Topic 2 of 6

Second-order linear equations

Step 20 · Differential equations topics

Solve constant-coefficient second-order equations with the characteristic equation, and use them to describe springs that oscillate, die away or resonate.

The characteristic equation

For ay″ + by′ + cy = 0, try $y = e^{rx}$ . Each derivative brings down a factor r, so the equation becomes ar² + br + c = 0, the characteristic equation. Its roots decide the solutions.

Roots	General solution
Two real roots	$c_1e^{r_1x} + c_2e^{r_2x}$
One repeated root r	$(c_1 + c_2x)e^{rx}$
Complex roots α ± βi	$e^{\alpha x}(c_1\cos\beta x + c_2\sin\beta x)$

Example: Real roots

Solve y″ − 5y′ + 6y = 0.

r² − 5r + 6 = (r − 2)(r − 3), so r = 2 and r = 3.

Answer $y = c_1e^{2x} + c_2e^{3x}$

Example: Complex roots

Solve y″ + 4y = 0 with y(0) = 1 and y′(0) = 0.

r² + 4 = 0 gives r = ±2i.
$y = c_1\cos 2x + c_2\sin 2x$ , and the starting values give c₁ = 1 and c₂ = 0.

Answery = cos 2x

Damped oscillations

A mass on a spring with friction obeys my″ + by′ + ky = 0. The damping b decides the motion: complex roots mean it oscillates while dying away, and real roots mean it creeps back without oscillating.

Underdamped motion: the spring oscillates inside a shrinking envelope (dashed) and settles at 0.

b² − 4mk	Behavior
Negative	Underdamped: oscillates and dies away
Zero	Critically damped: the fastest return, no oscillation
Positive	Overdamped: a slow return, no oscillation

Example: Underdamped

Solve y″ + 2y′ + 5y = 0.

r² + 2r + 5 = 0 gives r = −1 ± 2i.

Answer $y = e^{-x}(c_1\cos 2x + c_2\sin 2x)$

Example: Critically damped

For which b is y″ + by′ + 9y = 0 critically damped?

Critical damping needs b² − 36 = 0.

Answerb = 6

Forced oscillations

A driving force f(t) makes the equation nonhomogeneous: my″ + by′ + ky = f(t). Its general solution is the unforced solution plus one particular solution, often found by guessing a function shaped like f.

Example: A constant force

Solve y″ + 4y = 8.

Guess a constant y = A: 4A = 8, so A = 2.
Add the unforced solution.

Answer $y = 2 + c_1\cos 2x + c_2\sin 2x$

Example: Resonance

Why does y″ + 4y = cos 2t produce growing oscillations?

The driving frequency, 2, matches the spring's own frequency.
A particular solution is $\frac{t}{4}\sin 2t$ , whose size grows with t.

AnswerThe force pushes in step with the motion: resonance

Try it yourself

Work each one on paper first, then open the answer to check.

Solve y″ − y = 0.
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$y = c_1e^{x} + c_2e^{-x}$
Solve y″ + 9y = 0.
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$y = c_1\cos 3x + c_2\sin 3x$
Solve y″ + 4y′ + 4y = 0.
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$y = (c_1 + c_2x)e^{-2x}$
Is y″ + y′ + 10y = 0 underdamped, critically damped or overdamped?
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Underdamped: 1 − 40 is negative
What is the natural frequency of y″ + 25y = 0?
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5
Find a particular solution of y″ + y = 3.
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y = 3