Math 4B Practice Final Problems Solutions posted at clas.ucsb.edu/vince
1) Find the solution to the following differential equations 1) Find the solution to the following differential equations. If initial values are not given, find the general solution:
2) One solution to the following differential equation is given 2) One solution to the following differential equation is given. Use reduction of order to find a second linearly independent solution.
A mass of 100 kg is attached to a long spring suspended from the ceiling. When the mass comes to rest at equilibrium, the spring has been stretched 20 cm. The mass is then pulled down 40 cm below the equilibrium point and released. The system has damping, with coefficient b=500 N-s/m . Find the equation of motion for the mass. Is the system underdamped, critically damped, or overdamped? Now suppose a forcing function f(t)=100cos(7t) is applied. Find the resulting steady-state solution.
4) Find the general solution for the given systems of differential equations.
5) Find the solution to the given initial value problem.
6) Find the critical point of the given system, and determine its type and stability.
7) The motion of a damped pendulum can be described by the given differential equation. θ is the angle that the pendulum makes with the downward direction at time t. Convert this second-order equation into a system of first-order equations, then find the critical points and analyze their stability. Use the following variables, and restrict the analysis to the interval 0θ<2π.
8) The given linear system models the interaction of two species 8) The given linear system models the interaction of two species. Find the critical points of the system, and find the linearized system at each critical point. Determine the stability of each critical point.