Solving Matrix of Differential Equations With Initial Values

In summary, the problem is asking to solve a matrix of differential equations with given initial values. The eigenvalues are found to be -4 and 3, with eigenvectors of 1 and 1 for -4 and 1 and 3/2 for -3. The equations are then written out and solved for the constants C1 and C2. However, there was a mistake in entering the answer into the online homework program, causing the incorrect result to be marked as wrong.
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Homework Statement


Solve the matrix of differential equations with given initial values.

dx/dt= (-6 2) x
(-3 -1)

Initial value is x(0) = -2
-5

Homework Equations



(A-λI)=o


The Attempt at a Solution



My eigenvalues are -4 and 3

My eigenvectors for -4 are 1 and 1 and the eigenvectors for -3 are 1 on the top row and 3/2 on the bottom.

I write out the equations to look like:

C1e^-4t + C2e^-3t
C1e^-4t + 3/2C2e^-3t

I have IVs of -2 on the top row and -5 on the bottom row. To plug these in correctly,do I use the top row values for the top row equation and same for the bottom? If so, I'm getting C1=4 and C2=-6 but this is wrong in our online homework program.
 
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I figured out what I was doing wrong. It was a matter of entering the answer into the online homework program incorrectly, not so much that the answers were wrong.
 

What is a matrix of differential equations with initial values?

A matrix of differential equations with initial values is a set of equations that describe the relationship between multiple variables and their derivatives. The initial values are the known values of the variables at a specific starting point, which are used to solve the equations.

What is the purpose of solving a matrix of differential equations with initial values?

The purpose of solving a matrix of differential equations with initial values is to find the values of the variables at any given point in time. This allows scientists to make predictions and model the behavior of systems with changing variables.

What techniques are used to solve a matrix of differential equations with initial values?

The most common techniques used to solve a matrix of differential equations with initial values are the matrix exponential method, the Laplace transform method, and numerical methods such as Euler's method or the Runge-Kutta method.

What are some real-world applications of solving a matrix of differential equations with initial values?

Solving a matrix of differential equations with initial values has many real-world applications, such as predicting population growth, modeling the spread of diseases, analyzing chemical reactions, and simulating weather patterns. It is also commonly used in engineering for designing and optimizing control systems.

What challenges may arise when solving a matrix of differential equations with initial values?

Some challenges that may arise when solving a matrix of differential equations with initial values include the complexity of the equations, the sensitivity to initial conditions, and the potential for errors in the input data. It may also be difficult to find closed-form solutions for some systems, requiring the use of numerical methods.

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