1 Review 4: L22-L28 1. (1) Use variation of parameters to find
( )1vtand ( )2vt(up to a constant) that form a particular solution to the nonhomogeneous equation, given two linear independent solutions to the corresponding homogeneous equation. (2) Find and simplify (if possible) a particular solution, ( )pyt. (3) Give the general solution to the nonhomogeneous equation. ()25512515;51,tttytyyt eytye−−′′′+−−==−=2. The equation and a non-trivial solution ( )1ytto the equation are given. Find a second linearly independent solution. ( )211240,t ytyyytt−′′′−−=()0t>. 3. A mass-spring system is governed by the differential equation 30yy′′+=, ( )01y=, ( )01y′= −The initial displacement here is 1 m to the right of the equilibrium position (positive) and the initial velocity is 1 m/sec to the left (negative). (a) Determine the equation of the motion in terms of the sine function along with its amplitude, period, natural frequency, angular frequency, phase angle (πϕπ−<≤), and phase shift, ϕω−. Specify units for each of the quantities. (b) How long does it take after release to pass through the equilibrium position? (c) Graph the equation of the motion. 4. Assume that the motion of a mass-spring system with damping is governed by 2290d ydybydtdt++=; ( )01y=, ( )01y′=ϕ.