Math Problem Statement

For the initial value problem , d/dt [x y] = [2 1 0 2] [x y], x(0)=1, y(0)=0, what are the values of x(1) and y(1)?

Solution

We are solving a system of linear differential equations using matrix exponentiation. The system is given as:

\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix},$$ with the initial conditions: $$x(0) = 1, \quad y(0) = 0.$$ To solve this, we compute the **matrix exponential** for $$t = 1$$, apply it to the initial conditions, and find the values of $$x(1)$$ and $$y(1)$$. ### Step 1: Matrix Exponential Let $$A = \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}$$. The matrix exponential is given by: $$e^{At} = I + At + \frac{A^2 t^2}{2!} + \frac{A^3 t^3}{3!} + \dots$$ #### Eigenvalues of $$A$$ The eigenvalues of $$A$$ are both $$\lambda = 2$$, which comes from solving $$\det(A - \lambda I) = 0$$. #### General Form of $$e^{At}$$ for Jordan Block For this matrix $$A$$, which is in a Jordan block form, we use the result: $$e^{At} = e^{2t} \begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix}.$$ ### Step 2: Apply Initial Conditions The solution is: $$\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = e^{At} \begin{bmatrix} x(0) \\ y(0) \end{bmatrix}.$$ Substitute $$x(0) = 1$$ and $$y(0) = 0$$: $$\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = e^{2t} \begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix}.$$ Simplify: $$\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = e^{2t} \begin{bmatrix} 1 \\ 0 \end{bmatrix}.$$ ### Step 3: Evaluate at $$t = 1$$ At $$t = 1$$, the exponential factor is: $$e^{2t} = e^2.$$ The resulting solution is: \[ \begin{bmatrix}

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Math Problem Analysis

Mathematical Concepts

Linear Differential Equations
Matrix Exponentiation
Eigenvalues and Eigenvectors
Jordan Form

Formulas

Matrix exponential: e^{At} = I + At + (A^2 t^2) / 2! + ...
Jordan block matrix exponential: e^{At} = e^{λt} * (I + Nt), where N is the nilpotent part

Theorems

Matrix Exponential Theorem
Jordan Canonical Form Theorem

Suitable Grade Level

Undergraduate Mathematics