By Brian H. Chirgwin and Charles Plumpton (Auth.)
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Extra info for A Course of Mathematics for Engineers and Scientists. Volume 2
Since the two kinds of curve are given by the single formula (1) the confocal conies are said to form a 'self-orthogonal' family. Example. 4. Find the orthogonal trajectories of the family rl = c cos 26. Hence the differential equation of the family is and the differential equation of the orthogonal trajectories is which integrates to give Exercises 1:6 Find the equations of the families of curves which have the following properties. (Letters refer to Fig. ) 1. The subtangent TN is of constant length k.
Usually the method does not provide the solution so easily, if at all. Example 2. We use the Taylor's series method for and find the solution for which y = 1, when x = 0. From the equation itself wefind,by putting* = 0,thsLt(dyldx)Xin,Q = 2- Hence O0)9 = 1, O^o = 2. ) It should be noticed that the method will not give the solution for values of x beyond either of the singular points. § 1 : 12] FIRST ORDER DIFFERENTIAL EQUATIONS 59 Exercises 1:11 1. The function y = y(x) satisfies the differential equation and initial condition Find the Taylor series for y in powers of x as far as the term in x4.
E. In the second situation the amount remaining is ~x0 and the required time is T where Example 2. A tank contains V cm3 of brine in which there is S g of salt. Water is pumped into the tank at the rate of u cm3 s _ 1 and brine is pumped out at the rate of v cm3 s _ 1 . Assuming perfect mixing throughout, formulate and solve the differential equation for the amount s g of salt remaining in the tank after / s. Consider also the special case u = v. § 1 : 8] FIRST ORDER DIFFERENTIAL EQUATIONS 41 The volume of liquid in the tank alters with time, unless u = v, so we suppose that this volume is y cm3 at time t.