Question

Find all semistationary solutions of the following system of differential equations.

dx/dt= x + x^2+y^2
dy/dt= y - xy

Answer 1

The semistationary solutions of the given differential equation system are found by setting the derivatives equal to zero, which yields the points (0,0) and (-1,0) as the solutions.

Step-by-step explanation:

To find all semistationary solutions of the given system of differential equations, we must set the derivatives dx/dt and dy/dt to zero to find the equilibrium points. The system is as follows:

• dx/dt = x + x^2 + y^2
• dy/dt = y - xy

To find the equilibria, we solve the following equations simultaneously:

• 0 = x + x^2 + y^2
• 0 = y - xy

From the second equation, we either have y = 0 or x = 1. Inserting y = 0 into the first equation yields the solutions x = 0 or x = -1. If x = 1, then we insert this into the first equation and obtain y^2 = -2, which has no real solutions. Therefore, the semistationary solutions are at the points (0,0) and (-1,0).