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Problem 4 (25 points) Consider the following initial value p…

Problem 4 (25 points) Consider the following initial value problem associated with the response of a nonlinear spring written in non-dimensional state-space form:                         du / dt = v                         dv / dt = -ω2 ( 1 + ( u / ubar )2 )                                                                          with ω = 1 and ubar = 2 The initial conditions are specified as u(0) = 1 and v(0) = -1 Perform the first two steps of a second order Runge-Kutta method with a step size of Δt = 0.10 to determine approximate solutions at time t = 0.20 Note: This step size is only for illustrative purposes. In practice, a much smaller Δt  would be selected.

Problem 3 (25 points) For each case defined below, write a s…

Problem 3 (25 points) For each case defined below, write a sample ordinary differential equation (ode), along with any required additional conditions. Include the identification of the independent and dependent variables. An initial value problem defined by a second order linear homogeneous ode with constant coefficients having a non-trivial solution. A boundary value problem involving a fourth order linear nonhomogeneous ode. A first order homogeneous nonlinear ode.

Q3:  What are some of the recent developments in technology…

Q3:  What are some of the recent developments in technology for teaching foreign/second languages? Discuss at least three of these that you would use, and present arguments for using them in light of what we know about theories of second language acquisition and how these technologies can support the acquisition process.