General Question About a System of DE's

  • #1
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Main Question or Discussion Point

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)
 

Answers and Replies

  • #2
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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.
transmini said:
(I have a question current posted in the calculus section about splitting the derivative if anyone cares to answer when this technique DOES work)
 
  • #3
81
1
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!
 
  • #4
105
4
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.
 
  • #5
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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.
 
  • #6
105
4
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.
 
  • #7
33,639
5,301
As far as I know solving simple matrix exponentials is taught very early on at universities.
Yes, that's about right.
Strum said:
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.
 
  • #8
HallsofIvy
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And, not all matrices can be diagonalized.
 
  • #9
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4
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.
 

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