Advanced book on Mathematics Olympiad

3.4 Equations with Functions as Unknowns 195

3.4.3 Ordinary Differential Equations of Higher Order

The field of higher-order ordinary differential equations is vast, and we assume that you
are familiar at least with some of its techniques. In particular, we assume you are familiar
with the theory of linear equations with fixed coefficients, from which we recall some
basic facts. A linear equation with fixed coefficients has the general form

an

dny

dxn

+···+a 2

d^2 y

dx^2

+a 1

dy

dx

+a 0 =f(x).

Iff is zero, the equation is called homogeneous. Otherwise, the equation is called
inhomogeneous. In this case the general solution is found using the characteristic equation

anλn+an− 1 λn−^1 +···+a 0 = 0.

Ifλ 1 ,λ 2 ,...,λrare the distinct roots, real or complex, of this equation, then the general
solution to the homogeneous differential equation is of the form

y(x)=P 1 (x)eλ^1 x+P 2 (x)eλ^2 x+···+Pr(x)eλrx,

wherePi(x)is a polynomial of degree one less than the multiplicity ofλi,i= 1 , 2 ,...,r.
If the exponents are complex, the exponentials are changed into (damped) oscillations
using Euler’s formula.
The general solution depends onnparameters (the coefficients of the polynomials),
so the space of solutions is ann-dimensional vector spaceV. For an inhomogeneous
equation, the space of solutions is the affine spacey 0 +Vobtained by adding a particular
solution. This particular solution is found usually by the method of the variation of the
coefficients.
We start with an example that exploits an idea that appeared once on a Putnam exam.

Example.Solve the system of differential equations

x′′−y′+x= 0 ,

y′′+x′+y= 0

in real-valued functionsx(t)andy(t).

Solution.Multiply the second equation byithen add it to the first to obtain

(x+iy)′′+i(x+iy)′+(x+iy)= 0.

With the substitutionz=x+iythis becomes the second-order homogeneous linear
differential equationz′′+iz′+z=0. The characteristic equation isλ^2 +iλ+ 1 =0,
with solutionsλ 1 , 2 =−^1 ±

√

5

2 i. We find the general solution to the equation