Xii Differential Equations Assignment

ASSIGNMENT CLASS XII DIFFERENTIAL EQUATIONS 1. Determine the order and degree of each of the following differential equations: 1  d2y  (i ) 2  2   9 y   4e  x x  dx  2

3

 d 2 y   dy  (iv )  2      0  dx   dx 

2

2

(ii ) x 1  y dx  y 1  x dy  0 2

 d2y  d 2 y  dy  2 (v) 2  3    x log  2  dx  dx   dx 

dy  dy  (iii ) y  x  a 1    dx  dx  2

2

2

 d 2 y   dy   d2y  (vi )  2      x sin  2   dx   dx   dx 

2. Form the differential equations from the following family of curves:

(i ) y  c  x  c 

2

(ii ) y 2  a  b  x  b  x 

(iii ) y 2  2ay  x 2  a 2

2

(iv )  x  a   2 y 2  a 2 (v) y  a sin  x  b  (vi) xy  Ae x  Be  x  x 2 3. Find the differential equation of all the circles in the first quadrant which touch coordinate axes. d 2 y dy 4. Show that y  ae 2 x  be  x is a solution of the differential equation  2y  0 . dx 2 dx d2y 2 5. Show that y  A cos nx  B sin nx is a solution of the differential equation n y  0. dx 2 1 d2y dy 6. Show that y  e m cos x is a solution of the differential equation 1  x 2  2  x  m 2 y  0 . dx dx 2 d y dy B 7. Show that y  Ax  , x  0 is a solution of the differential equation x 2 2  x  y  0 . x dx dx 2 d y dy 8. Show that y  e x  e 2 x is a solution of the differential equation  3  2 y  0 , y (0) 1, y ' (0)  3 . 2 dx dx 9. Solve the following differential equations: dy dy  dy  (i ) 1  x 2   3 x 2   6  cos 1 x (ii )  e x  y  x 2e  y (iii )  1  x  y  xy dx dx  dx  (iv ) cos x 1  cos y  dx  sin y 1  sin x  dy  0

(vi)

dy x  2 y 1  dx x  2 y  1

(v) x cos y dy   xe x log x  e x  dx

(vii ) x 1  y 2 dx  y 1  x 2 dy  0

(viii ) y 1  x 2 

dy  x 1  y 2  dx

 dx  ( x) cos 1    x  y  dy  2 2 ( xi ) x 1  y  dy  y 1  x  dy  0, given that y  0, when x  1

(ix ) y  x

( xii )

dy dy    a  y2   dx dx  

dy  y sin 2 x , given that y (0) 1 ( xiii ) 1  y 2  1  log x  dx  x dy  0, given that when x  1, y 1 dx

10. Solve the following differential equations: dy 3 x  2 y dy (i )  (ii ) x 2  2 xy  y 2 dx dx 2 x  3 y

(v)  3 yx  y 2  dx   x 2  xy  dy

(iii )  x3  y 3  dy  x 2 y dx  0

(vi ) 2 xy dx   x 2  2 y 2  dy  0

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dy y  y  x tan dx x

(vii) x dy  y dx  x 2  y 2 dx

dy dy  y  log y  log x 1 (ix ) 2 xy  y 2  2 x 2  0 , y (1)  2 dx dx dy y y      ( x ) x sin    x  y sin   , y (1)  dx 2 x  x (viii ) x

(iv ) x

11. Solve the following differential equations: dy dy dy (i ) 4  8 y  5e 3 x (ii ) x  y  2 x 3  0 (iii ) 1  x 2   2 xy   x 2  2  x 2  1 dx dx dx dy dy 2 dy (iv ) 1  x 2   2 xy  x 2  4 (v)  x 2 1  2 xy  2 (vi) sin x  y cos x  sin 2 x cos x dx dx dx x 1 dy dy x  y cos x   (vii )   (viii ) x  y  x cos x  sin x , y   1 dx dx 1  sin x 2

(ix )

dy    y cot x  2 x  x 2 cot x , y    0 dx 2

( x)

dy  2 y  e 2 x sin x , y (0)  0 dx

ANSWERS 1. (i ) 2,1 (ii )1,1

(iii )1, 2

(iv ) 2, 2

3

(v) undefined, undefined

(vi) 2, undefined

2

2

2

d y dy dy  dy   dy   dy   dy  2. (i )    4 y  x  2 y  (ii ) xy 2  x    y  0 (iii )  x 2  2 y 2     4 xy  x 2  0 dx dx dx  dx   dx   dx   dx  2 2 d y d y dy dy (iv ) x 2  2 y 2  4 xy (v ) 2  y  0 (vi) xy  x 2  2  x 2  2 dx dx dx dx 2 2 2 dy  x3 1 2  dy    3.  x  y   1       x  y  9. (i ) y  x 3  6sin 1 x   cos 1 x   c (ii ) e y  e x   c   dx    dx  2 3  

(iii ) log 1  y  x 

x2 c 2

4 (vi ) 2  y  x   log 3 x  6 y  1  c 3 (ix )  x  a 1  ay   cy

x3  log y  c 3 y3

 y (vii) sin 1    log x  c  x 5 11. (i ) y   e3 x  ce2 x 4

(iv ) y 1  x 2  

( xi )1  x 2  2 1  y 2 

 y (iv ) x sin    c x

(v) log y 

(viii ) log y  log x  cx

(ii ) y  x 3  cx

(viii) y  sin x

( xii ) log y 

y 10. (i ) 3log  x 2  y 2   4 tan 1    c x

x x2  4  2 log x  x 2  4  c 2

2c  x 2 (vii ) y  2 1  sin x 

(viii ) 1  x 2 1  y 2   c

(vii ) 1  x 2  1  y 2  c

 x y  ( x ) y  tan  c  2 

 1 1  ( xiii ) y  tan    1  log x   4 2 2  (iii ) 

(v) sin y  e x log x  c

(iv ) 1  sin x 1  cos y   c

y  3log x  c x

(ix ) y 

1 1  cos 2 x  2

(ii ) y  cx  x  y 

(vi )3 x 2 y  2 y 3  c

x y , x  0,  e ( x ) log x  cos   1  log x x

(iii ) y  x 1  x 2   tan 1 x 1  x 2   c 1  x 2 

(v) y  x 2  1  log

2 (ix ) y  x  4sin x

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2

x 1 c x 1

1 (vi) y sin x  sin 3 x  c 3

( x ) ye 2 x  1  cos x