# Partially Decoupled System with 3 variables

• MHB
• Omzyma
In summary, this individual is having trouble getting x(t) to finish solving the system and is looking for guidance on what to do next.
Omzyma
Hello!

I have the following initial value problem:
$x' = x + 2y + 3z$
$y' = 4y + 5z$
$z' = 6z$

All I'm looking to do is find the general solution to this system, and as long as I'm doing this correctly I have these answers:

$y(t) = K_2e^{4t} + \tfrac{5K_1}{2}e^{6t}$
$z(t)=K_1e^{6t}$

But I'm having trouble getting x(t) to finish this up. Any help on what my next steps should be?

Omzyma said:
Hello!

I have the following initial value problem:
$x' = x + 2y + 3z$
$y' = 4y + 5z$
$z' = 6z$

All I'm looking to do is find the general solution to this system, and as long as I'm doing this correctly I have these answers:

$y(t) = K_2e^{4t} + \tfrac{5K_1}{2}e^{6t}$
$z(t)=K_1e^{6t}$

But I'm having trouble getting x(t) to finish this up. Any help on what my next steps should be?
I agree with your solutions to z(t) and y(t). What's the difficulty? You will have
$$\displaystyle x' = x + 2 \left ( K_2 e^{4t} \right ) + 3 \left ( \dfrac{5}{2} K_1 e^{6t} \right )$$

You've shown you can solve the homogeneous equation. So pick your particular solution: $$\displaystyle x_p = A \left ( e^{4t} \right )+ B \left ( e^{6t} \right )$$

-Dan

topsquark said:
I agree with your solutions to z(t) and y(t). What's the difficulty? You will have
$$\displaystyle x' = x + 2 \left ( K_2 e^{4t} \right ) + 3 \left ( \dfrac{5}{2} K_1 e^{6t} \right )$$

You've shown you can solve the homogeneous equation. So pick your particular solution: $$\displaystyle x_p = A \left ( e^{4t} \right )+ B \left ( e^{6t} \right )$$

-Dan
Thanks for your reply!

Picking $$\displaystyle x_p = A \left ( e^{4t} \right )+ B \left ( e^{6t} \right )$$ as my particular solution I ended up with $$\displaystyle 3A \left ( e^{4t} \right )+ 5B \left ( e^{6t} \right ) = 2K_2 \left ( e^{4t} \right ) + 3K_1 \left ( e^{6t} \right )$$

At this point do I solve for both A and B simultaneously, making either A or B as function of the other? Or should I just make like terms equal to each other and solve for A and B individually?

Omzyma said:
Thanks for your reply!

Picking $$\displaystyle x_p = A \left ( e^{4t} \right )+ B \left ( e^{6t} \right )$$ as my particular solution I ended up with $$\displaystyle 3A \left ( e^{4t} \right )+ 5B \left ( e^{6t} \right ) = 2K_2 \left ( e^{4t} \right ) + 3K_1 \left ( e^{6t} \right )$$

At this point do I solve for both A and B simultaneously, making either A or B as function of the other? Or should I just make like terms equal to each other and solve for A and B individually?
If you have $$\displaystyle 3A \left ( e^{4t} \right )+ 5B \left ( e^{6t} \right ) = 2K_2 \left ( e^{4t} \right ) + 3K_1 \left ( e^{6t} \right )$$ true for all t then $$\displaystyle 3A = 2 K_2$$ and $$\displaystyle 5B = 3 K_1$$.

-Dan

topsquark said:
If you have $$\displaystyle 3A \left ( e^{4t} \right )+ 5B \left ( e^{6t} \right ) = 2K_2 \left ( e^{4t} \right ) + 3K_1 \left ( e^{6t} \right )$$ true for all t then $$\displaystyle 3A = 2 K_2$$ and $$\displaystyle 5B = 3 K_1$$.

-Dan
So that would mean $$\displaystyle A=\tfrac{2}{3}$$ and $$\displaystyle B = \tfrac{3}{5}!$$

Thanks for your help. Just needed a little nudge I guess.

## What is a partially decoupled system with 3 variables?

A partially decoupled system with 3 variables is a mathematical model that describes the relationship between three variables, where the equations for each variable are partially independent from each other. This means that the variables are not completely isolated, but there is some level of interaction between them.

## What is the purpose of studying partially decoupled systems with 3 variables?

The study of partially decoupled systems with 3 variables allows scientists to understand the complex relationships between multiple variables in a system. This can be useful in fields such as biology, economics, and physics, where there are often multiple factors that influence a particular outcome.

## How do you determine if a system is partially decoupled with 3 variables?

To determine if a system is partially decoupled with 3 variables, you must first identify the equations for each variable and then analyze the level of independence between them. If the equations are partially independent, with some level of interaction between the variables, then the system can be considered partially decoupled.

## What are the advantages of using a partially decoupled system with 3 variables?

One advantage of using a partially decoupled system with 3 variables is that it allows for a more accurate representation of real-world systems, where variables are often interdependent. This can lead to a better understanding of how different factors impact a system and can help in making predictions and decisions.

## What are some real-world applications of partially decoupled systems with 3 variables?

Partially decoupled systems with 3 variables have a wide range of applications, including in fields such as ecology, climate science, and economics. They can be used to model the relationships between different species in an ecosystem, the interactions between different climate factors, and the effects of various economic policies on a country's economy.

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