Unformatted text preview: s means to find an orthogonal trajectory of g(x,y,c) we must:
Step 1: Derive g(x,y,c) and solve for y’.
Step 2: Take the negative reciprocal of y’, this will be our new slope of our orthogonal trajectory.
Step 3: Solve the differential equation y’=(negative reciprocal slope) to get your orthogonal trajectory. Example 1
Determine the differential equation that represents the orthogonal trajectory for the solution of the following differential equation: xy '2 x e y
xy ' e y 2 x
e y 2x
This means that the perpendicular equation will represent: x
e 2x Example 2 Determine the orthogonal trajectories to the following for a function that has the following ODE: e x y ' y 2 e y ' y 2
x e x y' 2 y
Orthogonal Trajectory ODE: ( y 2) y ' dx e x dx dy
( y 2) dx e x dx dx
( y 2)dy e x dx y2 2 y ex c
y' y2 y2 2 y ex c
2 ( y 2) y ' e x y 2 4 y 2e x c Example 3
Find the orthogonal trajectory that goes through (2,0) for the following relation xy y 3 xy y 3 xy ' y ' y xy ' y y ' 0 y ' ( x 1) y y' y
x 1 Taking the perpendicular to this slope to find the ODE that represents the orthogonal trajectory:
y' y yy ' x 1
y x 1
dx y dy
dx x 1dx
dx ydy x 1dx
2 2 y
2 y2 x2 2x c sub in (2,0)
0 2 2 2 2( 2) c 0 44c
8 c y2 x2 2x 8...
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This note was uploaded on 02/07/2014 for the course MATH 1004 taught by Professor Mark during the Summer '00 term at Carleton CA.
- Summer '00
- Calculus, SmartPen Lectures, Lecture Notes, MATH1004, Kyle Harvey