Math basics › Differential equations
Second-order linear equations
Try y = eʳˣ
For ay″ + by′ + cy = 0, trying y = eʳˣ turns the equation into ar² + br + c = 0, the characteristic equation.
Two different real roots
Roots r₁ and r₂ give y = C₁e^(r₁x) + C₂e^(r₂x).
A repeated root
One root r, twice, gives y = (C₁ + C₂x)eʳˣ. The extra x keeps the two solutions different.
Complex roots
Roots α ± βi give y = e^(αx)(C₁ cos βx + C₂ sin βx). The real part makes it grow or shrink; the imaginary part makes it wave.