$x,$$
$$\eqalign{ & x + yy' + g + fy' = 0......\left( 1 \right) \cr & 1 + y{'^2} + yy'' + fy'' = 0......\left( 2 \right) \cr & 3y'y'' + yy''' + fy''' = 0......\left( 3 \right) \cr & \left( 1 \right)y''' - \left( 2 \right)y''{\text{ gives}} \cr & y''' + y{'^2}y''' - 3y'y'{'^2} = 0 \cr & \Rightarrow y''' = \frac{{3y'y'{'^2}}}{{1 + y{'^2}}} \cr} $$

Releted MCQ Question on
Calculus >> Differential Equations

Releted Question 1

A solution of the differential equation $${\left( {\frac{{dy}}{{dx}}} \right)^2} - x\frac{{dy}}{{dx}} + y = 0$$ is-

A. $$y=2$$

B. $$y=2x$$

C. $$y=2x-4$$

D. $$y = 2{x^2} - 4$$

View Answer

Releted Question 2

If $${x^2} + {y^2} = 1,$$ then

A. $$yy'' - 2{\left( {y'} \right)^2} + 1 = 0$$

B. $$yy'' + {\left( {y'} \right)^2} + 1 = 0$$

C. $$yy'' + {\left( {y'} \right)^2} - 1 = 0$$

D. $$yy'' + 2{\left( {y'} \right)^2} + 1 = 0$$

View Answer

Releted Question 3

If $$y\left( t \right)$$ is a solution $$\left( {1 + t} \right)\frac{{dy}}{{dt}} - ty = 1$$ and $$y\left( 0 \right) = - 1,$$ then $$y\left( 1 \right)$$ is equal to-

A. $$ - \frac{1}{2}$$

B. $$e + \frac{1}{2}$$

C. $$e - \frac{1}{2}$$

D. $$\frac{1}{2}$$

View Answer

Releted Question 4

If $$y = y\left( x \right)$$ and $$\frac{{2 + \sin \,x}}{{y + 1}}\left( {\frac{{dy}}{{dx}}} \right) = - \cos \,x,\,y\left( 0 \right) = 1,$$
then $$y\left( {\frac{\pi }{2}} \right)$$ equals-

A. $$\frac{1}{3}$$

B. $$\frac{2}{3}$$

C. $$ - \frac{1}{3}$$

D. $$1$$

View Answer

The differential equation which represents the three parameter family of circles $${x^2} + {y^2} + 2gx + 2fy + c = 0{\text{ is :}}$$

Releted MCQ Question on
Calculus >> Differential Equations

A solution of the differential equation $${\left( {\frac{{dy}}{{dx}}} \right)^2} - x\frac{{dy}}{{dx}} + y = 0$$ is-

If $${x^2} + {y^2} = 1,$$ then

If $$y\left( t \right)$$ is a solution $$\left( {1 + t} \right)\frac{{dy}}{{dt}} - ty = 1$$ and $$y\left( 0 \right) = - 1,$$ then $$y\left( 1 \right)$$ is equal to-

If $$y = y\left( x \right)$$ and $$\frac{{2 + \sin \,x}}{{y + 1}}\left( {\frac{{dy}}{{dx}}} \right) = - \cos \,x,\,y\left( 0 \right) = 1,$$
then $$y\left( {\frac{\pi }{2}} \right)$$ equals-

Practice More Releted MCQ Question on
Differential Equations

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The differential equation which represents the three parameter family of circles $${x^2} + {y^2} + 2gx + 2fy + c = 0{\text{ is :}}$$

Releted MCQ Question on Calculus >> Differential Equations

A solution of the differential equation $${\left( {\frac{{dy}}{{dx}}} \right)^2} - x\frac{{dy}}{{dx}} + y = 0$$ is-

If $${x^2} + {y^2} = 1,$$ then

If $$y\left( t \right)$$ is a solution $$\left( {1 + t} \right)\frac{{dy}}{{dt}} - ty = 1$$ and $$y\left( 0 \right) = - 1,$$ then $$y\left( 1 \right)$$ is equal to-

If $$y = y\left( x \right)$$ and $$\frac{{2 + \sin \,x}}{{y + 1}}\left( {\frac{{dy}}{{dx}}} \right) = - \cos \,x,\,y\left( 0 \right) = 1,$$ then $$y\left( {\frac{\pi }{2}} \right)$$ equals-

Practice More Releted MCQ Question on Differential Equations

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Releted MCQ Question on
Calculus >> Differential Equations

If $$y = y\left( x \right)$$ and $$\frac{{2 + \sin \,x}}{{y + 1}}\left( {\frac{{dy}}{{dx}}} \right) = - \cos \,x,\,y\left( 0 \right) = 1,$$
then $$y\left( {\frac{\pi }{2}} \right)$$ equals-

Practice More Releted MCQ Question on
Differential Equations