Stuck trying to solve a non homogeneous differential equation

In summary, the conversation is about solving a system of differential equations with initial conditions and an interval of integration. The attempt at a solution involves trying different methods such as substitution and finding an integrating factor, but it is ultimately suggested to use the fact that the variables are separable and use Volterra-Lotka equations to solve the system.
  • #1
dooogle
21
0

Homework Statement



dx/dt=x(1-2y) t(0)=0 x(t(0))=1

dy/dt=-y(1-2x), t(0)=0 y(t(0))=2

inerval of integration= [0,40]

Homework Equations



The Attempt at a Solution



i let x=p(t) so t=(p^-1)(x)

dy/dx=(∂y/∂(p^-1))∂(p^-1)/∂x

dy/dx=(-y(1-2x))/(x(1-2y))

since this equation is non homogeneous cannot make a substitution y=vx

the equation can be rearranged into the form

M(x,y)+N(x,y)dy/dx=0

where M(x,y)=y+2xy

and N(x,y)=x-2xy

but the equation is not exact as there is no function Ψ(x,y) such that

Ψx(x,y) =M(x,y) and Ψy(x,y)=N(x,y)

ive tried finding an integrating factor to make the equation exact using the formula

dμ/dx=μ((My-Nx)/N)

but My=1+2x

Nx=1-2y

((My-Nx)/N)=(1+2x-1-2y)/x-2xy

so

please could you tell me if i have made mistakes or point me in the direction i should be going

thank you for your time

dooogle
 
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  • #2
In your dy/dx=(-y(1-2x))/(x(1-2y))

the variables are separable i.e. you get

(1/x - 2) dx = (-1/y + 2) dy ->

d(ln x - 2x) = d(- ln y + 2y)

Should be able to finish.

Let us know, with pics if possible. :smile:

I think these are a case of Volterra-Lotka equations.
 

1. How do I know if a differential equation is non-homogeneous?

A differential equation is considered non-homogeneous if it contains terms that are not functions of the independent variable. These terms can be constants or functions of the dependent variable.

2. What is the general approach for solving a non-homogeneous differential equation?

The general approach for solving a non-homogeneous differential equation involves finding a particular solution and a complementary function. The particular solution is a specific function that satisfies the non-homogeneous part of the equation. The complementary function is a general solution to the homogeneous form of the equation.

3. How do I find a particular solution for a non-homogeneous differential equation?

To find a particular solution, you can use the method of undetermined coefficients or the method of variation of parameters. The method of undetermined coefficients involves guessing a particular solution based on the form of the non-homogeneous term, while the method of variation of parameters involves finding a particular solution by varying the parameters of the complementary function.

4. Can a non-homogeneous differential equation have more than one solution?

Yes, a non-homogeneous differential equation can have multiple solutions. This is because the complementary function, which is a general solution to the homogeneous form of the equation, can have different forms depending on the initial conditions.

5. Are there any real-life applications of solving non-homogeneous differential equations?

Yes, non-homogeneous differential equations are used in various fields of science and engineering, such as physics, chemistry, biology, and economics. They can be used to model real-life phenomena, such as population growth, chemical reactions, and electrical circuits.

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