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By Rudolph E. Langer

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Extra resources for A First Course in Ordinary Differential Equations

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1 by the system dx/di = k{x,y)Q{x,y), dy/di = -k{x, y)P{x, y), with any chosen function k{x,y). T h e integration of this system is generally different for different choices oik{x, y). However, such different choices only determine effectively different parameters t. 1) is always obtained. Example 22. T o integrate the differential equation V l dx-\- {x + 2y] dy = 0. 22 is i n this case dx/dt = x-\-2y, dyjdi = - V l - y-. F r o m the second of the last two equations we find a first integral relation y — cos \t + c\\.

2 is to be used in any given case depends upon the variables i n terms of which the problem is to be solved. 2) is appropriate, for the variables are v and t. If velocity is required i n terms of distance, or vice versa, the third form is clearly the one to be used. W h e n a body falls i n a resisting medium like air or water, the law by which the force of resistance is related to the velocity varies with a number of circumstances, such as the density of the medium, the shape of the body, the magnitude of the velocity.

11) ds"" = r''dQ'^-\-dr''. These are the essential formulas for dealing with the geometrical problems to be considered. Example 10. T o find the equations of the curves through the point with the polar coordinates (1, ir/2) for which every normal line is at the distance 1 from the pole. 48 Applications T h e perpendicular from the pole upon the normal line has the direction of the tangent line. It therefore makes with the radius vector the angle ^ or — i^j according as ^ is acute or obtuse. F o r the curves sought we have thus 1 / r = cos ^, or 1 / r = cos ( r — W e can write these relations tan 1^ = ± V ' r ^ — 1.

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

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