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Module 4 Quick Reference

Second-Order Linear ODEs

Characteristic Equation: ar² + br + c = 0

D = b²-4acRootsSolution y(t)Behavior
D > 0r1, r2c1er1t + c2er2tExponential
D = 0r (double)(c1+c2t)ertCritically damped
D < 0α±βieαt(c1cosβt+c2sinβt)Oscillation

Undetermined Coefficients

y = yh + yp

Guess Table

poly → poly same degree

eαt → Aeαt

sin/cos → A cos + B sin

Overlap with yh? Multiply by t

Variation of Parameters

yp = -y1∫y2g/W dt + y2∫y1g/W dt

Requirements

  • Standard form (coeff of y''=1)
  • Know y1, y2
  • W = y1y2'-y2y1'
  • Works for ANY g(t)

Key Reminders

IVP Strategy

  • Find yh first
  • Find yp
  • y = yh + yp
  • Apply ICs to FULL solution

Wronskian

W = y1y2' - y2y1'

W ≠ 0: linearly independent

2nd order needs 2 ICs

n-th order has n constants