Example of systems of the differential linear equations

• MHB
• cbarker1
In summary: I hope that helps!In summary, the process of eliminating x from the system involves multiplying the first equation by 4 and applying (D-3) to the second equation, and then adding them together to find the given expression.
cbarker1
Gold Member
MHB
Dear Everybody,

I have a question about an example:
"Solve the system:

$x'(t)=3x(t)-4y(t)+1$
$y'(t)=4x(t)-7y(t)+10t$

We write the system using the operator notation:

$(D-3)[x]+4y=1$
$-4x+(D+7)[y]=10t$

We can eliminate x from this system by adding 4 times the first equation to $(D-3)$ applied to the equation. This gives
$(16+(D-3)(D+7))[y]=4*1+(D-3)[10t]=14-30t$"

How does this step work: "We can eliminate x from this system by adding 4 times the first equation to $(D-3)$ applied to the equation."

Thanks
Cbarker1

Cbarker1 said:
Dear Everybody,

I have a question about an example:
"Solve the system:

$x'(t)=3x(t)-4y(t)+1$
$y'(t)=4x(t)-7y(t)+10t$

We write the system using the operator notation:

$(D-3)[x]+4y=1$
$-4x+(D+7)[y]=10t$

We can eliminate x from this system by adding 4 times the first equation to $(D-3)$ applied to the equation. This gives
$(16+(D-3)(D+7))[y]=4*1+(D-3)[10t]=14-30t$"

How does this step work: "We can eliminate x from this system by adding 4 times the first equation to $(D-3)$ applied to the equation."

Thanks
Cbarker1

Hi Cbarker1,

Let's break it up into smaller steps.
We multiply the first equation by 4.
And we apply $(D-3)$ to the second equation:

$4(D-3)[x]+16y=4*1$
$-4(D-3)[x]+(D-3)(D+7)[y]=(D-3)[10t]$

$16y+(D-3)(D+7)[y]=4*1+(D-3)[10t]$

And from here we find the given expression.

What is an example of systems of differential linear equations?

An example of a system of differential linear equations is the Lotka-Volterra predator-prey model, which describes the interaction between two species in an ecosystem. It consists of two differential equations, one for the population of the predator species and one for the population of the prey species.

What makes a system of differential linear equations different from other types of differential equations?

A system of differential linear equations is a set of equations that relate the rates of change of multiple variables to each other. Unlike other types of differential equations, which typically only involve one variable, systems of differential linear equations involve multiple variables and their interdependence.

How are systems of differential linear equations solved?

Systems of differential linear equations can be solved using various methods, such as substitution, elimination, or matrix methods. The specific method used depends on the complexity and form of the equations in the system.

What applications do systems of differential linear equations have in science?

Systems of differential linear equations have many applications in science, particularly in fields such as physics, engineering, and ecology. They can be used to model complex systems and predict their behavior over time, making them useful for understanding and solving real-world problems.

Are there any limitations to using systems of differential linear equations?

Although systems of differential linear equations are powerful tools for modeling and analyzing complex systems, they do have some limitations. They may not accurately represent nonlinear relationships or systems with a large number of variables. Additionally, the accuracy of the solutions may be affected by the accuracy of the initial conditions and the assumptions made in the model.

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