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Unformatted text preview: Applications of First-Order Ordinary Differential Equations Shubhodeep Mukherji October 6, 2010 1 Orthogonal Trajectories Let F ( x,y,c ) = 0 be a given one parameter family of curves. A curve which intersects this family of curves at right angles is called an orthogonal trajectory of the given family. A general outline of the steps required to solve this type of problem is given below. 1). From the equation F ( x,y,c ) of the given family of curves, find the differential equation dy dx = f ( x,y ) of this family. In finding the differential equation, be sure to eliminate the parameter c during the process. 2). Let’s recall the equations of the form y = mx + b . In this family of equations, the slope of the perpendicular line was always the negative reciprocal,- 1 m . Similarly, we must take the negative reciprocal of the differential equation dy dx = f ( x,y ) found in step 1. So, we replace f ( x,y ) by- 1 f ( x,y ) . This yields the differential equation dy dx =- 1 f ( x,y ) of the orthogonal trajectories. 3). Solve the differential equation found in step 2. Thus, obtaining the desired family, G ( x,y,c ) = 0, of orthogonal trajectories. We will now utilize this knowledge in solving an example problem. Example 1 Find the orthogonal trajectories of the family of curves, y = cx 3 . 1 2 Oblique Trajectories Let F ( x,y,c ) = 0 be a given one parameter family of curves. A curve which intersects this family of curves at constant angle...
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