MTH 253 Calculus (Other Topics) Chapter 9 – Mathematical Modeling with Differential Equations Section 9.4 – Second-Order Linear Homogeneous Differential Equations; The Vibrating String Copyright © 2006 by Ron Wallace, all rights reserved.

The Study of Differential Equations The development and study of methods for solving differential equations and IVP’s. Categorizing Differential Equations Explicit methods  Find solutions and general solutions Numerical methods  Find a set of ordered pairs that approximate the solution of an IVP. Applications From 9.1 Order

Second-Order DE OR

Second-Order Linear DE OR

Second-Order Linear Homogeneous DE OR

Second-Order Linear Homogeneous DE with Constant Coefficients OR

Second-Order Linear Homogeneous DE with Constant Coefficients What functions have first and second derivatives such that a linear combination of these can give zero? Look at the derivatives of these functions. (first & second order)

Second-Order Linear Homogeneous DE with Constant Coefficients Auxiliary Equation

Second-Order Linear Homogeneous DE with Constant Coefficients The General Solution Requires two arbitrary constants (2 nd order DE) Linear combinations of solutions are solutions. Consider two solutions: y 1 and y 2 Linear combination: c 1 y 1 + c 2 y 2 Therefore, need two linear independent solutions. Linear Independent = one is not a constant multiple of the other

Second-Order Linear Homogeneous DE with Constant Coefficients Consider Three Cases The roots are distinct real numbers. There is only one real root. The roots are complex (a±bi).

Second-Order Linear Homogeneous DE with Constant Coefficients Consider Three Cases The roots are distinct real numbers (m 1 & m 2 ).

Second-Order Linear Homogeneous DE with Constant Coefficients Consider Three Cases There is only one real root (m).

Second-Order Linear Homogeneous DE with Constant Coefficients Consider Three Cases The roots are complex (a±bi).

Second-Order Linear Homogeneous DE with Constant Coefficients IVP ’s require two conditions Why? Two constants must be determined.