By Rudolph E. Langer

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**Additional info for A First Course in Ordinary Differential Equations**

**Sample text**

9 is -1 + X+ = a(x){2x + V ) -b{y)y{\ +x-\-y'^\. T h e term — 1 on the left must be matched on the right, and this suggests the trial of b{y)y — 1, that is, b{y) = \/y. T o match the term B^"^ on the left, we may then try a{x) = 1. These trials actually do fulfill the equation. 8, the function e^^^°* which can be written, ye^, is an integrating factor. PROBLEMS For each of the following differential equations, find an integrating factor, and, by using it, integrate the equation. 31. ydx - dy = 0.

W e are presently engaged in developing techniques i n differential equations, not i n obtaining results usable i n aviation. 9. T o the pilot of the pursued plane 0 of F i g . 4, the path of the pursuing plane A is a curve expressible by an equation either i n the polar coordinates (r, 6) or i n the rectangular coordinates (x — x*, y — y*). With reference to the ground, A has a velocity whose eastward component dx/dt is made up of the air-speed component V cos (x — 0) and the wind component Vx.

W i t h c determined so that x = 0, when t = 0, this equation yields log 200 - 15x 200 - 8x ^ -^kt T h e fact that x = 2, when / = 1, serves to evaluate k. 200 - 15x • 200 - 8x W e find from it . 85 = / log — ^ 92 This result can be written as my\ 2oo{i - 15 - 8(M) PROBLEMS 21, A mixing tank contains 100 gal. of fresh water at / = 0. A solution containing 7 lb. of salt per gal. is then added to it at the rate of 3 g a L / s e c , and the resulting mixture in the tank is drawn off at the same rate.