Home
Class 12
MATHS
The function y=f(x) is the solution o...

The function `y=f(x)` is the solution of the differential equation `(dy)/(dx)+(x y)/(x^2-1)=(x^4+2x)/(sqrt(1-x^2))` in `(-1,1)` satisfying `f(0)=0.` Then `int_((sqrt(3))/2)^((sqrt(3))/2)f(x)dx` is (a) `( b ) (c) (d)pi/( e )3( f ) (g)-( h )(( i )sqrt(( j )3( k ))( l ))/( m )2( n ) (o) (p)` (q) (b) `( r ) (s) (t)pi/( u )3( v ) (w)-( x )(( y )sqrt(( z )3( a a ))( b b ))/( c c )4( d d ) (ee) (ff)` (gg) (c) `( d ) (e) (f)pi/( g )6( h ) (i)-( j )(( k )sqrt(( l )3( m ))( n ))/( o )4( p ) (q) (r)` (s) (d) `( t ) (u) (v)pi/( w )6( x ) (y)-( z )(( a a )sqrt(( b b )3( c c ))( d d ))/( e e )2( f f ) (gg) (hh)` (ii)

A

`pi/3-sqrt(3)/2`

B

`pi/3-sqrt(3)/4`

C

`pi/6-sqrt(3)/4`

D

`pi/6-sqrt(3)/2`

Text Solution

Verified by Experts

The correct Answer is:
B
`(dy)/(dx) + x/(x^(2)-1)y=(x^(4)+2x)/sqrt(1-x^(2))`
This is a linear differential equation.
`I.F. =e^(intx/(x^(2)-1)dx)`
`e^(1/2"ln "|x^(2)-1|)=sqrt(1-x^(2))` (`therefore x in (-1,1))`
Therefore, solution is:
`ysqrt(1-x^(2))=int(x(x^(3)+2))/sqrt(1-x^(2)).sqrt(1-x^(2))dx`
or `ysqrt(1-x^(2))=int(x^(4)+2x)dx=x^(5)/5+x^(2)+C`
Since, `f(0)=0,C=0`
`therefore f(x) sqrt(1-x^(2))=x^(5)/5+x^(2)`
`rArr f(x) = x^(5)/(5sqrt(1-x^(2))+x^(2)/sqrt(1-x^(2))`
`therefore int_(-sqrt(3//2))^(sqrt(3)/2) f(x)dx= int_(sqrt(3)/2)^(sqrt(3)/2) (x^(5)/(5sqrt(1-x^(2))+x^(2)/sqrt(1-x^(2))))dx`
`=2int_(0)^(sqrt(3)//2)(x^(2)/sqrt(1-x^(2)))dx`
`rArr 2int_(0)^(pi//3) sin^(2)theta(d)theta`
`=2int_(0)^(pi//3)(1-cos2theta)/(2)(d)theta`
`=2[theta/2-(sin2theta)/4]_(0)^(pi//3)`
`=2(pi/6) -2(sqrt(3))/8 = pi/3-sqrt(3)/4`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIAL EQUATIONS

    CENGAGE ENGLISH|Exercise Multiple Correct Answerts type|5 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE ENGLISH|Exercise Single Correct Answer Type|37 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE ENGLISH|Exercise Archives|12 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE ENGLISH|Exercise Multiple correct answers type|11 Videos

  • DIFFERENTIATION

    CENGAGE ENGLISH|Exercise Archives|14 Videos

Similar Questions

Explore conceptually related problems

Solution of (dy)/(dx)+2x y=y is (a) ( b ) (c) y=c (d) e^( e ) (f) x-( g ) x^((( h )2( i ))( j ) (k))( l ) (m) (n) (b) ( o ) (p) y=c( q ) e^( r ) (s) (t) x^((( u )2( v ))( w )-x (x))( y ) (z) (aa) (c) ( d ) (e) y=c( f ) e^(( g ) x (h))( i ) (j) (k) (d) ( l ) (m) y=c (n) e^( o ) (p)-( q ) x^((( r )2( s ))( t ) (u))( v ) (w) (x)

The solution of differential equation (2y+x y^3)dx+(x+x^2y^2)dy=0 is (a) ( b ) (c) (d) x^(( e )2( f ))( g ) y+( h )(( i ) (j) x^(( k )3( l ))( m ) (n) y^(( o )3( p ))( q ))/( r )3( s ) (t)=c (u) (v) (b) ( w ) (x) x (y) y^(( z )2( a a ))( b b )+( c c )(( d d ) (ee) x^(( f f )3( g g ))( h h ) (ii) y^(( j j )3( k k ))( l l ))/( m m )3( n n ) (oo)=c (pp) (qq) (c) ( d ) (e) (f) x^(( g )2( h ))( i ) y+( j )(( k ) (l) x^(( m )4( n ))( o ) (p) y^(( q )4( r ))( s ))/( t )4( u ) (v)=c (w) (x) (d) None of these

The solution of the differential equation (x^2y^2-1)dy+2xy^3dx=0 is (a) ( b ) (c)1+( d ) x^(( e )2( f ))( g ) (h) y^(( i )2( j ))( k )=c x (l) (m) (b) ( n ) (o)1+( p ) x^(( q )2( r ))( s ) (t) y^(( u )2( v ))( w )=c y (x) (y) (c) ( d ) (e) y=0( f ) (g) (d) ( h ) (i) y=-( j )1/( k )(( l ) (m) x^(( n )2( o ))( p ))( q ) (r) (s) (t)

If x(t) is a solution of ((1+t)dy)/(dx)-t y=1 and y(0)=-1 then y(1) is (a) ( b ) (c)-( d )1/( e )2( f ) (g) (h) (i) (b) ( j ) (k) e+( l )1/( m )2( n ) (o) (p) (q) (c) ( d ) (e) e-( f )1/( g )2( h ) (i) (j) (k) (d) ( l ) (m) (n)1/( o )2( p ) (q) (r) (s)

If y=f(x) is solution of differentiable equation (dy)/(dx)=y/x((1-3x^(2)y^(3)))/((2x^(2)y^(3)+1)) , then ( f(x).x)^(3) is equal to

The solution of the differential equation (x+(x^3)/(3!)+(x^5)/(5!)+)/(1+(x^2)/(2!)+(x^4)/(4!)+)=(dx-dy)/(dx+dy)i s (a) ( b ) (c)2y( d ) e^(( e ) (f)2x (g))( h )=C( i ) e^(( j ) (k)2x (l))( m )+1( n ) (o) (p) ( q ) (r)2y( s ) e^(( t ) (u)2x (v))( w )=C( x ) e^(( y ) (z)2x (aa))( b b )-1( c c ) (dd) (ee) ( f f ) (gg) y( h h ) e^(( i i ) (jj)2x (kk))( l l )=C( m m ) e^(( n n ) (oo)2x (pp))( q q )+2( r r ) (ss) (d) None of these

A curve passing through (2,3) and satisfying the differential equation int_0^x ty(t)dt=x^2y(x),(x >0) is (a) ( b ) (c) (d) x^(( e )2( f ))( g )+( h ) y^(( i )2( j ))( k )=13 (l) (m) (b) ( n ) (o) (p) y^(( q )2( r ))( s )=( t )9/( u )2( v ) (w) x (x) (y) (c) ( d ) (e) (f)(( g ) (h) x^(( i )2( j ))( k ))/( l )8( m ) (n)+( o )(( p ) (q) y^(( r )2( s ))( t ))/( u )(( v ) 18)( w ) (x)=1( y ) (z) (d) ( a a ) (bb) x y=6( c c ) (dd)

The differential equation of all non-horizontal lines in a plane is (a) ( b ) (c) (d)(( e ) (f) d^(( g )2( h ))( i ) y)/( j )(( k ) d (l) x^(( m )2( n ))( o ))( p ) (q) (r) (s) (b) ( t ) (u) (v)(( w ) (x) d^(( y )2( z ))( a a ) x)/( b b )(( c c ) d (dd) y^(( e e )2( f f ))( g g ))( h h ) (ii)=0( j j ) (kk) (c) ( d ) (e) (f)(( g ) dy)/( h )(( i ) dx)( j ) (k)=0( l ) (m) (d) ( n ) (o) (p)(( q ) dx)/( r )(( s ) dy)( t ) (u)=0( v ) (w)

