Solving System of 2 ODEs: Analytical Solution

[trentar]

hallo I ams earching analytical solution for system of two ODE in next form

x*(dy/dr) - (y*y/r) = constant1*r*r*x*x*x
x*(dy/dr)+(x*y/r)=constant2*r*r*x*x*(r*constant3-y)

where x(r) and y(r). conditions are x=constant4 at r=constant5 and y=0 at r=constant5

thnx

r.

 

[HallsofIvy]

(r^2x^3 is simpler than r*r*x*x*x)

Do both equations have "dy/dr"? If neither equation involves dx/dr, you can treat x as a constant. In fact, if we write the two equations as x(dy/dr)= y^2/r+ C₁r^2x^3 and x(dy/dr)= -xy/r+ C₂r^2x^2(C₃- y) and, because the two left sides are equal, equate the two right sides: y^2/r+ C₁r^2x^3= -xy/r+ C₂r^2x^2(C₃- y). You can solve that for y as a function of both x and r and then put that back into the equation to get a single equation for y.

 

[trentar]

hallo like this:
x*(dy/dr) - (y*y/r) = constant1*r*r*x*x*x
x*(dx/dr)+(x*y/r)=constant2*r*r*x*x*(r*constant3-y)

where x(r) and y(r). conditions are x=constant4 at r=constant5 and y=0 at r=constant5