By M.V. Makarets, V. Yu. Reshetnyak.
1. First Order Differential Equations --
2. N-th Order Differential Equations --
3. Linear moment Order Equations --
4. platforms of Differential Equations --
5. Partial Equations of the 1st Order --
6. Nonlinear Equations and balance --
7. Calculus of adaptations --
8. solutions to difficulties.
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Extra resources for Ordinary differential equations and calculus of variations : book of problems
Example text
E**dy + feye? - i sin i ) dx = 0. 50. y ' i c o s i + yfxsin I + cosx) = 1. Find a particular solution by inspection; find a solution when the term not involving y is replaced by zero; and write down the general solution. 51. y' - y = 2. 52. y' + y = 2e . 53. xy'= 1. 54. y' = y + l. 55. tf+y = *• + !. ," (10) where n is a constant but not necessary an integer, known as Bernoulli's equation, was studied in 1695 by the Swiss mathematician Jacob Bernoulli (1654-1705). We rule out cases n = 0 and n — 1, for which the equation is already linear.
3 J 49. e**dy + feye? - i sin i ) dx = 0. 50. y ' i c o s i + yfxsin I + cosx) = 1. Find a particular solution by inspection; find a solution when the term not involving y is replaced by zero; and write down the general solution. 51. y' - y = 2. 52. y' + y = 2e . 53. xy'= 1. 54. y' = y + l. 55. tf+y = *• + !. ," (10) where n is a constant but not necessary an integer, known as Bernoulli's equation, was studied in 1695 by the Swiss mathematician Jacob Bernoulli (1654-1705). We rule out cases n = 0 and n — 1, for which the equation is already linear.
28. (y + xy) dx - xHy = 0; f t = (*»»)"'•. 29. (3xy + y ) + (x + xy)y' = 0; ft m at. 30. arV + x (1 + j , ) y' = 0; a = ( i y ) - ' . /siny „ \ , / cosy + 2 e cosr\ _ J 1 7 1 3 -1 31. I — - - 2 e - ' s m x \ d x + I 2 I = 0 ; i* = ye* 32. y d i + (2x-ye")dy = 0; u. = y. ^ + *£j 33. ( l dx + (2xy + dy = 0; u = i . 34. ( i - i y - y ) d r + ( r y - y + i ) dy = 0; u, = — ; u = ry x' — y' 35. (y - x)e-'dx + xe"*dy = 0; ft = e*. 1 1 2 3 3 36. (x - y)dx + f> + y)dy = I}; ** = u ( ^ 7 ^ ) • 37. (i - «v) <** + - * ) * "o; /• - / » ( ^ ) • In each of Problems 38 through 72 find an integrating factor or change the variables, and solve the given equation.