Analytical solution of these coupled differential equations

In summary, the conversation is about a student who is struggling with a differential equations homework problem involving coupled equations. The student has tried to solve it using matrix exponentiation but is not successful. Another user suggests decoupling the equations and provides steps for solving it in a simpler way.
  • #1
gursimran
36
0

Homework Statement


I don't know how to type math equations of I have included a image file. Take initial conditiona as [0 1]

Homework Equations


attachment.php?attachmentid=36370&stc=1&d=1307816145.png


The Attempt at a Solution


No idea
 

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  • #2
"No idea" isn't really an attempt...

What do you know of differential equations? Did you already solve things like this?
 
  • #3
This is your homework problem- obviously your teacher thinks you should know how to do it yourself! Another problem is there there are several different ways to do this (finding eigenvalues and eigenvectors of a matrix or reducing to a single second order equation, or ...) and we have no idea which you have been taught.
 
  • #4
My teacher dinnt give me this. I figured out this on internet. I'm bad in differential equations. I thought these coupled diff equations can be solved simultaneously and then integration but no such luck cos of one being exactly opp of other ..
 
  • #5
gursimran said:
My teacher dinnt give me this. I figured out this on internet. I'm bad in differential equations. I thought these coupled diff equations can be solved simultaneously and then integration but no such luck cos of one being exactly opp of other ..

First read http://en.wikipedia.org/wiki/Matrix_exponential then, the interesting part is the section of "applications" which solves a problem very much like yours!
 
  • #6
Or failing matrix exponentiation, you can decouple them; Differentiate the top equation to find:
[tex]
\frac{d^{2}A}{dt^{2}}=-10\frac{dA}{dt}+50\frac{dB}{dt}
[/tex]
Now use the second equation to substitute for dB/dt, this will leave you with a B in your equation which can be gotten rid of by using the first equation. This will leave you with a second order differential equation for A.
 
  • #7
hunt_mat said:
Or failing matrix exponentiation, you can decouple them; Differentiate the top equation to find:
[tex]
\frac{d^{2}A}{dt^{2}}=-10\frac{dA}{dt}+50\frac{dB}{dt}
[/tex]
Now use the second equation to substitute for dB/dt, this will leave you with a B in your equation which can be gotten rid of by using the first equation. This will leave you with a second order differential equation for A.

oh thanks a lot, this is a much better and simple solution.
 

FAQ: Analytical solution of these coupled differential equations

What are coupled differential equations?

Coupled differential equations are a set of differential equations that are connected by a shared variable. This means that the equations cannot be solved independently and must be solved simultaneously.

Why is it important to find analytical solutions for coupled differential equations?

Analytical solutions for coupled differential equations provide a complete and exact solution to the equations, which is essential for understanding the behavior of the system being modeled. It also allows for the prediction of future states and behaviors.

What is the process for finding analytical solutions for coupled differential equations?

The process involves solving the equations simultaneously using various mathematical techniques such as substitution, elimination, or integration. It also requires a thorough understanding of the equations and their relationship to each other.

What are some common techniques used for solving coupled differential equations analytically?

Some common techniques include separation of variables, Laplace transforms, and matrix methods. These techniques can be used to simplify the equations and solve them in a step-by-step manner.

What are some real-world applications of analytical solutions for coupled differential equations?

Coupled differential equations are used to model various physical phenomena, such as chemical reactions, population dynamics, and fluid flow. Analytical solutions allow scientists to make predictions and better understand these systems, which can have practical applications in fields such as engineering, biology, and economics.

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