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Basics of Second Order, Ordinary Differential Equations (continued)

If the right hand side of the d.e. is zero, the d.e. is said to be homogeneous.

a y’’ + by’ + cy = 0 where a, b, and c are constants

If you assume an “exponential” such as y(x) = e ^rx and substitute it into this

d.e., then the result (factoring out e^rx ) is:

The characteristic equation is ar² + br + c = 0

This characteristic equation has four possible roots using the quadratic formula

for this second order quadratic, algebraic equation. See the table below.

The solution using the appropriate roots is called the “complementary solution”

of the d.e. Note that the values of the roots depend on the values of the constants

a, b, and c.

Possibilities for roots, r Complementary solution, y_c

distinct real roots r₁ and r₂

y_c = C₁ exp( r₁x) + C₂ exp(r₂x)

repeated roots real r₁ and r₁

y_c = C₁ exp( r₁x) + C₂ x exp(r₁x)

distinct imaginary roots, ir₁ and ˗ ir₁

y_c = C₁ sin r₁ x+ C₂ cos r₁ x

complex roots r₁ ± i r₂

y_c = e^r₁^x ( C₁ sin r₂ x+ C₂ cos r₂ x )

repeated complex roots r₁ ± i r₁

y_c = e^r₁^x ( C₁ sin r₁ x+ C₂ x cos r₁ x )

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