Applications of First Order ODE

Applications of First Order ODE - Applications of...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 4

Applications of First Order ODE - Applications of...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online