Orthogonal Trajectories

OrthogonalCurves.html

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St. Michael the Archangel defend us in battle. Be our protection against the wickedness and snares of the Devil. May God rebuke him, we humbly pray, and do thou most Heavenly host, by the power of God cast into hell Satan and all the evil spirits who roam around the world seeking the ruin of souls. Amen.

Orthogonal Curves ....

Properties:

  • * curves belonging to a family of differential equations.
  • * ....a way of going back and forth between equation and family of solution curves.
    •

    Remember that a family of curves can represent solutions in differential equations. Presented here below are curves that are tangent to the y-axis at the origin. x^2 + y^2 = 2cx the goal is to take derivatives with respect to x on both sides solving the result for 2c (yielding 2x + 2y(dy/dx) = 2c). The resulting equation will yield a term (at least one) with dy/dx. Now take the original equation and solve it in for 2c (yielding x + (y^2)/x = 2c)

    Now, as shown above equate the two terms for 2c to each other, or 2x + 2y(dy/dx) = x + (y^2)/x, and solve this expression for dy/dx = (y^2 - x^2)/2yx.

    "In summary we see that we can pass back and forth between a differential equation and its family of solution curves."(Krantz)

    You can go to DE Ex2 here...D.E. Ex2.

    You can go to DE Ex1 here ...D.E. Ex1.

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