Solving Linear Differential Equations

Setting up the system

I’m piecing together a system of first-order linear differential equations, noting the initial conditions. This involves expressing the solutions solely in terms of real numbers.

Piecing together solution format

I’m analyzing the solution instructions, which require clarity, definition, and a structured step-by-step approach, including a heading "Subject or sub-subject" and specifics in each step.

Formulating the system

I am setting up the system in matrix form to simplify solving linear differential equations by pinpointing eigenvalues and eigenvectors.

Calculating eigenvalues

First, I calculated the determinant, set it to zero, and found the eigenvalues. Then, I computed the eigenvectors for each eigenvalue and formulated the general solution.

Mapping the journey

OK, let me see. I’m outlining the process of solving a system of first-order linear differential equations with eigenvalues and eigenvectors. This involves understanding the problem, applying formulas, and detailing the solution step-by-step.

Applying eigenvalues and eigenvectors

OK, let me see. The problem involves solving a system of first-order linear differential equations using eigenvalues and eigenvectors, with initial conditions given. This will allow us to determine functions $h_1(t)$ and $h_2(t)$.

Structuring the system

I’m piecing together the system into matrix form with $\mathbf{y}$ and $A$ to find eigenvalues and eigenvectors for the general solution.

Crunching the numbers

I’m working out the determinant of $A - \lambda I$ to find eigenvalues. Expanding and combining terms, it’s interesting to see the matrix's properties unfold.

Revisiting the equation

I’m working through the product $( -13 - \lambda )( 3 - \lambda )$. Initially, I calculated $-39 + 16 \lambda - \lambda^2$, but I’m careful to ensure accuracy with each step.

Crunching the numbers

I expanded $(-13 - \lambda)(3 - \lambda)$ and added 68. This gives a characteristic equation for eigenvalues.

Evaluating eigenvalues

I determined that the eigenvalues of matrix $A$ are $-5 \pm 2i$ by solving the quadratic equation derived from $\det(A - \lambda I) = 0$.

Breaking down the matrix

I’m subtracting $\lambda I$ from matrix $A$ to get the matrix $A - \lambda I$, which is $\begin{pmatrix} -8 - 2i & 2 \\ -34 & 8 - 2i \end{pmatrix}$.

Solving for $v_2$ in terms of $v_1$

I’m simplifying the first equation to solve for $v_2$ as $(4 + i) v_1$. Substituting this into the second equation, I verify that the left side matches the right side, confirming the relationship.

Confirming the equation

Interestingly enough, establishing that $v_2 = (4 + i) v_1$ aligns with the equation $34 v_1 = 34 v_1$.

Constructing the general solution

I’m working on building the general solution by combining real and imaginary parts of eigenvectors, forming $\mathbf{y}(t) = e^{\alpha t} [ \mathbf{C_1} \cos(\beta t) + \mathbf{C_2} \sin(\beta t) ]$.

Adjusting my approach

I’m piecing together the general solution, ensuring $\alpha = -5$ and $\beta = 2$, to find $\mathbf{y}(t)$.

Mapping out the solution

I'm thinking about expressing the solution for each component, considering the constants $c_1$ and $c_2$. This involves trigonometric terms and vector constants to ensure precision.

Formulating the solution

I’m incorporating the vectors $\mathbf{p}$ and $\mathbf{q}$ into the general solution, $\mathbf{y}(t)$, by combining constants and adjusting for initial conditions.

Weighing the options

I’m mapping out whether to use $\mathbf{y}(t) = e^{-5 t} [ \mathbf{p} \cos(2 t) - \mathbf{q} \sin(2 t) ]$ or explore $y_1(t)$ and $y_2(t)$ directly.

Rewriting the solution

I’m thinking through a new way to express $\mathbf{y}(t)$ using constants $\mathbf{A}$ and $\mathbf{B}$, and providing simpler forms for $y_1(t)$ and $y_2(t)$ with cosine and sine functions, while noting their interdependence.

