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Differential Equations, 1 Mark questions:, 1. Define a differential equation., It is an equation containing derivatives., 2. Define order of a differential equation., It is highest order of derivative appearing in the given equation., 3. Define degree of a differential equation., It is the highest power of highest ordered derivative appearing in the given, equation., 4. Define general solution of a differential equation., It is solution of given differential equation and it contains arbitrary constants., 5. Define particular solution of a differential equation., It is that solution of given differential equation and is free from arbitrary constants., 2 Marks questions:, 1. Form a differential equation of family of, i. Straight lines with slope = m and passing through origin., Consider a straight line with slope = m and passing through origin, i.e. y = mx -----(1),  y1 = m -------(2), , ii., ,  y = x y1, , Circles with centre on y-axis and passing through origin., Consider x 2  y 2  2 fy  0    1  2 x  2 yy1  2 fy 1  0,  x  yy1 ,   f,  2  , 1,  y , , , , , ,  x 2  y 2  2 x  yy1  0, , 2. Solve the following by using separation of variables., i. x dy  y dx  dx  dy,  d xy   d x  y , , ii., , a  y2, dy, , 0, dx, 1 x2, , , iii., ,  d xy    d x  y   xy  x  y  c, , dy, , , , 1 y, , 2, , dx, , , , 1 x, , 2, ,  0  sin 1 y  sin 1 x  c, , 1  x  dy,  1  y   0, dx, 2, , 2, , 1

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 y1 , , , , 2y, x, x, , P, , 2, x, ,  , , Pdx  2 log x  log x 2, , I . f is e, , 2, , 3. y1  y cot x  2 x  x 2 cot x, P  cot x, , , , , , Pdx, , 2,  e logx   x 2, , x4, x4, 2, x dx ,  yx , c, 4, 4, , Solution is y. x   x .x dx  , 2, , Qx, , ;, , 3, , ; y 2  0, , Q  2 x  x 2 cot x, , ;, , Pdx   cot x dx  log sin x, , e, ,  Pdx, ,  e logsin x  sin x, , , , , , Solution is y. sin x   2 x sin x  x 2 cot x sin x dx,  2 x sin x dx   x 2 cos x dx  2  x sin x dx   x 2 d sin x ,  2 x sin x dx  x 2 sin x   2 x sin x dx, ,  y sin x  x 2 sin x  c, , given, y  2  0  Q   2  c  c   2  y sin x  x 2 sin x   2, , 4. y1  3 y  e2 x, P3, , , , Q  e 2 x dx, , ;, , Pdx   3dx  3x, , I. f .  e, ,  Pdx, , Solution is y. e3 x   e3 x e  2 x dx, , 5., , x  3 y y, 2, , 1, ,  y , y  0  y1 , ,  e3 x,   e x dx  e x  y . e3 x  e x  c, , y, x  3y2, , dx x  3 y 2, dx x, ,    3y, dy, y, dy y, 1, P, ; Q  3y,  Pdy   log y  log y 1, y, , , ,  , , 1, 1, I . f .  e log y  , y, 1, 1, x,  Solution is x.   . 3 y dy  3   3 y  c, y, y, y, , 6. y1  2 y tan x  sin x, P  2 tan x, , , , ;, , Q  sin x, , Pdx   2 tan x dx  2 log cos x  log cos x , , 2, , 2, I . f .  e logcos x   cos 2 x, , Solution is y. cos 2 x   sin x . cos 2 x dx ,    cos x  d cos x   y cos 2 x   cos 3 x  c, 2, , 4

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7., , x  y  dy  1, dx, , , , , , , , dx,  x y, dy, , dx,  x  y P  1 ; Q  x  y, dy, , Pdy    1dy   y, , I. f . e, ,  Pdy, , , ,  e y, , Sol. is x . e  y   e  y . y dy   y d  e  y, , , ,   ye  y    e  y   ye  y  e  y  x e  y  e  y  y  1  c, , 8. 1  x 2 y1  2 xy ,  y1 , , , , 1, 1 x2, , , , 2x, 1, y, 2, 1 x, 1 x2, , , , , , , , P , , , , , , 2x,  log 1  x 2, 1 x2, , Pdx  , ,  ,  y 1  x   tan, , , , 2x, 1 x2, , I. f . e, , 1, , 1, , 1  x , , 2 2, ,  Pdx, ,  1 1x  dx   1 1x, , Sol. is y . 1  x 2   1  x 2 ., 2, , ; Q, , 2, , 2, , , , 2,  e log1 x   1  x 2, , dx, , xc, , 9. y1  3 y cot x  sin 2 x, P  3 cot x ; Q  sin 2 x, , , , , , Pdx  3 cot x  3 log sin x  log sin 3 x, , I. f . e, ,  Pdx, , , ,  e logsin x   sin x , 3, , 3, , , , , , Sol. is y . sin x    sin 1 x . 2 sin x cos x dx, 3, , , , 10., , 1, , cos x dx,  2  sec x tan x dx  2 sec x, sin 2 x, , dx, x, tan 1 y, , , dy 1  y 2 1  y 2, P , , , , 1, 1 y2, , tan 1 y, a  y2, , 1,  Pdx, 1, tan 1 y, dy, , tan, y, , I, ., f, ., , e, , e, 1 y2, , Pdy  , , Sol. is x .e, , ; Q, , tan 1 y, ,  , ,  e, , tan 1 y, , tan 1 y, dy   t et dt , t  tan 1 y, 2, 1 y, ,   t d et  te t   et dt  te t  et  xe tan, , 1, , y, ,  tan 1 y  e tan, , 5, , 1, , y, , c, , 

