Consider the (scalar) ODE '(t) = ax(t) with initial data (0) = IO.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. Consider the (scalar) ODE \( x'(t) = ax(t) \) with initial data \( x(0) = x_0 \). Show that \( x(t) = x_0 e^{at} \) solves the preceding equation.
Let \( B \) be an \( n \times n \) matrix, with real or complex entries, and \( I \) the \( n \times n \) identity matrix. The matrix exponential is the \( n \times n \) matrix given by the (convergent) sum
\[
e^B := I + B + \frac{B^2}{2} + \frac{B^3}{6} + \cdots = \sum_{k=0}^{\infty} \frac{B^k}{k!}.
\]
Similarly, we can define an \( n \times n \) matrix which is a function of \( t \) by
\[
e^{tB} := I + tB + t^2 \frac{B^2}{2} + t^3 \frac{B^3}{6} + \cdots = \sum_{k=0}^{\infty} \frac{t^k B^k}{k!}.
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Transcribed Image Text:1. Consider the (scalar) ODE \( x'(t) = ax(t) \) with initial data \( x(0) = x_0 \). Show that \( x(t) = x_0 e^{at} \) solves the preceding equation.
Let \( B \) be an \( n \times n \) matrix, with real or complex entries, and \( I \) the \( n \times n \) identity matrix. The matrix exponential is the \( n \times n \) matrix given by the (convergent) sum
\[
e^B := I + B + \frac{B^2}{2} + \frac{B^3}{6} + \cdots = \sum_{k=0}^{\infty} \frac{B^k}{k!}.
\]
Similarly, we can define an \( n \times n \) matrix which is a function of \( t \) by
\[
e^{tB} := I + tB + t^2 \frac{B^2}{2} + t^3 \frac{B^3}{6} + \cdots = \sum_{k=0}^{\infty} \frac{t^k B^k}{k!}.
\]
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