General Question About a System of DE's

[transmini]

We were looking at some examples of systems of differential equations in class a couple days ago, and some of them were simple looking systems. One such was

$\frac{dr}{dt} = 4j$
$\frac{dj}{dt} = r$

Why is it that we can't treat this as a scenario where both sides of each equation could just be integrated immediately or something like in physics where the first equation would become $dr = 4jdt$ and then integrate? (I have a question current posted in the calculus section about splitting the derivative if anyone cares to answer when this technique DOES work)

 

[Mark44]

We were looking at some examples of systems of differential equations in class a couple days ago, and some of them were simple looking systems. One such was

$\frac{dr}{dt} = 4j$
$\frac{dj}{dt} = r$

Why is it that we can't treat this as a scenario where both sides of each equation could just be integrated immediately or something like in physics where the first equation would become $dr = 4jdt$ and then integrate?

If j and r were constants, you could do this, but the implication is that both j and r are unknown functions of t.
It's exactly the same as trying to evaluate this integral: $\int f(x) dx$, without knowing anything about f.

Also, if this were your system...
$\frac{dr}{dt} = 4r$
$\frac{dj}{dt} = j$

... then you could separate each equation like so:
$dr = 4rdt \Rightarrow \frac{dr}r = 4 dt \Rightarrow \int \frac{dr}{r} = 4 \int dt \Rightarrow \ln|r| = 4t + C$
Exponentiating both sides would yield $|r| = Ae^{4t}$, where $A = e^C$.
The other equation could be solved in a like manner.

(I have a question current posted in the calculus section about splitting the derivative if anyone cares to answer when this technique DOES work)

 

[transmini]

If j and r were constants, you could do this, but the implication is that both j and r are unknown functions of t.
It's exactly the same as trying to evaluate this integral: $\int f(x) dx$, without knowing anything about f.

Also, if this were your system...
$\frac{dr}{dt} = 4r$
$\frac{dj}{dt} = j$

... then you could separate each equation like so:
$dr = 4rdt \Rightarrow \frac{dr}r = 4 dt \Rightarrow \int \frac{dr}{r} = 4 \int dt \Rightarrow \ln|r| = 4t + C$
Exponentiating both sides would yield $|r| = Ae^{4t}$, where $A = e^C$.
The other equation could be solved in a like manner.

Perfect, that explanation helped a lot. Thanks!

 

[Strum]

Well couldn't you rewrite on vector form and then integrate to get a matrix exponential?
Set $v =(r,j)$ such that we get the equation
$\frac{d}{dt}v = (0,4;1,0)v$,
and then integrate.

 

[Mark44]

Well couldn't you rewrite on vector form and then integrate to get a matrix exponential?
Set $v =(r,j)$ such that we get the equation
$\frac{d}{dt}v = (0,4;1,0)v$,
and then integrate.

It's a lot more complicated than this. If you integrate as you suggest you get a solution that contains this factor:
$$e^{\begin{bmatrix} 0 & 4 \\ 1 & 0 \end{bmatrix}~t}$$
Expanding that exponential is difficult, as it involves an infinite number of terms, each requiring a higher power of that matrix.

The usual technique involves finding a diagonal matrix D that is similar to the matrix above; that is, a matrix $D = P^{-1}AP$, where the eigenvalues of matrix A appear on the diagonal in D, and the eigenvectors of A are the columns of P. For more information, see any differential equations textbook that has a section on solution of systems of diff. equations using matrix diagonalization.

 

[Strum]

As far as I know solving simple matrix exponentials is taught very early on at universities. I would not call it more complicated but certainly less used.

 

[Mark44]

As far as I know solving simple matrix exponentials is taught very early on at universities.

Yes, that's about right.

I would not call it more complicated but certainly less used.

It's quite complicated in comparison to matrix addition and multiplication, and even finding the inverse of a matrix. For diagonalization, you need to find the eigenvalues, find the eigenvectors, form a matrix whose columns are the eigenvectors, find the inverse of that matrix, and so on.

 

[HallsofIvy]

And, not all matrices can be diagonalized.

 

[Strum]

But you can still find the matrix exponential. At any rate. The question was why can't we just integrate both sides and my answer is: you can but you will have to solve something different instead.