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Statement problems:, 1. Find equation of a curve which passes through origin given that slope at any point, on it = sum of coordinates., given, , dy,  x y, dx, , dy, yx, dx, , y 0   0, , &, , P  1, , Qx, , ;, ,  Pdx    1dx   x  I . f .  e, , , , Pdx, ,  e x, , , , , , Sol. is y . e  x . x dx, ,  ye  x   x d  e  x, ,   xe  x   e  x dx, , ye  x   xe  x  e  x  c, , given y 0  0 0  0  1  c c  1 ye  x   x  1e  x  1, , 2. Find the equation of a curve which passes through (0, 2) given that sum of, coordinates at any point exceeds slope at that point by 5., dy, 5, dx, , Given, x  y , , , ,   Pdx    1 dx   x, , P  1 ; Q  x  5, , I. f . e, ,  Pdx, , dy,  y x 5, dx, ,  Sol. is ye  x    x  5e  x dx, ,  ex, ,  , ,    x d e  x  5 e  x dx   xe  x    e  x dx  5e  x,  ye  x   xe  x  e  x  5e  x  c, , given, y 0  22  0  1  5 .1  c  c  8, ,  y e  x  e x  x  6   8, , 3. Find the equation of curve which passes through (0, 1) given that slope of that at, any point = sum of abscissa and product of coordinates., dy,  x  xy, dx, , Given,, P  x, ,   Pdx  , , 2, , 2, , 2, ,  Sol. is y . e  x, , 2, , , , d ex, , 2, , 2, ,  e, , given, y 0  1,  y ex, , 2, , dy,  xy  x, dx, , ; Qx, , I . f .  ex,   ex, , , , 2, ,  e  x, ,  x2 2, , 2, , 2, ,  x2,  x dx , 2, ,   e  x 2 dx, 2, , c, , 1, e 0  e 0  c  c  8, 2, , 2, , 2, , 6

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4. Find the equation of the curve which passes through (0, 1) given that slope at any,  dy ,  satisfies (x – y) (dx + dy) = dx –dy.,  dx , , point on it , , Given, x  y dx  dy   dx  dy, , d x  y ,  x  y  log x  y   c, x y, given, y 0  1,  0  1  log 1  c  c  1  x  y  log x  2  1, ,   dx  dy   , , 5. Find the equation of curve which passes through (0, 2) given that product of, ordinate and slope at that point = abscissa., Given,, , ydy, x, dx, ,   y dy   x dx, given, y 0  2, , , y2 x2, , c, 2, 2, 2, ,  2, ,  0 c  c  2, 2, , , y2 x2, ,  2  x2  y2  4  0, 2, 2, , 6. Find the equation of a curve which passes through (1, 1)., Given, x, , given, x, , dy,  x  2, dx, ,  y  2, , dy,   x  2 y  2, dx, , , , dy,  x2, , dx, , y2  x , ,  2,  log  y  2   1  dx  x  2 log x  c,  x, given, y 1  1, log 3  1  2 log x  c,  c  log 3  1 log  y  2  x  2 log x  log 3  1, , 7. At any point P on a curve slope =2 (slope segment joining P & A (-4, -3). Find its, equation if it passes through (-2, 1), dy, dy, dx,  x3,  2,  2,  , dx, y3, x4,  x4, log  y  3  2 log x  4  c, Given,, , given, y  2  1, ,  log y  2 log 2  c c  0, , log  y  3  2 log x  4, 8., , In a bank principal, increases continuously at the rate of 5% per year. In how many years, 2, of Rs.100 doubles itself? Use log e  0.6931, , 7

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dp, 5, dp, t, , .P   20,   dt  log P , c, dt 100, P, 20, P  et 20. ec  P  Ket 20, Given, , given P0  1000, , 1000  ke0 k  1000, , P  1000et 20 2000  1000et 20  2  et 20 log 2 ,  t  20 log 2  200.6931  13.862, , t, 20, , 9. Find the equation of a curve whose differential equations is y1= ex sinx given that it, passes through origin., y1  e x sin x, , , , dy,  e x sin x, dx, ,   dy   e x sin x dx, , x, ex, sin x  cos x   c But y0  0 c  1 2  y  e sin x  cos x   1, 2, 2, 2, dy, 10. Find equation of a curve, whose differential equation,  1 x2 1 y 2, dx, given that it passes through A (0, ½), , y  , , , , given,, , , , , , , dy,  1 x2 1 y2, dx, , , , , , , , 1 1 y , x3,   x   c, log , 2 1 y , 3, , given y 0  1 2, c , , , , dy,  1  x 2 dx, 1 y2 , , 1 11 2 ,   00c,  log, 2  1  1 2 , , 1, 1 1 y , x3 1,   x   log 3, log 3  log , 2, 2 1 y , 3 2, , 8, , , , 