ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL

Unit 9 -- Application: Orthogonal Trajectories

Outline of Unit 9

Initialization

> restart;

> with( DEtools ):

> with( plots ):

> with( linalg ):

Warning, the name changecoords has been redefined

Warning, the name adjoint has been redefined

Warning, the protected names norm and trace have been redefined and unprotected

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The family of circles centered at ( ,0) with radius can be described as , or with

> F := unapply( simplify( (x-c)^2+y^2-c^2 ), (x,y) );

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Representative examples of this family of curves can be plotted with

> plotF := C -> implicitplot( eval(F(x,y),c=C), x=-20..20, y=-10..10, scaling=constrained ):

> display( seq( plotF(c), c=-9..9) );

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To find the curves orthogonal to one of these circles, the first step is to find an ODE that has the circle as a solution. Differentiating the equation for the circle as a level curve gives

> odeF := diff( F(x,y(x))=0, x );

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The orthogonal trajectories are the curves whose slope is the negative reciprocal of the slope of the original curve. Thus, a differential equation for the orthogonal trajectories is

> orthog_traj_ode := eval( odeF, diff(y(x),x)=-1/diff(y(x),x) );

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This simplifies to

> odeG := isolate( orthog_traj_ode, diff(y(x),x) );

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While this ODE is not exact,

> odeadvisor( odeG, [exact] );

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it does have an integrating factor that makes it exact

> intfactor( odeG );

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or note that the differential equation for the orthogonal trajectories is linear. In either case, the solution is

> infolevel[dsolve] := 3:

> dsolve( odeG, y(x), [exact] );

> infolevel[dsolve] := 0:

Classification methods on request

Methods to be used are:   [exact]

Trying to isolate the derivative dy/dx...

Successful isolation of dy/dx

----------------------------

* Tackling ODE using method:   exact

   -> Trying classification methods

   trying exact

   exact successful

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Thus, the orthogonal trajectories to the circle are the straight lines through the center of the circle.

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