Initial Value Linear system of DQ's

In summary, the unique solution to the given linear system of differential equations is v(t)=C1e^(-2t)+C2e^(6t) and w(t)=C3e^(-2t)+C4e^(6t), where C1=-11, C2=10, C3=22, and C4=-25. The eigenvalues are -2 and 6, and the corresponding eigenvectors are [-1,2] and [-2,5]. To obtain the constants, the initial values are set equal to the sum of the constants for each equation, resulting in C1+C2=-1 and C3+C4=-3.
  • #1
scorpius1782
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Homework Statement


The unique solution of the linear system of differential equations
##\frac{dv}{dt}=-34v+ -16w, v(0)=-1##
##\frac{dw}{dt}=80v+ 38w, w(0)=-3##
is: (enter the smaller of the eigenvalues first, and note that all entries here are integers)
##v(t)= C_1 e^{-2t}+C_2 e^{6t}##
##w(t)= C_3 e^{-2t}+C_4 e^{6t}##

I plugged in the exponential values since they're easy to get and not my problem.
Since I already know the answer to this practice problem:
##C_1=-11##
##C_2=10##
##C_3=22##
##C_4=-25##

Homework Equations





The Attempt at a Solution



I just can't figure out how they get the constants.

The eigenvalues are -2 and 6. And the eigenvectors are [-1,2] and [-2, 5]

I thought I was suppose to set the vectors in a matrix and set equal to the initial values but this doesn't work in anyway I've tried at all. I see that C1+C2=-1 and that the other two constants add up to -3 but I have no clue how they picked out those numbers. The example we did in class only had 1 constraint and was an annoyingly simple problem.

I've done everything I can think of to extract the method but am just missing the method. I'm sure it will be very simple. If anyone can please help me I'd appreciate it.
 
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  • #2
Nevermind, solved it.
 

FAQ: Initial Value Linear system of DQ's

What is an initial value linear system of DQ's?

An initial value linear system of DQ's (differential equations) is a mathematical model that describes the relationship between a set of variables and their rates of change over time. It is called "linear" because the equations involved are linear, meaning that the variables are raised to the first power and do not include products or powers of variables.

What is the difference between initial value and boundary value problems for linear systems of DQ's?

An initial value problem for a linear system of DQ's is one in which the values of the variables are known at a specific starting time, and the goal is to find the values of the variables at a later time. A boundary value problem, on the other hand, involves finding the values of the variables at specific points or boundaries rather than at specific times.

What are the steps for solving an initial value linear system of DQ's?

The steps for solving an initial value linear system of DQ's include: 1) setting up the system of equations, 2) finding the general solution to the equations, 3) using the initial values to find the specific solution, and 4) checking the solution for accuracy.

How are initial conditions used in solving initial value linear systems of DQ's?

Initial conditions refer to the known values of the variables at a specific starting time in an initial value problem. These values are used to find the specific solution to the equations and determine the behavior of the system over time.

What are some real-life applications of initial value linear systems of DQ's?

Initial value linear systems of DQ's are commonly used in physics, engineering, and economics to model real-life systems such as population growth, chemical reactions, and electrical circuits. They are also used in fields such as medicine and biology to model the behavior of biological systems like the spread of diseases or reactions within the body.

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