Numerical errors in a numerical solution of coupled non-linear ODEs with a linear center

NOTE: This may be more of a "non-linear dynamics" problem than an ODE one. Here goes... I've recently been toying around with this system: x' = y*e^{-(x^2+y^2)} y' = -x*e^{-(x^2+y^2)} // (where "e^" denotes the exponential function) I've noticed strange behavior that I can't seem to explain. I used a program I found online (PPlane) to graph this system. It _appears_ to consist of a number of closed circular orbits around the origin, as well as isolated fixed points (lying outside the closed orbit with the largest radius). What I _think_ I know is that this system is reversible with a linear center (used the Jacobian), so we should theoretically expect these closed orb... click for more

Differential Equation by Separation of Variables

Could someone please help me with the problem and show me ALL the steps. (See attached file for full problem description)

Differential Equation by vseperation of Variables

Could someone please help me with the problem and show me ALL the steps. (See attached file for full problem description)

Differential Equation by Seperation of Variables

Could someone please help me with the problem and show me ALL the steps. (See attached file for full problem description)

Use any method for xy'=2y-3x

2. Use any method for xy'=2y-3x