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Higher Order Ordinary Differential Equations

In this chapter, we begin by solving homogeneous linear ordinary differential equations with constant coefficients by characteristic equations. Then we solve the Euler equations and exact equations. The method of undetermined coefficients for solving inhomogeneous linear ordinary differential equations is presented, as well as the method of variation of parameters for solving second-order inhomogeneous linear ordinary differential equations. In addition, we introduce the power series method to solve variable-coefficient linear ordinary differential equations and study the Bessel equation in detail.

2.1 Basics This section deals with homogeneous linear ordinary differential equations with constant coefficients, the Euler equations, and exact equations. A second-order homogeneous linear ordinary differential equation with constant coefficients is of the form ay  + by  + cy = 0,

a, b, c ∈ R.

(2.1.1)

To find the general solution, we assume that y = eλt is a solution of (2.1.1), where λ is a constant to be determined. Substituting it into (2.1.1), we get aλ2 eλt + bλeλt + ceλt ∼ aλ2 + bλ + c = 0,

(2.1.2)

which is called the characteristic equation of (2.1.1). If the above equation has two distinct real roots λ1 and λ2 , then the general solution of (2.1.1) is y = c1 eλ1 t + c2 eλ2 t ,

(2.1.3)

where c1 and c2 are arbitrary constants. When (2.1.2) has two complex roots r1 ± r2 i, then the real part and imaginary part of e(r1 +r2 i)t are solutions of (2.1.1). X. Xu, Algebraic Approaches to Partial Differential Equations, DOI 10.1007/978-3-642-36874-5_2, © Springer-Verlag Berlin Heidelberg 2013

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Higher Order Ordinary Differential Equations

So the general solution of (2.1.1) is y = c1 er1 t sin r2 t + c2 er1 t cos r2 t.

(2.1.4)

If (2.1.2) has a repeated root r, the general solution of (2.1.1) is y = (c1 + c2 t)ert .

(2.1.5)

Example 2.1.1 The general solution of the equation y  − 2y  − 3y = 0

(2.1.6)

y = c1 e3t + c2 e−t

(2.1.7)

is

because λ = 3 and λ = −1 are real roots of the characteristic equation λ2 − 2λ − 3 = 0. Moreover, the general solution of the equation y  − 4y  + 13y = 0

(2.1.8)

y = c1 e2t sin 3t + c2 e2t cos 3t

(2.1.9)

is

because λ = 2 + 3i and λ = 2 − 3i are roots of the characteristic equation λ2 − 4λ + 13 = 0. Furthermore, the general solution of the equation y  + 6y  + 9y = 0

(2.1.10)

y = (c1 + c2 t)e−3t .

(2.1.11)

is

In general, the algebraic equation bn λn + bn−1 λn−1 + · · · + b0 = 0

(2.1.12)

is called the characteristic equation of the differential equation bn y (n) + bn−1 y (n−1) + · · · + b0 y = 0, If (2.1.12) has a real root r with multiplicity m, then   cm−1 t m−1 + · · · + c1 t + c0 ert

br ∈ R.

(2.1.13)

(2.1.14)

is a solution of (2.1.13) for arbitrary c0 , c1 , . . . , cm−1 ∈ R. When r1 + r2 i is a complex root of (2.1.12) with multiplicity m, then   cm−1 t m−1 + · · · + c1 t + c0 er1 t sin r2 t (2.1.15)

2.1 Basics

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and   am−1 t m−1 + · · · + a1 t + a0 er1 t cos r2 t

(2.1.16)

are solut