Orthogonal Trajectories
- To: mathgroup at smc.vnet.net
- Subject: [mg45156] Orthogonal Trajectories
- From: "Albireo" <predmore.at.comcast.dot.net at giganews.com>
- Date: Thu, 18 Dec 2003 06:55:32 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
17 December 2003
I am working with the family of functions
y[x,C] = x / ( x^2 + C)
where x is the independent variable and C>0 is a parameter.
I am interested in the family of functions which are orthogonal to the above
function.
y_ortho[x,D] = f[x,D]
where x is the independent variable and D is a parameter.
The family of functions
y[x,C]
has the differential equation
y'[x] = y[x]/x - 2 (y[x])^2
The orthogonal trajectories would have the negative inverse for their
differential equation
y_ortho'[x] = -1/y'[x] = x / (2 x (y_ortho[x])^2 - y_rtho[x])
Using DSolve on y'[x]
DSolve[y'[x] == y[x]/x - 2y[x]^2, y[x], x]
gives
y[x] -> x / (x^2 + C[1])
as expected.
When I use DSolve on y_ortho'[x], there is no solution
\!\(DSolve[\(y'\)[x] == x\/\(2\ x\ y[x]\^2 - y[x]\), y[x], x]\)
gives the above statement back.
If any of you know how to solve the differential equation
y_ortho'[x] = x / (2 x (y_ortho[x])^2 - y_rtho[x])
I would appreciate your input.
I have looked through a few books on differential equations and haven't been
able to find an answer.
Thanks,
Read