Stewart Calculus · 8th Edition
Chapter 17: Second-Order Differential Equations
Every key formula from Stewart Calculus Chapter 17, in one searchable page. Click a card to study it — worked examples included.
17.1 · HOMOGENEOUS EQUATIONS
ay″ + by′ + cy = 0 ⇒ ar² + br + c = 0
r₁ ≠ r₂ real: y = c₁er₁x + c₂er₂x
Repeated r: y = c₁erx + c₂xerx
r = α ± βi: y = eαx(c₁ cos βx + c₂ sin βx)
Repeated r: y = c₁erx + c₂xerx
r = α ± βi: y = eαx(c₁ cos βx + c₂ sin βx)
WORKED EXAMPLES
y″ − 3y′ + 2y = 0 ⇒ r = 1, 2:
y = c₁ex + c₂e2x.
y″ + 4y = 0 ⇒ r = ±2i:
y = c₁ cos 2x + c₂ sin 2x.
y = c₁ex + c₂e2x.
y″ + 4y = 0 ⇒ r = ±2i:
y = c₁ cos 2x + c₂ sin 2x.
General solution
y = c₁y₁ + c₂y₂ with y₁, y₂ linearly independent — neither a constant multiple of the other.
WORKED EXAMPLE
y″ − 4y′ + 4y = 0: (r − 2)² = 0 ⇒
y = c₁e2x + c₂xe2x.
y = c₁e2x + c₂xe2x.
17.2 · NONHOMOGENEOUS
ay′′ + by′ + cy = G(x): y = yh + yp
Undetermined coefficients: guess from G‘s form;
if it overlaps yh, multiply the guess by x.
if it overlaps yh, multiply the guess by x.
WORKED EXAMPLE
y″ + y = 2x: try yp = Ax + B ⇒ yp = 2x;
y = c₁ cos x + c₂ sin x + 2x.
y = c₁ cos x + c₂ sin x + 2x.
Guess guide
poly × eαx ⇒ same form · trig ⇒ A cos βx + B sin βx
resonance ⇒ × xs, smallest s making the guess new
resonance ⇒ × xs, smallest s making the guess new
WORKED EXAMPLE
y″ − y = ex: ex is in yh ⇒ try Axex;
2Aex = ex ⇒ yp = xex/2.
2Aex = ex ⇒ yp = xex/2.
mx″ + cx′ + kx = 0
c² − 4mk > 0 overdamped · = 0 critically damped · < 0 underdamped (oscillates).
Resonance: forcing frequency = natural frequency.
Resonance: forcing frequency = natural frequency.
Overdamped: c² > 4mk (no oscillation); critically damped: =.
Frequency & period
ω = √(k/m), T = 2π/ω · underdamped: e−ct/2m(A cos ω₁t + B sin ω₁t)
ω₁ = √(4mk − c²)/(2m).
Forced vibration
mx″ + cx′ + kx = F(t); small c + forcing near ω ⇒ practical resonance (large swings).
17.4 · SERIES SOLUTIONS
Assume y = ∑n=0∞ cnxn;
plug in and match coefficients.
plug in and match coefficients.
Works near ordinary points — yields the Taylor series of the solution.
KEY NOTES
- Differentiate term-by-term, shift indices to align powers of x.
Ordinary vs singular points
Ordinary: coefficients analytic at x₀; singular otherwise (e.g. x = 0 for Bessel).
The method
Substitute ∑cnxn, shift indices so every term is xn, match coefficients ⇒ recurrence.
WORKED EXAMPLE
y′ = y: ∑(n+1)cn+1xn = ∑cnxn ⇒
cn+1 = cn/(n+1) ⇒ y = c₀ex.
cn+1 = cn/(n+1) ⇒ y = c₀ex.
KEY NOTES
- The series converges at least as far as the nearest singular point.
- It IS the Taylor series of the true solution.
INITIAL VALUE PROBLEMS
Second order needs BOTH y(x₀) and y′(x₀).
WORKED EXAMPLE
y″ − 3y′ + 2y = 0, y(0) = 1, y′(0) = 0:
c₁ + c₂ = 1, c₁ + 2c₂ = 0 ⇒
y = 2ex − e2x.
c₁ + c₂ = 1, c₁ + 2c₂ = 0 ⇒
y = 2ex − e2x.
Series about ordinary x₀: y = ∑cn(x−x₀)n.
Existence & uniqueness
Coefficients continuous near x₀ ⇒ exactly one solution.
Boundary value problems
y given at TWO x-values: can have 0, 1, or infinitely many solutions.
WORKED EXAMPLE
y″ + y = 0, y(0) = 2, y′(0) = 3:
c₁ = 2, c₂ = 3 ⇒ y = 2 cos x + 3 sin x.
c₁ = 2, c₂ = 3 ⇒ y = 2 cos x + 3 sin x.
Mistakes that cost points
- Repeated root: don’t forget the ×x term
- Guess overlaps yh ⇒ multiply by x (or x²)
- Complex roots ⇒ trig, NOT exponentials alone
- IVP needs two conditions: y AND y′
- Check: plug y back into the DE!
- r = α ± βi ⇒ keep the eαx factor!
- Trig guesses need BOTH A cos and B sin
- c² − 4mk < 0 oscillates; > 0 does not
- s in xs counts overlaps: yh overlap ⇒ s ≥ 1