A12 Second Order Differential Equations

Second-order differential equations (by using explicit representation) can be written y'' = f (x,y,y'), where f is a function defined on G c R3 having real values. A function F : Ix ^ R defined by x ^ y = F (x) is a solution of the differential equation if it is twice differentiable and if for any x e Ix, (x,F(x),F'(x)) e G and F''(x) = f (x, F(x), F'(x)). A solution denoted FA is said solution of the problem of initial values (x0,y0,y0) e G, if (x0,FA(x0),F'A(x0)) = (x0,y0,y0). Resolve a second-order differential equation is usually more difficult than a first-order equation. There are many methods to resolve particular types of equation. Generally we look for to reduce an equation of second-order to a system of first-order equations by a change of variables. Within this framework, differential equations of order 2 are very important.

A.12.1 General Resolution of Linear Differential Equations of Second-Order

A linear second-order differential equation can be written y'' + a1(x) ■ y' + a0(x) ■ y = s(x), where a0, a1, s are continuous functions on Ix having real values. If s = 0 on any Ix, the equation becomes homogeneous; By contrast it is said to be non-homogeneous in the opposite case. For the general resolution we use the notion of linear independence of two functions f 1 : Ix x R and f2 : Ix x R: Both functions f1 and f2 are said linearly independent if the functional relation on Ix c1 ■ f1 + c2 ■ f2 = 0 where (c1, c2) e R2 requires c1 = c2 = 0. We define the Wronskian32 of two functions by W(F1,F2)(x) = F1(x) ■ F'(x) — F'(x) ■ F2(x). We show that two solutions (of the homogeneous equation) defined on Ix, F1 and F2 are linearly independent if for any x e Ix, W(F1, F2)(x) = 0 and

Proposition. (1) There exist two linearly independent solutions F1 and F2 defined on Ix for the homogeneous equation y'' + a1 (x) ■ y' + a0(x) ■ y = 0. The set of solutions of the homogeneous equation is the set of Fh(x) = a1 F1 (x) + a2F2(x) with (a1, a2) e R2. (2) There exists one solution Fp defined on Ix for the non-homogeneous equation. The set of solutions of the non-homogeneous equation is the set of Fnh (x) = Fp (x) + a1 F1 (x) + a2F2 (x) with (a1, a2) e R2. (Furthermore we know that any problem of initial conditions admits a single solution.)

A.12.2 Resolution of Linear Homogeneous Equations

Then we are looking for two linearly independent solutions. If we do not know at the beginning the nonzero solution, the problem becomes very difficult; this is why we often use in practice approached solutions (when the tables or intuitions are not

32 Wronskian: An n x n matrix whose /throw is a list of the (i — 1)st derivatives of a set of functions fi,..., fn; ordinarily used to determine linear independence of solutions of linear homogeneous differential equations.

sufficient). By contrast, if we know already a solution F1 (which furthermore is not canceled on Ix) the following method makes it possible to calculate another solution F2 independent of F1. We pose F2 (x) = v(x) ■F1 (x), and v is then solution of the differential equation: v" = — I tfif-vjj • V. The integration is easy and we take for example F2(x) = F\(x) JXQ 1 3 exp f{Q ai(u)di?j dt with jco € Ix.

A.12.3 Particular Solution of a Non-Homogeneous Equation

If F1 and F2 are two linearly independent solutions of the corresponding homogeneous equation, it is possible to find a particular solution of the non-homogeneous equation by the method of variation of constants by posing Fp(x) = v1 (x) ■ F1 (x) + v2(x) ■ F2(x). Then it suffices to resolve both differential equations: V1(x) =

s(x)f2(x) „„a ,/ (Y\ _ I s(x)Fi (x) W(FuF2)(x) dnu V1 W - ^W(Fx,F2)(x) •

A.12.4 Linear Differential Equations of Second-Order with Constant Coefficients

We are interested here in the equations of the type: y'' + p-y1 + q • y = s(x) where p, q are two real numbers. First, we resolve the homogeneous equation y" + p • y' + q • y = 0. So, we consider the characteristic equation z2 + pz + q = 0: its resolution leads to distinguish three cases (see Fig. A.21):

(a) p2 - 4q > 0: two distinct real solutions z1 and z2.

(c) p2 - 4q < 0: two complex (conjugate) solutions z1 and z2.

The solutions of the homogeneous equation are then:

(c) Fh(x) = eRez1x (a1 cos(Imz1 x) + a2sin(Imz1x)).

In order to find a particular solution of the non-homogeneous equation, it is possible to use the methods of variation of constants. However there exist simpler methods if s is a trigonometric function, exponential or an entire series.

Let us consider a (standard) linear homogeneous equation of second order with constant coefficients {p>0,q> 0), y" + p •/ + q-y = 0 where p= jl,q=jl,x = t (r G R+,m e R+,D e R+). Fig. A.21 Qualitative graph of solutions

General solution

For initial conditions (0,0, v0)

—r± a/?"2 — 4mD 2m where z2 < z1 < 0 (b) p2 - 4q = 0, i.e. r2 = 4mD:

(c) p2 - 4q < 0, i.e. r2 < 4mD: F(x) = eReZ1X(a1 cos Imz1 x + a2sinIm z1 x), a1 + a2 = 0 A a1 Z1 + a2 Z2 = V0 v0 v0

Imz1

x sin-x

Particular solution: r -(without friction)

0 0