The solution of the differential equation {1+xsqrt((x^2+y^2))}dx+{sqrt((x^2+y^2))-1}ydy=0 is equal to (a) ( b ) (c) (d) x^(( e )2( f ))( g )+( h )(( i ) (j) y^(( k )2( l ))( m ))/( n )2( o ) (p)+( q )1/( r )3( s ) (t) (u) (v)(( w ) (x) (y) x^(( z )2( a a ))( b b )+( c c ) y^(( d d )2( e e ))( f f ) (gg))^(( h h ) (ii) (jj)3/( k k )2( l l ) (mm) (nn))( o o )=c (pp) (qq) (rr) ( s s ) (tt) x+( u u )(( v v ) (ww) y^(( x x )3( y y ))( z z ))/( a a a )3( b b b ) (ccc)+( d d d )1/( e e e )2( f f f ) (ggg) (hhh) (iii)(( j j j ) (kkk) (lll) x^(( m m m )2( n n n ))( o o o )+( p p p ) y^(( q q q )2( r r r ))( s s s ) (ttt))^(( u u u ) (vvv) (www)1/( x x x )2( y y y ) (zzz) (aaaa))( b b b b )=c (cccc) (dddd) (eeee) ( f f f f ) (gggg) x-( h h h h )(( i i i i ) (jjjj) y^(( k k k k )2( l l l l ))( m m m m ))/( n n n n )2( o o o o ) (pppp)+( q q q q )1/( r r r r )3( s s s s ) (tttt) (uuuu) (vvvv)(( w w w w ) (xxxx) (yyyy) x^(( z z z z )2( a a a a a ))( b b b b b )+( c c c c c ) y^(( d d d d d )2( e e e e e ))( f f f f f ) (ggggg))^(( h h h h h ) (iiiii) (jjjjj)3/( k k k k k )2( l l l l l ) (mmmmm) (nnnnn))( o o o o o )=c (ppppp) (qqqqq)

The solution of the differential equation {1+xsqrt((x^2+y^2))}dx+{sqrt((x^2+y^2))-1}ydy=0 is equal to (a) ( b ) (c) (d) x^(( e )2( f ))( g )+( h )(( i ) (j) y^(( k )2( l ))( m ))/( n )2( o ) (p)+( q )1/( r )3( s ) (t) (u) (v)(( w ) (x) (y) x^(( z )2( a a ))( b b )+( c c ) y^(( d d )2( e e ))( f f ) (gg))^(( h h ) (ii) (jj)3/( k k )2( l l ) (mm) (nn))( o o )=c (pp) (qq) (rr) ( s s ) (tt) x+( u u )(( v v ) (ww) y^(( x x )3( y y ))( z z ))/( a a a )3( b b b ) (ccc)+( d d d )1/( e e e )2( f f f ) (ggg) (hhh) (iii)(( j j j ) (kkk) (lll) x^(( m m m )2( n n n ))( o o o )+( p p p ) y^(( q q q )2( r r r ))( s s s ) (ttt))^(( u u u ) (vvv) (www)1/( x x x )2( y y y ) (zzz) (aaaa))( b b b b )=c (cccc) (dddd) (eeee) ( f f f f ) (gggg) x-( h h h h )(( i i i i ) (jjjj) y^(( k k k k )2( l l l l ))( m m m m ))/( n n n n )2( o o o o ) (pppp)+( q q q q )1/( r r r r )3( s s s s ) (tttt) (uuuu) (vvvv)(( w w w w ) (xxxx) (yyyy) x^(( z z z z )2( a a a a a ))( b b b b b )+( c c c c c ) y^(( d d d d d )2( e e e e e ))( f f f f f ) (ggggg))^(( h h h h h ) (iiiii) (jjjjj)3/( k k k k k )2( l l l l l ) (mmmmm) (nnnnn))( o o o o o )=c (ppppp) (qqqqq)