By Brian H. Chirgwin and Charles Plumpton (Auth.)
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Extra resources for A Course of Mathematics for Engineers and Scientists. Volume 2
E. as / -• oo, u -*• c which is the terminal velocity. [This can be seen directly from eqn. (3) which implies that the acceleration -* 0 when u -* c] § 1 : 7] FIRST ORDER DIFFERENTIAL EQUATIONS 37 Note that the resistance always opposes the motion. 42) whatever the sign of v (= x). 44) if v < 0. In this case the motions for v > 0 and v < 0 must be considered separately. This is so in Example 3 above. Exercises 1:7 1. A particle of unit mass is projected vertically upwards under gravity in a resisting medium, whose terminal velocity is V.
Express this statement as a differential equation and solve it for N in terms of t and any necessary constants. Initially the output was 1,000 articles per day but after 50 days it has dropped to 950 articles per day. Calculate how much longer the machine will be kept in use if it is to be discarded as soon as its output falls to 500 articles per day. 14. A radioactive substance disintegrates in accordance with the equation amfat — - km, where m is the mass remaining at time / and k is a constant.
Show by Picard's method that the series solution of the differential equation for which y — 0 when x = 0 is as follows : 2. By Picard's method obtain the first four non-vanishing terms of a power series in x satisfying the equation with the condition that y — 2 when x = 0. 001 of the correct value. § 1 : 11] FIRST ORDER DIFFERENTIAL EQUATIONS 57 3. 3 with that obtained in (i). 4. By Picard's method obtain an approximate solution in series, as far as the term in x7> of dy/dx = 1 - xy2, given that y = 0 when * = 0.
A Course of Mathematics for Engineers and Scientists. Volume 2 by Brian H. Chirgwin and Charles Plumpton (Auth.)