Solve 2nd Order ODE Mirror for Parallel Reflection from Origin

In summary, the mirror has an equation y=y(x) that states that whenever a ray of light emanates from the origin it reflects parallel to the x-axis. To reflect from the mirror, the rays must make equal angles with the tangent line.
  • #1
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Homework Statement


A curved mirror of equation y=y(x) has that property that whenever a ray of light emanates from the origin it reflects parallel to the x-axis. Find the equation of the mirror


Don't even know how to get started on this, Don't need a solution just need some starting hints / help!

thank you
 
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  • #2
Let y= f(x) be the equation of the mirror. At point [itex](x_0, f(x_0))[/itex], the tangent line to the curve is given by [itex]y= f'(x_0)(x- x_0)+ f(x_0)[/tex]. The line parallel to the x-axis has equation [itex]y= f(x_0)[/itex] and the equation of the line from the origin to the point on the curve has [itex]y= f(x_0)x/x_0[/itex].

To "reflect from the mirror" those two lines must make equal angles with the tangent line.
 
  • #3
HallsofIvy said:
Let y= f(x) be the equation of the mirror. At point [itex](x_0, f(x_0))[/itex], the tangent line to the curve is given by [itex]y= f'(x_0)(x- x_0)+ f(x_0)[/tex]. The line parallel to the x-axis has equation [itex]y= f(x_0)[/itex] and the equation of the line from the origin to the point on the curve has [itex]y= f(x_0)x/x_0[/itex].

To "reflect from the mirror" those two lines must make equal angles with the tangent line.

How do I come up with this information my self and how do you know the the origin to the point on the curve has equation [itex]y= f(x_0)x/x_0[/itex]?
 
  • #4
Bump!, I understand how you got the equation at the point to the curve now.
I don't understand how to use these three equations to make an ODE. I know that the angles must be equal but I don't understand how they relate to the question. I also know I can take a tangent line at the origin, to the intersection of y = [itex]y= f(x_0)[/itex] and that would be a equation that has the same slope has my tangent line at [itex]x_0[/itex]
 
  • #5
bump! same question still
 

Related to Solve 2nd Order ODE Mirror for Parallel Reflection from Origin

1. What is a 2nd Order ODE?

A 2nd Order ODE (Ordinary Differential Equation) is a mathematical equation that involves an unknown function and its derivatives up to the second order. It is commonly used to model physical phenomena in various fields such as physics, engineering, and economics.

2. How do you solve a 2nd Order ODE?

The general method for solving a 2nd Order ODE is by using techniques such as separation of variables, variation of parameters, or the method of undetermined coefficients. The specific method used depends on the form of the equation and the initial conditions given.

3. What is the "Mirror for Parallel Reflection from Origin" in 2nd Order ODEs?

The "Mirror for Parallel Reflection from Origin" is a concept used in solving 2nd Order ODEs with boundary conditions that involve the function and its derivatives evaluated at the origin. It involves reflecting the solution of the equation across the origin to obtain a new solution that satisfies the boundary conditions.

4. Why is solving 2nd Order ODEs important?

Solving 2nd Order ODEs is important because it allows us to model and understand real-world phenomena in a mathematical way. Many physical systems and processes can be described by these equations, making it essential for scientists and engineers to be able to solve them and make predictions.

5. What are some applications of solving 2nd Order ODEs?

2nd Order ODEs have a wide range of applications in various fields such as mechanics, electrical circuits, population dynamics, and quantum mechanics. They are used to model and predict the behavior of physical systems, making them essential in scientific research and engineering design.

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