Expressing the general solution

I’m working through expressing $y_2(t)$ in terms of $y_1(t)$ using standard theory for systems with complex eigenvalues and eigenvectors. This involves exponential functions and trigonometric terms.

Breaking down the solution

I’m combining constants into $c_1$ and $c_2$, then expanding $\mathbf{y}(t)$ component-wise to see how $\mathbf{p}$ and $\mathbf{q}$ combine with sines and cosines.

Breaking down the processes

I’m decomposing the expressions for $y_1(t)$ and $y_2(t)$, combining exponential and trigonometric elements to streamline and group terms effectively.

Setting initial values

I determined $y_1(0)$ and $y_2(0)$, finding $c_1 = 2$. Now, I'll solve $y_2(0)$ to pinpoint $c_2$.

Breaking down the equation

First, I substitute $c_1 = 2$ into $y_2(0) = 4 c_1 + c_2 = 13$, arriving at $c_2 = 5$. This leads me to values for $c_1$ and $c_2$, which I'll use to formulate the solutions.

Breaking down the solutions

I derived two distinct solutions for $y_1(t)$ and $y_2(t)$ by substituting specific values and simplifying each term. The results included exponential decay modulated by trigonometric functions.

Subject: Solving a System of First-Order Linear Differential Equations Using Eigenvalues and Eigenvectors

Step 1: Understanding the Problem

Given:

We have the system of differential equations:

We need to find: The explicit functions $y_1(t)$ and $y_2(t)$ that satisfy both the differential equations and the initial conditions.

Explanation:

To solve this system, we'll use the method of eigenvalues and eigenvectors. By expressing the system in matrix form and finding the eigenvalues and eigenvectors of the coefficient matrix, we can construct the general solution.

Step-by-Step Solution:

1. Write the System in Matrix Form:

Let $\mathbf{y} = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$. Then the system becomes:

where $A = \begin{pmatrix} -13 & 2 \\ -34 & 3 \end{pmatrix}$.

2. Find the Eigenvalues of Matrix $A$:

Compute the characteristic equation $\det(A - \lambda I) = 0$:

Expanding:

Solving the quadratic equation:

3. Find the Corresponding Eigenvectors:

For $\lambda = -5 + 2i$, solve $(A - \lambda I)\mathbf{v} = 0$:

which simplifies to:

Let $\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$. From the first row:

Thus, the eigenvector is $\mathbf{v} = v_1 \begin{pmatrix} 1 \\ 4 + i \end{pmatrix}$.

Step 2: Constructing the General Solution

Explanation:

With complex eigenvalues and eigenvectors, the solution involves exponential, sine, and cosine functions.

Step-by-Step Solution:

1. Express the Eigenvector in Terms of Real and Imaginary Parts:

Separate $\mathbf{v}$ into real and imaginary parts:

2. Write the General Solution:

The general solution is:

3. Express $y_1(t)$ and $y_2(t)$ Separately:

• For $y_1(t)$: $$y_1(t) = e^{-5t} \left[ c_1 \cos(2t) + c_2 \sin(2t) \right].$$
• For $y_2(t)$: $$y_2(t) = e^{-5t} \left[ (4 c_1 + c_2) \cos(2t) + (4 c_2 - c_1) \sin(2t) \right].$$

4. Apply Initial Conditions to Solve for Constants:

At $t = 0$:

• $y_1(0) = c_1 = 2$.
• $y_2(0) = 4 c_1 + c_2 = 13$.

Solving:

5. Write the Final Expressions for $y_1(t)$ and $y_2(t)$:

• $y_1(t) = e^{-5t} \left[ 2 \cos(2t) + 5 \sin(2t) \right]$.
• $y_2(t) = e^{-5t} \left[ 13 \cos(2t) + 18 \sin(2t) \right]$.

Conclusion:

Therefore, the solutions to the differential equations are: