Second order equations as coupled systems

Higher Level

Here is a mass on a spring: x″ + ax′ + 4x = 0. Move the damping and watch both pictures change together. The top shows x against time; the bottom is the phase portrait from 5.17.

a = 1.00

–eigenvalues

–phase portrait

–in words

Zero damping never settles. A little damping wobbles in. A lot of damping slides home without crossing once.

The substitution

Let y = dxdt. Then dydt is x″, and the second order equation becomes two first order ones.

x″ + ax′ + bx= 0 dxdt= y that is the definition, and it is the first equation dydt= −bx − ay just the original equation rearranged

That is a coupled system with matrix [[0, 1], [−b, −a]], so everything from 5.17 applies, and Euler from 5.16 works on it unchanged.

Worked example

Write x″ + 3x′ + 2x = 0 as a coupled system and classify it.

system= [[0, 1], [−2, −3]] trace= −3, determinant = 0×(−3) − 1×(−2) = 2 λ= −3 ± √(9 − 8)2 = −1 and −2 real, distinct, both negative

A stable node. The mass returns to rest without oscillating at all, which is what "overdamped" means. Set the slider above to a = 3 and look.

What the damping does

a	Eigenvalues	Phase portrait	The mass
0	±2i	centre	oscillates forever
small	complex, negative real part	stable spiral	wobbles, fading out
large	real, both negative	stable node	slides home, no wobble

The changeover happens when a² = 4b, which is when the square root turns real. With b = 4 that is a = 4. Below it the system turns, above it the system does not, and the phase portrait says so at a glance.

In an examination the second order equation will be given to you. You are not expected to derive it from a physical situation, only to convert it, solve it numerically or classify it.

Your turn

1. Written as a coupled system, x″ + 5x′ + 6x = 0 has matrix [[0, 1], [p, q]]. What is p?

2. x″ + 2x′ + 5x = 0 gives eigenvalues −1 ± 2i. The motion is:

an oscillation that never fades. an oscillation that fades away. a slide back to rest with no oscillation. a growing oscillation.

3. With no damping at all, x″ + 4x = 0, the phase portrait is:

a stable node. a centre, with closed loops. a saddle point. a stable spiral.

Where the marks go

Writing the substitution down, y = dx/dt, and then both equations, is the method mark. Jumping straight to a matrix skips it.

Watch the signs when rearranging. x″ = −bx − ay, so the bottom row is minus b and minus a, not b and a.

Classify and then say what it means physically. "Stable node, so the mass returns to rest without oscillating" is the complete answer.

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