# Question:Coupled Initial Conditions for Differential Equations

## Question:Coupled Initial Conditions for Differential Equations

Maple

Hi there, I am ultimately trying to solve a system of differential equation numerically that reads

diff(y(x),x,x) + (2/x)diff(y(x),x) + y(x)(1-(y(x))^2) = (F(x) - G(x)/x^2)
(1/2 - Xi(x))F(x) = -diff(G(x),x) - (1/x)G(x)
(1/2 + Xi(x))G(x) = diff(F(x),x) - (1/x)F(x)

My initial idea to tackle this system would be to solve the second pair of equations with an approximate Xi(x) and then feed the solution back into the first one and proceed to solve it. If the approximation is close enough after sufficient iterations the solutions should converge to the true solutions. However, when trying to solve the second pair of solutions, I am only able to impose initial conditions when x is a small value, say 0.01. In that case the equations basically read

diff(G(x),x)/G(x) = -(1/x)
diff(F(x),x)/F(x) = 1/x

and so I would like to impose initial conditions that look like

ICs := D(G)(0.01)/G(0.01) = -(1/0.01), D(F)(0.01)/F(0.01) = 1/0.01

However upon reading the help file it mentions that such coupled initial conditions cannot be imposed on a numeric application of dsolve. For example,

dsolve({diff(y(x),x,x) + y(x) = 0,D(y)(0)/y(0) = 1, y(0) = 1})

returns the correct solution of y(x) = sin(x) + cos(x), but

dsolve({diff(y(x),x,x) + y(x) = 0,D(y)(0)/y(0) = 1, y(0) = 1},numeric)

disallows such specification of initial conditions. Is there a way to make Maple work with such initial conditions numerically, since no analytic solutions would be available for the problem I'm trying to solve. Thanks in advance for the replies!

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