** 24.T Second-order homogeneous linear differential equations
**

** Reference: ** Stewart 15.5, 6.7

** Keywords: ** Second-order equations, characteristic equation,
homogeneous equations, hyperbolic sine and cosine functions

** Learning Objectives: ** The student will be able to:

- Recognize second-order linear differential equations
- Solve equations of the form
*ay" + by' + cy*= 0, where*a, b, c*are constants, by looking for solutions of the form*y=e*, where^{rx}*r*is a constant - Handle the three cases of solutions to the characteristic equation
*ar*= 0: real and unequal roots, real and equal roots, and complex roots^{2}+ br + c - Use linear combinations of solutions to obtain the general solution
- Solve initial value problems involving these differential equations
- Express solutions using hyperbolic sine and cosine, when appropriate

** 24.R Second-order nonhomogeneous linear equations;
Applications of second-order ODEs**

** Reference:** Stewart 15.6, 15.7

** Keywords: ** Nonhomogeneous, method of undetermined coefficients,
harmonic motion, damped and undamped vibrations, spring-mass system,
RLC series circuit

** Learning Objectives:** The student will be able to:

- Solve nonhomogeneous linear differential equations by the method of undetermined coefficients
- Solve initial value problems for second-order nonhomogeneous equations
- Set up and solve initial value problems for vibrating spring-mass systems, with and without damping, and with and without an applied force.
- Set up and solve initial value problems for series RLC electrical circuits