Lesson 3a - Orthogonal Trajectories

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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 y' x This means that the perpendicular equation will represent: x y' y 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 2 y y' x e Perpendicular Slope 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 2 ex 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: x 1 y' y yy ' x 1 dy y x 1 dx y dy dx x 1dx dx ydy x 1dx 2 2 y x xc 2 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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