Orthogonal trajectories

Orthogonal trajectories - Orthogonal Trajectories...

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Orthogonal Trajectories We have seen before (see separable equations for example) that the solutions of a differential equation may be given by an implicit equation with a parameter something like This is an equation describing a family of curves. Whenever we fix the parameter C we get one curve and vice-versa. For example, consider the families of curves where m and C are parameters. Clearly, we may change the names of the variables and still have the same geometric curves. For example, the above families define the same geometric object as Note that the first family describes all the lines passing by the origin (0,0) while the second the family describes all the circles centered at the origin (including the limit case when the radius 0 which reduces to the single point (0,0)) (see the pictures below). Systematic Inspection One Method for Solving Differential www.sysins.com/blog/?p=1 Curves: Official Site Discover a gym where women change their lives - 30 minutes at a time. www.Curves.com Curve fitting in Excel XLfit - powerful curve fitting in Excel. Try XLfit free for 30 days. www.ExcelCurveFitting.com Orthogonal Trajectories http://www.sosmath.com/diffeq/first/orthogonal/orthogonal.html 1 of 11 13/01/2010 9:27 PM

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and Orthogonal Trajectories http://www.sosmath.com/diffeq/first/orthogonal/orthogonal.html 2 of 11 13/01/2010 9:27 PM
In this page, we will only use the variables x and y . Any family of curves will be written as One may ask whether any family of curves may be generated from a differential equation? In general, the answer is no. Let us see how to proceed if the answer were to be yes. First differentiate with respect to x , and get a new equation involving in general x , y , , and C . Using the original equation, we may able to eliminate the parameter C from the new equation. Example.

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This note was uploaded on 02/28/2011 for the course MATH 101 taught by Professor Duke during the Spring '11 term at University of Ottawa.

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Orthogonal trajectories - Orthogonal Trajectories...

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