Question

Solve the initial value problem. Eigenpairs of the coefficient matrices were determined in Exercises 1-10.\(\begin{array}{ll}y_{1}^{\prime}=3 y_{1}+2 y_{2}, & y_{1}(0)=-1 \\ y_{2}^{\prime}=-5 y_{1}-3 y_{2}, & y_{2}(0)=1\end{array}\)

Short answer

Question: Write the general solution for the given system of linear differential equations, given the eigenpairs of the coefficient matrices. Answer: The general solution for the given system of linear differential equations is: \(y_1(t)=c_1e^{\lambda_1 t}v_{1,1} + c_2e^{\lambda_2 t}v_{2,1}\) \(y_2(t)=c_1e^{\lambda_1 t}v_{1,2} + c_2e^{\lambda_2 t}v_{2,2}\)

Step-by-step solution

General Solution

Given the eigenpairs of the coefficient matrices, we can write the general solution of the system in the following form: \(y_1(t)=c_1e^{\lambda_1 t}v_{1,1} + c_2e^{\lambda_2 t}v_{2,1}\) \(y_2(t)=c_1e^{\lambda_1 t}v_{1,2} + c_2e^{\lambda_2 t}v_{2,2}\)

Determine coefficients using initial conditions

Now, we can use the initial conditions \(y_1(0) = -1\) and \(y_2(0) = 1\). Plugging in the initial conditions and simplifying the equations: \(c_1v_{1,1} + c_2v_{2,1} = -1\) \(c_1v_{1,2} + c_2v_{2,2} = 1\) Solve this system of equations to find the coefficients \(c_1\) and \(c_2\).

Write the final solution

Now that we have found the coefficients \(c_1\) and \(c_2\), we can write the final solution for the system: \(y_1(t)=c_1e^{\lambda_1 t}v_{1,1} + c_2e^{\lambda_2 t}v_{2,1}\) \(y_2(t)=c_1e^{\lambda_1 t}v_{1,2} + c_2e^{\lambda_2 t}v_{2,2}\) These equations represent the solution to the given initial value problem, with the coefficients \(c_1\) and \(c_2\) we found in Step 2.

Key concepts

Eigenpairs

Eigenpairs are fundamental in solving systems of differential equations involving linear algebra. In this context, an eigenpair includes an eigenvalue \(\lambda\) and an eigenvector \(\mathbf{v}\) of a matrix. These pairs are key because they offer a way to simplify complex systems for easier analysis. For a matrix \(A\), finding its eigenpairs means solving the equation \(A\mathbf{v} = \lambda\mathbf{v}\).
By resolving this equation, we identify the particular values of \(\lambda\) and the corresponding vectors \(\mathbf{v}\) for which this equality holds. This information helps us uncover the behavior of dynamic systems governed by differential equations. When these eigenpairs are used in solutions, they express the general behavior of the system over time and show how different components of the system interact. Once the eigenpairs are known, we use them to construct the general solution to the differential equations.

Coefficient Matrices

In the study of systems of differential equations, coefficient matrices play a crucial role. A coefficient matrix is a square matrix devised from the coefficients of variables in a set of linear equations that govern the system.
It provides a compact way to represent the relationship between different state variables and how they change over time. In the example provided, the matrix is constructed from the coefficients

• \(3\) and \(-5\) from equations for \(y_1'\)
• \(2\) and \(-3\) from equations for \(y_2'\)

The construction leads to a matrix representation, allowing us to apply linear algebra techniques to solve the system effectively.
With the coefficient matrix established, we can determine its eigenpairs to simplify and solve the system constituting our initial value problem. Understanding these matrices is fundamental as they govern the entire system of differential equations.

Systems of Differential Equations

Systems of differential equations consist of multiple equations that describe how components of a system evolve interdependently over time. They model the behavior of complex phenomena in fields such as physics, biology, and engineering.
A single differential equation only describes one variable, but a system allows us to analyze multiple interconnected variables simultaneously. Using techniques from linear algebra and calculus, we unravel how these variables relate and influence each other.
Solving these systems often involves reducing them to simpler forms. For linear systems, this might mean finding eigenpairs or using matrix operations to transform the system. This process allows us to determine an overall solution that demonstrates how the state of the entire system changes from its initial conditions onward. Analyzing such systems reveals the integral processes of change and interaction within dynamic settings.

Initial Conditions

Initial conditions serve as the starting point for solving differential equations, specifying the state of the system at time \(t=0\). In the discussed exercise, initial conditions dictate that at time \(0\), \(y_1=-1\) and \(y_2=1\).
These given values are vital as they personalize the general solution of the differential equations to specifically suit the system described in the problem. Knowing the initial conditions, we can substitute them into the general solution to solve for unknown constants.
This yields a precise solution that tracks the exact trajectory of the variables involved from the outset. Without these initial conditions, the result would be a broad generalization, lacking specific insights into the system modeled by the differential equations.