2.6.3—Linear differential equations

For ay″+by′+cy=0, try y=e^{mx}; the auxiliary equation am²+bm+c=0 determines the complementary function. Distinct real roots give two exponentials, a repeated root gives (C₁+C₂x)e^{mx}, and complex roots give e^{αx}(C₁cosβx+C₂sinβx).

The form follows from the root type, not from memorised labels. Substitute the proposed solution back into the differential equation to check the signs and coefficients.

For y″−4y′+13y=0, m²−4m+13=0 gives m=2±3i, so y=e^{2x}(C₁cos3x+C₂sin3x).

Complex roots do not make y complex when real initial data are used; the conjugate pair combines into real sine and cosine terms.