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If y(x) satisfies the differential eq...

If `y(x)` satisfies the differential equation `y^(prime)-ytanx=2xs e c x` and `y(0)=0` , then (a) `( b ) (c) y(( d ) (e) (f)pi/( g )4( h ) (i) (j))=( k )(( l ) (m)pi^(( n )2( o ))( p ))/( q )(( r )8sqrt(( s )2( t ))( u ))( v ) (w) (x)` (y) (b) `( z ) (aa) (bb) y^(( c c )prime( d d ))( e e )(( f f ) (gg) (hh)pi/( i i )4( j j ) (kk) (ll))=( m m )(( n n ) (oo)pi^(( p p )2( q q ))( r r ))/( s s )(( t t ) 18)( u u ) (vv) (ww)` (xx) (c) `( d ) (e) y(( f ) (g) (h)pi/( i )3( j ) (k) (l))=( m )(( n ) (o)pi^(( p )2( q ))( r ))/( s )9( t ) (u) (v)` (w) (d) `( x ) (y) (z) y^(( a a )prime( b b ))( c c )(( d d ) (ee) (ff)pi/( g g )3( h h ) (ii) (jj))=( k k )(( l l )4pi)/( m m )3( n n ) (oo)+( p p )(( q q )2( r r )pi^(( s s )2( t t ))( u u ))/( v v )(( w w )3sqrt(( x x )3( y y ))( z z ))( a a a ) (bbb) (ccc)` (ddd)

A

(a) `y(pi/4)=(pi^(2))/(8sqrt2)`

B

(b) `y(pi/4)=(pi^(2))/18`

C

(c) `y(pi/3)=(pi^(2))/9`

D

(d) `y(pi/3)=(4pi^(2))/3=(2pi^2)/(3sqrt3)`

Text Solution

Verified by Experts

PLAN Linear differential equation under one variable.
`dy/dx +Py=Q, IF =e^(int Pdx)`
`therefore` Solution is , `y (IF)=intQ cdot (IF)dx+C`
`y'-y tanx=2x sec x and y(0)=0`
`rArr dy/dx-ytanx=2xsecs`
`therefore IF=inte^(-tanx)dx=e^(logabs(cos x))=cosx`
Solution is `y cdot cos x = int 2 x sec x cos x dx +C`
`rArr y cdot cosx=x^(2)+C`
As `y(0)=0 rArr C=0`
`therefore y=x^(2) sec x `
Now, `y(pi/4)=pi^(2)/(8sqrt2)`
` rArr y'(pi/4)=pi/sqrt2+pi^(2)/(8sqrt2)`
`y(pi/3)=(2pi^(2))/9 rArr y'(pi/3)=(4pi)/3+(2pi^(2))/(3sqrt2)`
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