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- Thread starter gursimran
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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.

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What do you know of differential equations? Did you already solve things like this?

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HallsofIvy

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gursimran

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gursimran said:

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!

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hunt_mat

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[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.

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gursimran

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hunt_mat said:

[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.

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.

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.

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.

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.

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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