You find orthogonal trajectories of differential equations by following these steps:
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Find the Differential Equation of the Family of Curves: Start with the given family of curves, expressed as an equation involving both the independent variable (often x) and the dependent variable (often y). Differentiate this equation implicitly with respect to x to obtain a differential equation that represents the entire family of curves.
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Replace dy/dx with -dx/dy: The slopes of orthogonal curves are negative reciprocals of each other. Therefore, replace dy/dx in the differential equation from step 1 with -dx/dy. This new differential equation will represent the family of orthogonal trajectories.
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Solve the New Differential Equation: Solve the differential equation obtained in step 2 to find the general solution, which will represent the family of orthogonal trajectories.
Example:
Let's find the orthogonal trajectories of the family of circles centered at the origin, given by the equation x² + y² = r².
- Find the Differential Equation:
- Differentiate the equation implicitly: 2x + 2y*(dy/dx) = 0.
- Solve for dy/dx: dy/dx = -x/y.
- Replace dy/dx with -dx/dy:
- -dx/dy = -x/y.
- Simplify: dx/dy = x/y.
- Solve the New Differential Equation:
- Separate variables: (1/x)dx = (1/y)dy.
- Integrate both sides: ln|x| = ln|y| + C.
- Solve for y: y = ±Cx.
Therefore, the orthogonal trajectories of the family of circles centered at the origin are the family of straight lines passing through the origin, represented by the equation y = ±Cx.
Practical Insights:
- Orthogonal trajectories have applications in various fields, including physics, engineering, and geometry.
- They are useful for understanding the behavior of systems that interact with each other in a perpendicular manner.
- For example, in electromagnetism, the electric field lines and magnetic field lines are orthogonal trajectories.
