Page 557 - Modelling in Transport Phenomena A Conceptual Approach

B.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 537

B.2 SECOND-ORDER LINEAR
DIFFE-NTIAL EQUATIONS

A general second-order linear differential equation with constant coefficients is writ-
ten as
&Y dy
a, - +a1 - + a2y = R(x) (B.2-1)
dx2 dx
If R(x) = 0, the equation

&Y + dy
dx
a, 2 a1 - + a2 y = 0 (B.2-2)
dx
is called a homogeneous equation.
The second-order homogeneous equation can be solved by proposing a solution
of the form
y = emx (B.2-3)

where m is a constant. Substitution of Eq. (B.2-3) into Eq. (B.2-2) gives

a,m 2 +alm+aa=O (B.2-4)
which is known as the charucteristic or auxiliary equation. Solution of the given
differential equation depends on the roots of the characteristic equation.

Distinct real roots

When the roots of Eq. (B.2-4), ml and m2, are real and distinct, then the solution
is
y = C1 emlx + C2 (B.2-5)

Repeated real roots

When the roots of Eq. (B.2-4), ml and 77x2, are real and equal to each other, i.e.,
ml = m2 = m, then the solution is

Y=(Ci +C2x)emz (B.2-6)

Conjugate complex roots

When the roots of Eq. (B.2-4), ml and m2, are complex and conjugate, i.e.,
m1,2 = a f ib, then the solution is
y = ear(Cl cos bx + C2 sin bx) (B.2-7)