Confused by dy/dt and dx/dt equations?

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In summary, the conversation discusses a pair of non-linear equations with two equilibrium solutions. The equations can be linearized in the vicinity of these solutions, but a closed form solution cannot be expected. The suggestion is made to try solving the equations in the form of a differential quotient.
  • #1
Rotan72
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dy/dt=xy+ay
dx/dt=bx+yx^2

I don't know how to solve the equations because I never took a class in diff equations when I was still in college

psss Thank you
 
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  • #2
If you had taken a class in diff equations, you would probably have learned not to try to solve things like that! That looks to me to be a pair of rather nasty non-linear equations. Generally speaking, one can't expect to get a closed form solution for non-linear equations.

I note that there are two equilibrium solutions: x= 0, y= 0, and x= a, y= -b/a are constant solutions.
In the vicinity of (0,0), the equation linearize to dy/dt= ay, dx/dt= bx.
In the vicinity of (a,-b/a), the equations linearize to dy/dt= -bx+ a2y and dx/dt= -bx/a + 2ay.
 
  • #3
Perhaps you could make more progress with

[tex]\frac {dy}{dx} = \frac {y (x + a)} {x (b + xy)}[/tex]
 

1. What do dy/dt and dx/dt represent in an equation?

Dy/dt and dx/dt represent the rates of change of the dependent and independent variables, respectively, in a differential equation. They are also known as the derivatives of the variables with respect to time.

2. How do I interpret dy/dt and dx/dt in a real-world scenario?

In a real-world scenario, dy/dt and dx/dt can represent the instantaneous rates of change of a quantity with respect to time. For example, if y represents the height of an object and t represents time, then dy/dt would represent the object's instantaneous velocity at a specific time.

3. Can I solve a differential equation with only dy/dt and dx/dt?

No, in order to solve a differential equation, you will need to have additional information such as initial conditions or boundary conditions. Dy/dt and dx/dt alone are not enough to solve a differential equation.

4. How do I find dy/dt and dx/dt in an equation?

To find dy/dt and dx/dt in an equation, you will need to use the rules of differentiation. For example, if y = x^2, then dy/dt = 2x and dx/dt = 1. You can also use implicit differentiation to find dy/dt and dx/dt in more complicated equations.

5. What is the relationship between dy/dt and dx/dt in a derivative equation?

The relationship between dy/dt and dx/dt can be described by the chain rule, which states that dy/dt = (dy/dx) * (dx/dt). This means that the rate of change of the dependent variable is equal to the rate of change of the independent variable multiplied by the rate of change of the dependent variable with respect to the independent variable.

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