6. Linear Second Order Differential Equations

First Order Linear Differential Equations

Definition

A first-order linear differential equation is one that can be put into the form

$y^{\prime }+P(x)y=Q(x)$

Second Order Linear Differential Equations

Definition

A second-order linear differential equation is one that can be put into the form

$a(x)y^{\prime \prime }+b(x)y^{\prime }+c(x)y=F(x)$

Linearity Property

$(a(x)y_1^{\prime \prime }+b(x)y_1^{\prime }+c(x)y_1)+(a(x)y_2^{\prime \prime }+b(x)y_2^{\prime }+c(x)y_2)=a(x)y^{\prime \prime }+b(x)y^{\prime }+c(x)y$

$k(a(x)y_1^{\prime \prime }+b(x)y_1^{\prime }+c(x)y_1)=a(x)(k{y}_1{)}^{\prime \prime }+b(x)(k{y}_1{)}^{\prime }+c(x)k{y}_1$

where $y=y_1+y_2$.

Homogeneous Equation

Definition

A second-order linear differential equation

$a(x)y^{\prime \prime }+b(x)y^{\prime }+c(x)y=F(x)$

is called homogeneous if $F(x)=0$, so we get

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

Linearity

If $y_1$ and $y_2$ both solve $a(x)y^{\prime \prime }+b(x)y^{\prime }+c(x)y=0$ then so does $y=y_1+y_2$.

Linear Combination

Theorem

If two functions $y_1$ and $y_2$ solve a homogeneous linear second order differential equation then so does any linear combination of them.

Definition

A linear combination of two functions $y_1$ and $y_2$ is a function

$y=c_1{y}_1+c_2{y}_2$

for some constants $c_1$ and $c_2$.

Linear Independence

Theorem

If two linearly independent functions $y_1$ and $y_2$ solve a homogeneous linear second order differential equation then the general solution is

$y=c_1{y}_1+c_2{y}_2$

Definition

Two functions $y_1$ and $y_2$ are called linearly independent if $y_1\ne \alpha y_2$ and $y_2\ne \beta y_1$ for any constants $\alpha$ and $\beta$.

Complementary Equation

Definition

For a second-order linear differential equation

$a(x)y^{\prime \prime }+b(x)y^{\prime }+c(x)y=F(x)$

we call $F(x)$ the driving term and we call

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

the complementary (homogeneous) equation.

Linearity - Non-Homogeneous Case

Theorem

If $y_1$ and $y_2$ are two solutions to a second order linear differential equation then

$y=y_2-y_1$

is a solution to the complementary equation.

General Solution

Theorem

If two linearly independent functions $y_1$ and $y_2$ solve the complementary equation to a second order linear differential equation and $y_p$ is some particular solution to that equation then the general solution is

$y=y_p+c_1{y}_1+c_2{y}_2$

Solving Strategy

Solving $a(x)y^{\prime \prime }+b(x)y^{\prime }+c(x)y=F(x)$:

1. Find two linearly independent solutions $y_1$ and $y_2$ to the complementary equation $a(x)y^{\prime \prime }+b(x)y^{\prime }+c(x)y=0$.
2. Find one particular solution $y_p$ to the actual equation.
3. The general solution is

$y=y_p+c_1{y}_1+c_2{y}_2$

Constant Coefficients

Definition

A second order linear differential equation with constant coefficients is a differential equation in the following form:

$a{y}^{\prime \prime }+b{y}^{\prime }+cy=F(x)$

where $a$, $b$, and $c$ are constants.