Study Guide: Linear Differential Equations
1. Linear Differential Equations
A linear differential equation is an equation of the form
fn (x)y (n) + · · · + f2 (x)y 00 + f1 (x)y 0 + f0 (x)y = g(x)
where f0 (x), f1 (x), . . . , fn (x) and g(x) are functions. For example, a second-order linear differential equation has the form
f2 (x)y 00 + f1 (x)y 0 + f0 (x)y = g(x),
and a third-order linear differential equation has the form
f3 (x)y 000 + f2 (x)y 00 + f1 (x)y 0 + f0 (x)y = g(x).
A linear differential equation is homogeneous if g(x) = 0. The solutions to a homogeneous
linear differential equation form a vector space known as the solution space. The dimension
of this vector space is equal to the order of the equation.
2. Constant-Coefficient Equations
A linear, homogeneous differential equation is constant-coefficient if the coefficient functions
f0 (x), f1 (x), . . . , fn (x) are constants. For example, a second-order constant-coefficient linear
homogeneous differential equation has the form
a1 y 00 + a2 y 0 + a3 y = 0,
where a1 , a2 , and a3 are constants. The best way to find a basis for the solution space of such
an equation is to look for solutions of the form y = erx . However, a few difficulties arise:
• Sometimes two of the values for r are imaginary, i.e. r = ±bi. In this case, the corresponding solutions are cos(bx) and sin(bx).
• More generally, sometimes two of the values for r are complex, i.e. r = a ± bi. In this
case, the corresponding solutions are eax cos(bx) and eax sin(bx).
• Sometimes the polynomial has a repeated root for r, i.e. it is divisible by (r − a)2 for
some value of a. In this case, the corresponding solutions are eax and xeax .