Second Order equations with Constant Coefficients

DEFINITION 8.3.1   Let $a$ and $b$ be constant real numbers. An equation

$$y^{\prime\prime} + a y^\prime + b y = 0$$ (8.3.1)

is called a SECOND ORDER HOMOGENEOUS LINEAR EQUATION WITH CONSTANT COEFFICIENTS.

Let us assume that $y = e^{{\lambda}x}$ to be a solution of Equation (8.3.1) (where ${\lambda}$ is a constant, and is to be determined). To simplify the matter, we denote

$$L(y) = y^{\prime\prime} + a y^\prime + b y$$

and

$$p({\lambda}) = {\lambda}^2 + a {\lambda}+ b.$$

It is easy to note that

$$L(e^{{\lambda}x}) = p({\lambda}) e^{{\lambda}x}.$$

Now, it is clear that $e^{{\lambda}x}$ is a solution of Equation (8.3.1) if and only if

$$p({\lambda}) = 0.$$ (8.3.2)

Equation (8.3.2) is called the CHARACTERISTIC EQUATION of Equation (8.3.1). Equation (8.3.2) is a quadratic equation and admits 2 roots (repeated roots being counted twice).

Case 1: Let ${\lambda}_1, {\lambda}_2$ be real roots of Equation (8.3.2) with ${\lambda}_1 \neq {\lambda}_2.$
Then $e^{{\lambda}_1 x}$ and $e^{{\lambda}_2 x}$ are two solutions of Equation (8.3.1) and moreover they are linearly independent (since ${\lambda}_1 \neq {\lambda}_2$ ). That is, $\{ e^{{\lambda}_1 x} , e^{{\lambda}_2 x} \}$ forms a fundamental system of solutions of Equation (8.3.1).

Case 2: Let ${\lambda}_1 = {\lambda}_2$ be a repeated root of $p({\lambda}) = 0.$
Then $p^\prime({\lambda}_1) = 0.$ Now,

$$\frac{d}{dx}(L(e^{{\lambda}x})) = L( x e^{{\lambda}x}) = p^\prime({\lambda}) e^{{\lambda}x} + x p({\lambda}) e^{{\lambda}x}.$$

But $p^\prime({\lambda}_1) = 0$ and therefore,

$$L(x e^{{\lambda}_1 x}) = 0.$$

Hence, $e^{{\lambda}_1 x}$ and $x e^{{\lambda}_1 x}$ are two linearly independent solutions of Equation (8.3.1). In this case, we have a fundamental system of solutions of Equation (8.3.1).

Case 3: Let ${\lambda}= {\alpha}+ i \beta$ be a complex root of Equation (8.3.2).
So, ${\alpha}- i \beta$ is also a root of Equation (8.3.2). Before we proceed, we note:

LEMMA 8.3.2   Let $y = u + i v$ be a solution of Equation (8.3.1), where $u$ and $v$ are real valued functions. Then $u$ and $v$ are solutions of Equation (8.3.1). In other words, the real part and the imaginary part of a complex valued solution (of a real variable ODE Equation (8.3.1)) are themselves solution of Equation (8.3.1).

Proof. exercise. height6pt width 6pt depth 0pt

Let ${\lambda}= {\alpha}+ i \beta$ be a complex root of $p({\lambda}) = 0.$ Then

$$e^{{\alpha}x} ( \cos(\beta x) + i \sin (\beta x) )$$

is a complex solution of Equation (8.3.1). By Lemma 8.3.2, $y_1 = e^{{\alpha}x} \cos(\beta x)$ and $y_2 = \sin (\beta x)$ are solutions of Equation (8.3.1). It is easy to note that $y_1$ and $y_2$ are linearly independent. It is as good as saying $\{ e^{{\lambda}x} \cos(\beta x) , e^{{\lambda}x} \sin(\beta x) \}$ forms a fundamental system of solutions of Equation (8.3.1).

EXERCISE 8.3.3  

1. Find the general solution of the follwoing equations.
   1. $y^{\prime\prime} - 4 y^\prime + 3 y = 0.$
   2. $2 y^{\prime\prime} + 5 y = 0.$
   3. $y^{\prime\prime} - 9 y = 0.$
   4. $y^{\prime\prime} + k^2 y = 0,$ where $k$ is a real constant.
2. Solve the following IVP's.
   1. $y^{\prime\prime} + y = 0, \; y(0) = 0, \; y^\prime(0) = 1.$
   2. $y^{\prime\prime} - y = 0, \; y(0) = 1, \; y^\prime(0) = 1.$
   3. $y^{\prime\prime} + 4 y = 0, \; y(0) = -1, \; y^\prime(0) = -3.$
   4. $y^{\prime\prime} + 4 y^\prime + 4 y = 0, \; y(0) = 1, \; y^\prime(0) = 0.$
3. Find two linearly independent solutions $y_1$ and $y_2$ of the following equations.
   1. $y^{\prime\prime} - 5 y = 0.$
   2. $y^{\prime\prime} + 6 y^\prime + 5 y = 0.$
   3. $y^{\prime\prime} + 5 y = 0.$
   4. $y^{\prime\prime} + 6 y^\prime +9 y = 0.$ Also, in each case, find $W(y_1, y_2).$
4. Show that the IVP
   $$y^{\prime\prime} + y = 0, \; y(0) = 0 \; {\mbox{ and }} \; y^\prime(0) = B$$
   has a unique solution for any real number $B.$
5. Consider the problem

   $$y^{\prime\prime} + y = 0, \; y(0) = 0 \; {\mbox{ and }} \; y^\prime(\pi) = B.$$ (8.3.3)

   
   
   Show that it has a solution if and only if $B = 0.$ Compare this with Exercise 4. Also, show that if $B = 0,$ then there are infinitely many solutions to (8.3.3).