Re: Orthogonal Trajectories
- To: mathgroup at smc.vnet.net
- Subject: [mg45176] Re: Orthogonal Trajectories
- From: "Bo Le" <bole79 at email.si>
- Date: Fri, 19 Dec 2003 06:57:48 -0500 (EST)
- References: <brs5rt$ine$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Hi! y1 ... original family y2 ... orthogonal family You have the problems because you didn't simplify the expression for y2', in which you should plug y1, and you'll see the DE gets solvable, ie. y2' = (#^2+C)^2 / (#^2-C) & The solution is cute. Borut Levart Slovenia "Albireo" <predmore.at.comcast.dot.net at giganews.com> wrote in message news:brs5rt$ine$1 at smc.vnet.net... > 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 > >