By Derek F. Lawden
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Extra info for A Course in Applied Mathematics, Vol.1, 2
28 A COURSE OF MATHEMATICS Solve the differential equations: 3 . 4 . 5. 6. Find the primitive and singular solution of each of the following differential equations : 7 . 9 11. 8 . 1 . 0 . 12. Solve the differential equations : 1 3 . 1 4 . 1 5 . 1 6 . 1 7 . 1 8 . 19. 1:6 Orthogonal trajectories and geometrical applications The general solution of a first order differential equation involves one arbitrary constant and hence represents a family of curves. r. 32) at right angles is called an orthogonal trajectory of that family.
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.
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. After time t (t < ctjVk) the velocity of the particle is given by Find the resisting force exerted on the particle when its velocity is v. Prove also that when its velocity is v it has risen a height x from the point of projection, where 2.