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Q4.2 Nearly linear oscillator II. See Q4.1; repeat this for the problem

with

Q4.3 Nearly linear oscillator III. See Q4.1; repeat this for the problem

where (> 0) is a constant (independent of and is an integrable
function; the initial condition is What condition must satisfy
if, on the basis of the evidence of your two-term expansion, the asymptotic
expansion is to be uniformly valid?
[Hint: remember that the general solution of

where is an arbitrary constant.]
Q4.4 A nearly linear oscillator with forcing. See Q4.1; repeat this for the problem

where and are constants, both independent of Find the general form
of the first term, given that Explain the consequences
of (a) (b) Also, write down the equations defining the slow
evolution of the solution in the cases: (c) (d) (See Q4.7,
4.8 for more details.)
Q4.5 A slowly varying linear oscillator. A damped, linear oscillator is described by the
equation

with and Obtain the complete description of the (general)
first term of a uniformly valid asymptotic expansion, for which you should use
the fast scale defined by (and the slow scale is
Q4.6 A coupled oscillatory system. An oscillation is described by the pair of equations

with and the initial conditions are Introduce
and use the method of multiple scales
and find, completely, the first term of a uniformly valid asymptotic expansions.