Second Order Linear Equations

Introduction: Seond-Order Equations

Recall: A differential equation is linear if it is linear in its dependent variable(s).  Thus, (x^2)y''+y=3 is linear, however (((x^2)(y''))^2)+y=3 is not linear.

Definition

A second order linear differential equation can be written in the form:

A(x)y''+B(x)y'+C(x)y=F(x)

where A, B, C, and F are continuous on an interval I.  If F(x)≡0 for all x∈I, then we say the equation is homogeneous.  Otherwise, it is non-homogeneous.

Theorem - Principle of Supperposition for Linear, Homogeneous Equations

Let (Subscript[y, 1]) and (Subscript[y, 2]) be two solutions to y''+p(x)y'+q(x)y=0.  Then,

y=(Subscript[c, 1])(Subscript[y, 1])+(Subscript[c, 2])(Subscript[y, 2])

is also a solution.

Definition

Two functions on an open interval I are linearly independent on I provided that neither is a constant multiple of the other.  Else they are linearly dependent.

Example

The functions
	A)	x, (1/2) x are linearly dependent.
	B)	x, (x^2) are linearly independent.

Definition

the Wronskian of f and g is

({{({{W(f,g)(x)=det(({{f(x), g(x)}, {f'(x), g'(x)}}))
=f(x) g'(x) - f'(x) g(x).}})}})

Theorem - Wronskians of Solutions

Suppose (Subscript[y, 1]) and (Subscript[y, 2]) are two solutions of y''+p(x) y'+q(x) y=0 on an open interval I on which both p(x) and q(x) are continuous.  Then,

({{({{(Subscript[y, 1]) and (Subscript[y, 2]) are linearly independent ⟺ W((Subscript[y, 1])}}), (Subscript[y, 2]))(x)≠0 at each point of I.}})

Example

Consider the DE a y''+b y' +c y=0, where a, b, and c are constants.  For what values of r is y(x)=(e^(r x)) a solution for this differential equation?

Second Order, Linear,  Homogeneous ODEs with Constant Coefficients

Definition

The characteristic equation to the homogeneous, linear, differential equation with (real) constant coefficients a y'' +b y'+c y=0 is a (r^2)+b r+c=0.

Note: The characteristic equation is a result of assuming a solutions of the form y(x)=exp(r x).

Theorem - Solutions to Second Order, Linear, Homogeneous ODE with Constant Coefficients:  Distinct and Real

If the roots (Subscript[r, 1]) and (Subscript[r, 2]) of the characteristic equation are distinct and real,
then y(x)=(Subscript[c, 1]) (e^((Subscript[r, 1]) x))+(Subscript[c, 2]) (e^((Subscript[r, 2])x)) is the general solution.

Example

Find the general solution to 3 y'' +2 y'-5 y=0.

Theorem - Solutions to Second Order, Linear, Homogeneous ODE with Constant Coefficients:  Repeated Roots

If the characteristic equation has roots r=(Subscript[r, 1])=(Subscript[r, 2]), 
then y(x)=(Subscript[c, 1]) (e^((Subscript[r, 1]) x))+(Subscript[c, 2]) x (e^((Subscript[r, 1])x))=((Subscript[c, 1])+(Subscript[c, 2]) x )(e^(r x)) is the general solution.

Example

Solve the IVP y''-6 y'+9 y=0,     y(0)=2,    y'(0)=-3.

Theorem - Solutions to Second Order, Linear, Homogeneous ODE with Constant Coefficients:  Complex Roots

If the characteristic equation has roots r=a±b i, 
then y(x)=(e^(a x))((Subscript[c, 1])cos(b x)+(Subscript[c, 2])sin(b x) is the general solution.

Example

Find the general solution to y''+y'+y=0.