常微分方程初步 二阶微分方程 特征方程 (英文)常微分方程初步 二阶微分方程 特征方程 (英文).pdfVIP

Introduction to Second Order ODEs Section 3.1 in Boyce and DiPrima Wenhan Wang Department of Mathematics University of Washington 10-15-2012 Wenhan Wang Department of Mathematics University of WashingtonSecond Order()ODE 10-15-2012 1 / 15 Second Order Equations and Vibration Models Consider a spring-mass system. The object with mass m is attached to a spring with spring constant k . Let y t be the displacement of the object from the equilibrium position. Assume also that there is a resistance force proportional to the velocity dy . dt Moreover, there is an external force F t. By Newton’s Second Law, we have 2 d y dy m ky F t dt2 dt Wenhan Wang Department of Mathematics University of WashingtonSecond Order()ODE 10-15-2012 2 / 15 Classification of Second Order Equations The general form of second order equation is 2 d y dy f t y dt2 dt We say that the second order equation is linear, if 2 d y dy dt2 g t p t dt qty Another form is Pty Qty Rty Gt If Gt 0, we call the equation homogeneous. Wenhan Wang Department of Mathematics University of WashingtonSecond Order()ODE 10-15-2012 3 / 15 Initial Conditions for Second Order Equations