Solve System of DEs: x', y', x^2, y^2

[accountkiller]

Homework Statement

Solve the system of differential equations:
x' = 10x - x^2 - yx , y' = 30y - 2xy - y^2

Homework Equations

The Attempt at a Solution

I tried solving the first equation for y and plugging it into the second, and vice-versa, but my answers get so complicated, like I got (-x'')/(x') + (10x'/2x) + (x'^2/x^2) - x' - x^2 + 30x - 109 = 0. How on Earth would I find the characteristic equation/roots of something like that?

Is there some easier way to solve this system that I didn't catch?

 

[mighty2000]

Hello,

Solving systems of differential equations can be a challenging task, especially when the equations are non-linear as in this case. However, there are some techniques that can help simplify the process.

One approach is to try and find a way to eliminate one of the variables, either x or y, from the system of equations. This can be done by manipulating the equations algebraically or by using substitution. Once one variable is eliminated, the resulting single differential equation can be solved using standard methods.

Another approach is to use numerical methods, such as Euler's method or Runge-Kutta methods, to approximate the solutions. These methods involve breaking down the system of equations into smaller steps and using iterative calculations to approximate the solution at each step.

If you are familiar with linear algebra, you can also try using matrix methods to solve the system. This involves writing the system of equations in matrix form and using techniques such as Gaussian elimination or matrix inversion to find the solution.

I hope these suggestions help you in solving the system of equations. If you are still having trouble, I would recommend seeking guidance from a colleague or a mathematics tutor. Good luck!