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Second Order equations with Constant Coefficients

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

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

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

and

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

It is easy to note that

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

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

$\displaystyle p({\lambda}) = 0.$ (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,

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

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

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

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

$\displaystyle 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

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

    has a unique solution for any real number $ B.$
  5. Consider the problem

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

next up previous contents
Next: Non Homogeneous Equations Up: Second Order and Higher Previous: Method of Reduction of   Contents
A K Lal 2007-09-12