The initial-value problemmx″ + cx′ + kx = F sinωt; x(0) = x0, x′(0) = x′0corresponding to...

The initial-value problem

mx″ + cx′ + kx = F sinωt; x(0) = x0, x′(0) = x′0

corresponding to a damped mechanical oscillator driven by the force F sin ωt, contains seven parameters: m,c,k,F,ω,x0, x′0. Nondimensionalize How many parameters are present in the nondimensionalized system?

Which is now exact. Thus

and

So

and

The latter gives

A′(y) = −e− y

so

A(y) = −ey + B.

Thus,

F(x, y) = xe3y + A(y) = xe3y + (−ey + B) = constant

or

xe3y − ey = C,(34)

where C is an arbitrary constant; (34) is the general solution of (29), in implicit form.

COMMENT. Can we solve (34) for y?If we let ey = z, then (34) is the cubic equation xz3 − z = C in z, and there is a known solution to cubic equations. If we can solve for z, then we have y as y = ln z. However, the solution of that cubic equation (as can be obtained using the Maple solve command) is quite a messy expression.