-307.28.1 Find the general solution to the system of DE

In summary, a system of DE is a set of equations that describe how variables change over time, often used to model complex systems in science and engineering. A general solution is a set of equations that satisfies all equations in a system of DE, representing all possible solutions. To find the general solution, each individual equation must be solved for its respective variable and then combined with constants. It is important as it allows for understanding and prediction of system behavior and provides a framework for finding specific solutions. In scientific research, the general solution is used to model and analyze data, as well as design experiments in various fields.
  • #1
karush
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Find the general solution to the system of differential equations
\begin{align*}\displaystyle
y'_1&=y_1+5y_2\\
y'_2&=-2y_1+-y_2
\end{align*}
why is there a $+-y_2$ in the given
ok going to take this a step at a time... so..$A=\left[\begin{array}{c}1 & 5 \\ -2 & -1 \end{array}\right]$
then
$\left[\begin{array}{c}1-\lambda & 5 \\ -2 & -1-\lambda \end{array}\right]
=\lambda^2+9$ ?
 
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  • #2
yes, \(\displaystyle \left|\begin{array}{cc}1- \lambda & 5 \\ -2 & -1- \lambda\end{array}\right|= (1- \lambda)(-1- \lambda)+ 10= -1- \lambda+ \lambda+ \lambda^2+ 10= \lambda^2+ 9= 0\).

Now, what are the values of \(\displaystyle \lambda\)?
 
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FAQ: -307.28.1 Find the general solution to the system of DE

1. What is a system of differential equations (DE)?

A system of differential equations is a set of equations that involve one or more unknown functions and their derivatives with respect to one or more independent variables. These equations are used to model relationships between different variables in a dynamic system.

2. How do you find the general solution to a system of DE?

To find the general solution to a system of DE, you need to solve each equation in the system separately and then combine the solutions into one general solution. This can be done using various methods such as substitution, elimination, or using linear algebra techniques.

3. What is the importance of finding the general solution to a system of DE?

The general solution to a system of DE allows us to find the behavior of the system over time and understand how different variables are related to each other. This is crucial in many fields of science, such as physics, chemistry, and engineering, where systems are constantly changing and evolving.

4. Can the general solution to a system of DE be unique?

No, the general solution to a system of DE may not always be unique. This depends on the initial conditions given for the system. If the initial conditions are not specified or are not enough to uniquely determine the solution, then the general solution will have multiple solutions.

5. Are there any applications of solving systems of DE in real life?

Yes, there are many real-life applications of solving systems of DE. Some examples include modeling population growth, predicting weather patterns, analyzing chemical reactions, and designing control systems for engineering processes. Systems of DE are also used in fields such as economics, biology, and neuroscience to understand complex systems and make predictions about their behavior.

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