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By Rudolph E. Langer

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Since at time / there is a total of x l b . of salt i n solution, the amount of salt per gallon at that time is x/200. T h e rate of withdrawal is thus 5x/200. T h e rate of replacement is T ^ - W e thus have the relation Problems in Rates dx 5 5A: ~ 20Ö It ~ 39 10* T h i s is a linear differential equation for x. Its general integral is x = 20 + ce^'^^^y and with c determined so that the condition x = 60 when / = 0 is fulfilled, the result is X = 20-\- 40e-'^^^. Example 5. A certain compound X is formed by the combination of 2 parts of a chemical U with 3 parts of a chemical W.

PROBLEMS Solve each of the following differential equations by use of the indicated change of variables, or as a Bernoulli equation. 55. 56. 57. (1 + 2JC + 2y| 2 + 4* + 2> + {2y - xylogx] 1 - e«] dy - 0, + 12x + 2y 1 dx-\-{l 1 +x dx -2x +2x+>|<6'-0. { 3 / - x^y] dx + {x* sin y + x* - 59. \4xy - 60. iy + V + dx-\-dy0, x^y 1 dx+{x-\- 61. {sin~* y — cos xjrfxH 62. {y + V l - 63. i2> + 2 x V 64. x V dx - 65. - y =» u log s. X 3xy^] dy = 0, = S/Uy y •= s. sm u, y = s esc u. dy ~ 0, AT =• J , 1^ r V „ .

36, A tank is in the form of a right circular cone with altitude 4 ft. and radius 5 ft. 5 applies, with k = - j ^ . Find the formula for the depth of water in the tank at time /, if the tank is mounted with its axis vertical and its apex downward, and if the hole is at the apex. 37- If the tank of problem 36 is mounted with its apex upward, and the whole is in the base, what is the formula for the time when the depthj^f water in the lank is^? 38. A tank is in the shape of a sphere with radius 4 ft.

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A First Course in Ordinary Differential Equations by Rudolph E. Langer


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