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}Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Solving Coupled Differential Equations Using Eigenvalues: dx/dt = 2x - 8y, dy/dt = x + 6y
Solve the IVP: Matrix Differential Equation with Eigenvalues
Solve a Differential Equation System with Initial Conditions x(0) = 1 and y(0) = 2
Solve the Linear System of Differential Equations with Eigenvalues and Eigenvectors
Solution and Stability of a Linear System of Differential Equations