Periodic Solution of a Certain Non-Linear Differential Equation

Extract

The equation to be solved is

where a, b, ƒ are real parameters. (1) is non-linear by virtue of the term by³. Suppose that the free end of a restoring device or spring is fixed to a mass m resting on a frictionless horizontal plane. Let the force-displacement relationship of the spring ƒ₁ = g(y), a function of the displacement y. If the mass is driven by an external force F cos ωt, the equation of motion is

or

In (l), g(y/m = ay + by³ and ƒ = F/m. If b = 0, the spring stiffness is ma, a constant, so (3) is linear. When b ≠ 0, the stiffness is not constant, but has the value mdg(y)/dy = m(a + 3by²). This increases or decreases with increase in y, according as b ≷ 0, and (1) is then non-linear. Putting a = (ω² + λ²), (1) becomes

This equation may be considered to symbolise a simple (linear) mass-spring system of free pulsatance ω, driven by a force ƒ cos ωt – (λ²y + by³).