工程數學(一)

臺北科技大學 115-1 學期課程，課號 362735。

課程大綱

本課程全面介紹常微分方程（ODEs）與積分轉換，此兩者皆為工程應用中不可或缺的基礎數學工具。課程內容主要分為兩大核心部分：（1）一階、二階及高階線性常微分方程的解析技巧與數學建模，並著重於如何從實際的工程與物理問題中推導出這些微分方程。（2）拉氏轉換（Laplace Transform）的理論與應用，特別聚焦於其高效求解高階非齊次微分方程、不連續訊號輸入以及聯立常微分方程組的能力 This course provides a comprehensive introduction to ordinary differential equations (ODEs) and integral transforms, which are fundamental mathematical tools for engineering applications. The curriculum is divided into two primary components: (1) The analytical techniques and modeling of first-order, second-order, and higher-order linear ordinary differential equations, emphasizing how to derive these equations from practical engineering and physical problems. (2) The theory and application of the Laplace transform, focusing on its capability to solve higher-order nonhomogeneous differential equations, discontinuous inputs, and systems of ODEs efficiently.

課程進度

Weeks 1–3: First-Order Ordinary Differential Equations. Weeks 4–5: First-Order Linear ODEs, Bernoulli Equations, and Engineering Applications. Weeks 6–7: Second-Order Homogeneous Linear ODEs. Week 8: Midterm Exam Weeks 9–10: Second-Order Nonhomogeneous Linear ODEs. Week 11: Higher-Order Linear ODEs and Euler-Cauchy Equations Weeks 12–13: Laplace Transform I: Fundamental Theory, Inverse Transforms, and Shifting Theorems Weeks 14–15: Laplace Transform II: Discontinuous Inputs (Step/Delta Functions), Convolution, and Solving ODE Systems. Week 16: Final Exam

評量方式

Homework & Quiz & Attendance (30%) Midterm exam (30%) Final exam (30%) Report (10%)

教材

【遵守智慧財產權觀念，請使用正版教科書，不得使用非法影印教科書】 使用外文原文書：是 Advanced Engineering Mathematics Erwin Kreyszig, Wiley 11th edition.