## 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