Question

Solve the initial value problem. $$ \mathbf{y}^{\prime}=\left[\begin{array}{lll} -2 & 2 & 1 \\ -2 & 2 & 1 \\ -3 & 3 & 2 \end{array}\right] \mathbf{y}, \quad \mathbf{y}(0)=\left[\begin{array}{r} -6 \\ -2 \\ 0 \end{array}\right] $$

Short answer

Based on the given matrix and initial value, the solution to the initial value problem for the system of first-order linear differential equations is: $$\mathbf{y}(t) = 7e^{7t}\begin{bmatrix} 0 \\ 1 \end{bmatrix} -e^{-7t}\begin{bmatrix} 1 \\ 0 \end{bmatrix}$$

Step-by-step solution

Find the eigenvalues of the given matrix

To find the eigenvalues, we need to solve the characteristic equation, which is obtained by setting the determinant of the matrix minus the eigenvalue times the identity matrix to zero: $$\begin{vmatrix} -7 - \lambda & 4 \\ -6 & 7 - \lambda \end{vmatrix} = 0$$ Expanding the determinant, we get: \((-7 - \lambda)(7-\lambda) - (-6)4 = \lambda^2 - 49\) Solving for the eigenvalues, we have: \(\lambda_1 = 7\) \(\lambda_2 = -7\)

Find the eigenvectors corresponding to the eigenvalues

We start by finding the eigenvector corresponding to \(\lambda_1 = 7\). Substitute the eigenvalue into the matrix equation \((A - \lambda I)\mathbf{v} = 0\): $$ \begin{bmatrix} -7 - 7 & 4 \\ -6 & 7 - 7 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = 0 $$ This simplifies to the system of linear equations: $$-14v_1 + 4v_2 =0$$ $$-6v_1 =0$$ Solving for \(v_1\) and \(v_2\) gives \(v_{11} = 0\) and \(v_{12} = 1\). This means, the eigenvector corresponding to \(\lambda_1\) is \(\mathbf{v}_1=\begin{bmatrix} 0 \\ 1 \end{bmatrix}\). Next, find the eigenvector corresponding to \(\lambda_2 = -7\): $$ \begin{bmatrix} -7 - (-7) & 4 \\ -6 & 7 - (-7) \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = 0 $$ This simplifies to the system of linear equations: $$4v_2 =0$$ $$-6v_1 + 14v_2 =0$$ Solving for \(v_1\) and \(v_2\) gives \(v_{21} = 1\) and \(v_{22} = 0\). This means, the eigenvector corresponding to \(\lambda_2\) is \(\mathbf{v}_2=\begin{bmatrix} 1 \\ 0 \end{bmatrix}\).

Set up and solve the general solution of the system

Based on the eigenvalues and eigenvectors found in steps 1 and 2, the general solution of the given system is: $$\mathbf{y}(t) = c_1e^{7t}\begin{bmatrix} 0 \\ 1 \end{bmatrix} + c_2e^{-7t}\begin{bmatrix} 1 \\ 0 \end{bmatrix}$$ Now, we apply the initial condition \(\mathbf{y}(0)=\left[\begin{array}{l}-1\\\ 7\end{array}\right]\) to determine the constants \(c_1\) and \(c_2\): $$\begin{bmatrix} -1 \\ 7 \end{bmatrix} = c_1\begin{bmatrix} 0 \\ 1 \end{bmatrix} + c_2\begin{bmatrix} 1 \\ 0 \end{bmatrix}$$ This gives us the equations \(c_2 = -1\) and \(c_1 = 7\). Substituting the constants back into the general solution gives the particular solution: $$\mathbf{y}(t) = 7e^{7t}\begin{bmatrix} 0 \\ 1 \end{bmatrix} -e^{-7t}\begin{bmatrix} 1 \\ 0 \end{bmatrix}$$ This is the solution to the initial value problem.

Key concepts

Eigenvalues

Eigenvalues are fundamental in understanding the behavior of linear transformations. They are scalars associated with a square matrix that, when multiplied by an eigenvector, don't alter its direction. In simple terms, finding the eigenvalues of a matrix involves solving the characteristic polynomial equation derived by subtracting a scalar \(\lambda\) from the diagonal entries of the matrix.

• Mathematically, this is expressed as: \det(A - \lambda I) = 0\.
• Where \(A\) is the matrix, \(I\) is the identity matrix, and \(\lambda\) represents the eigenvalues.

Determining eigenvalues is critical for systems of differential equations because they help analyze the stability and dynamics of solutions.

Eigenvectors

Eigenvectors work hand-in-hand with eigenvalues to provide deep insights into matrix transformations. An eigenvector corresponding to an eigenvalue \(\lambda\) is a non-zero vector that, when multiplied by the matrix \(A\), results in a scaled version of itself, specifically \(Av = \lambda v\).

• Once the eigenvalues are found, we solve \( (A - \lambda I)\mathbf{v} = 0 \) for non-trivial vectors \(\mathbf{v}\).
• The solutions to this equation indicate how the initial movement described by the vector \(\mathbf{v}\) scales under the transformation.

In real-world applications, eigenvectors often represent directions of action, major modes of oscillation, or principal components in systems analysis.

Linear Differential Equations

Linear differential equations are equations involving unknown functions and their derivatives. These equations highlight relationships where the unknown function's exponent is one, maintaining a linear relationship.

• In the context of matrices, they are expressed as \(\mathbf{y}' = A\mathbf{y}\), where \(\mathbf{y}\) is the vector of unknown functions.
• The linearity ensures the principle of superposition applies, meaning the sum of solutions is also a solution.

Linear differential equations simplify solving systems because their solutions can be constructed using basic techniques, such as finding eigenvalues and eigenvectors. They also often model real-world phenomena where effects are proportional to causes, such as in mechanical vibrations or electrical circuits.

System of Differential Equations

A system of differential equations comprises multiple equations involving derivatives of several functions. Such systems effectively model complex interactions between different variables.

• In matrix form, this can be expressed as a vector equation \(\mathbf{y}' = A\mathbf{y}\), where \(\mathbf{y}\) is a vector of unknown functions differentiated and \(A\) is a matrix of coefficients.
• These systems frequently use initial conditions to find unique solutions, which in turn identify how a system evolves over time.

Analyzing these systems with matrices makes it easy to apply concepts of linear algebra, like eigenvalues and eigenvectors, to find solutions. Interpretation of these solutions then offers insights into the dynamics of the system, allowing predictions of future behavior under given conditions.