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By Rudolph E. Langer

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Extra info for A First Course in Ordinary Differential Equations

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Iii) There is a constant wind whose components toward the east and north are, respectively, Vx and Vy. It need not concern us much that condition (i) is actually quite artificial unless 0 is fixed. 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*).

T h e i r combined weight is 192 l b . D u r i n g the first 10 sec, before the parachute opens, the air resistance amounts to kv^ w i t h A: = i n the foot-poundsecond system. Thereafter, while'the parachute is open, the resistance amounts to AID, with k = M. T o find the man's velocity of fall 15 sec. after the j u m p . W i t h the direction of increasing s taken as downward, the acting forces are 192 and —kv. 2 thus gives the differential equation 192 - yti/ = 6 dv/dt. 4 but with different values for k and ¿1 applies to each part of the fall.

7 - l*i dy = 0. dx + (6* + 1 0 / - Ij rf> = 0. dx + {* + 2y^ + 9} rfy = 0 Answers to Problems 1 127. dx - X Xhx + 6x2} 128. _ {2x - 31 2x^} dy = 0. j _ _ 3y dx = 0. 129. 2 V ^ r f x + V 1 - * ((y -= 0. 130. rfjr - V l - ((y =• 0. ANSWERS TO ODD-NUMBERED PROBLEMS 1. > = 2 + 5. sin^ y 1 -hex and y = 2. 3. >r» = 4 + 2 tan x = c. 1. y = log 11. fe-v'dy 9. {y + \ / l ~ T ? " | l o g x 13. ^2 „ _ logxp - 1. — tan * x}. =c H-i'il 15. log ( . » ' + 4 1 - -xj. { x - l - l ^ - ' + f. 17. > « X + log {2 - x)2 + c, and> = - 1 .

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


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