Chapter 1 - Second Order Linear Differential Equations

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1.1 Introduction

A general second order ODE has the form y'' = f(t, y, y'), where y is an unknown function and f: R^3 -> R.

The order of the differential equation is the highest derivative.

The equation is linear if the function f is linear in both arguments y and y' i.e. f(t, y, y') = a(t)y + b(t)y' + c(t). (No y^2/y.y' etc)

Definition 1.1 - A second order differential equation in the unknown y(t) is

a(t)y'' + b(t)y' + c(t)y = h(t) (1.1)

where a, b, c, h: I -> R are given functions on the interval I in R.

Equation (1.1) is called homogeneous if the source h(t)=0 for all t in R. Equation (1.1) is called constant coefficient if a(t), b(t), c(t) are all constants, otherwise the equation is called variable coefficients.

Theorem 1.3 - If the functions p(t), q(t), f(t): I -> R are continuous on a closer interval I in R, t_0 in I, and y_0, y'_0 in R are constants, then there exists a unique solution y:I -> R to the initial value problem (IVP)

y'' + p(t)y' + q(t)y = f(t), y(t_0) = y_0 and y'(t_0) = y'_0. (1.2)

1.2 Homogeneous Equations

To simplify the study of equation (1.2) we start with learning about properties of solutions to homogeneous equations.

1.2.1 Linear Operators

A function, such as f: R -> R defined by f(x) = x^2, takes a number as an input and squares it an the output.

One can think of differentiation in the a similar way, but with the inputs and outputs as functions rather than numbers.

Definition 1.7 - An operator L is called a linear operator if for every pair of functions y_1, y_2 and constants c_1, c_2 we have

L[c_1y_1 + c_2y_2] = c_1L[y_1] + c_2L[y_2].

Theorem 1.10 - If L is a linear operator and y_1, y_2 are solutions of the homogeneous equation L[y_1]=0, L[y_2]=0 then for every constants c_1, c_2 we have L[c_1y_1 + c_2y_2] = 0.

Definition 1.11 - Two continuous functions y_1, y_2 : I -> R are called linearly dependent on the interval I if there exists constants c_1, c_2 not both zero such that for all t in I we have c_1y_1(t) + c_2y_2(t) = 0.

y_1, y_2 are linearly independent on the interval I if they are not linearly dependent, that is if the only constants c_1, c_2 that for all t in I satisfy c_1y_1(t) + c_2y_2(t) =0 are the constants c_1=0, c_2=0.

Theorem 1.13 - If y_1 and y_2 are linearly independent solutions of the homogeneous equation L[y]=0 on an interval I in R, where L[y] = y'' + p(t)y' + q(t)y and p(t), q(t) are continuous functions on I, then there exists unique constants c_1, c_2 such that every solution y of the differential equation L[y]=0 on I can be written as a linear combination y(t) = c_1y_1(t) + c_2y_2(t).

Definition 1.14 - a) The functions y_1 and y_2 are fundamental solutions of the equation L[y]=0 if L[y_1]=0 and L[y_2]=0 and y_1 and y_2 are linearly independent.

b) The general solution of the homogeneous equation L[y]=0 is a two parameter family of functions y_gen given by y_gen(t) = c_1y_1(t) + c_2y_2(t) where the arbitrary constants c_1 and c_2 are the parameters of the family and y_1, y_2 are the fundamental solutions of L[y]=0.

1.3 The Wronskian Function

Definition 1.14' - The Wronskian of the differentiable functions y_1, y_2 is the function W(y_1, y_2) = y_1y'_2 - y_2y'_1.

Theorem 1.18 - Let y_1, y_2 : I -> R be solutions of L[y]=0 on I. If there exists one point t in I such that W(y_1, y_2)=0 then y_1, y_2 are linearly dependent.

Corollary 1.19 - Let y_1, y_2 : I -> R be both solutions of L[y]=0 on I. If y_1, y_2 are linearly independent on I then their Wronskian W(y_1, y_2) does not equal 0 for all t in I.

Theorem 1.20 - If y_1, y_2 are twice continuously differentiable solutions of L[y] = y'' + p(t)y' + q(t)y = 0, where p(t), q(t) are continuous on I, subset of R, then the Wronskian satisfies W' + p(t)W = 0.

1.4 Reduction of Order

This method provides a way to obtain a second solution to the homogeneous equation L[y]=0 if we already know one solution.

If y_1 is a solution of L[y] = y'' + p(t)y' + q(t)y = 0 we can use Abel's theorem to find the Wronskian W.

But W(y_1, y_2) = y_1y'_2 - y_2y'_1 so provided y_1 does not equal zero, we can find a first order ODE in y_2, which can be solved using an integrating factor to find y_2.