## Week 24

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=erx, where r is a constant
• Handle the three cases of solutions to the characteristic equation ar2 + br + c = 0: real and unequal roots, real and equal roots, and complex roots
• 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