Calculus/Systems of ordinary differential equations

We have already examined cases where we have a single differential equation and found several methods to aid us in finding solutions to these equations. But what happens if we have two or more differential equations that depend on each other? For example, consider the case where

$D_{t}x(t)=3y(t)^{2}+x(t)t$

and

$D_{t}y(t)=x(t)+y(t)$

Such a set of differential equations is said to be coupled. Systems of ordinary differential equations such as these are what we will look into in this section.

First order systems

A general system of differential equations can be written in the form

$D_{t}\mathbf {x} =\mathbf {F} (\mathbf {x} ,t)$

Instead of writing the set of equations in a vector, we can write out each equation explicitly, in the form:

$D_{t}x_{1}=F_{1}(x_{1},\ldots ,x_{n},t)$

$\vdots \!\;$

$D_{t}x_{i}=F_{i}(x_{1},\ldots ,x_{n},t)$

If we have the system at the very beginning, we can write it as:

$D_{t}\mathbf {x} =\mathbf {G} (\mathbf {x} ,t)$

where

$\mathbf {x} =(x(t),y(t))=(x,y)$

and

$\mathbf {G} (\mathbf {x} ,t)=(3y^{2}+xt,x+y)$

or write each equation out as shown above.

Why are these forms important? Often, this arises as a single, higher order differential equation that is changed into a simpler form in a system. For example, with the same example,

$D_{t}x(t)=3y(t)^{2}+x(t)t$

$D_{t}y(t)=x(t)+y(t)$

we can write this as a higher order differential equation by simple substitution.

$D_{t}y(t)-y(t)=x(t)$

then

$D_{t}x(t)=3y(t)^{2}+(D_{t}y(t)-y(t))t$

$D_{t}x(t)=3y(t)^{2}+tD_{t}y(t)-ty(t)$

Notice now that the vector form of the system is dependent on t since

$\mathbf {G} (\mathbf {x} ,t)=(3y^{2}+xt,x+y)$

the first component is dependent on t. However, if instead we had

$\mathbf {H} (\mathbf {x} )=(3y^{2}+x,x+y)$

notice the vector field is no longer dependent on t. We call such systems autonomous. They appear in the form

$D_{t}\mathbf {x} =\mathbf {H} (\mathbf {x} )$

We can convert between an autonomous system and a non-autonomous one by simply making a substitution that involves t, such as y=(x, t), to get a system:

$D_{t}\mathbf {y} =(\mathbf {F} (\mathbf {y} ),1)=(\mathbf {F} (\mathbf {x} ,t),1)$

In vector form, we may be able to separate F in a linear fashion to get something that looks like:

$\mathbf {F} (\mathbf {x} ,t)=A(t)\mathbf {x} +\mathbf {b} (t)$

where A(t) is a matrix and b is a vector. The matrix could contain functions or constants, clearly, depending on whether the matrix depends on t or not.