01 - Second Order Linear ODEs

Second-Order Linear ODE

$$L[y(x)]=f(x)$$

where $f$ is a forcing function and $L$ is a linear differential operator that is

$$\begin{array}{rl}L[y(x)]& \equiv P_2(x)y^{\prime \prime }(x)+P_1(x)y^{\prime }(x)+P_0(x)y(x)\\ & \equiv P_2(x)\frac{d^{2}y}{d{x}^2}+P_1(x)\frac{dy}{dx}+P_0(x)y(x)\end{array}$$

for some given coefficients $P_2(x),P_1(x)$ and $P_0(x)$

$L$ is linear

$$L[{\alpha }_1{y}_1(x)+{\alpha }_2{y}_2(x)]\equiv a_{1}L[y_1(x)]+{\alpha }_{2}L(y_2(x))$$

for some constants ${\alpha }_i$ and functions $y_i(x)$

Homogeneous and Inhomogeneous Versions of an ODE

Let $L$ be a linear differential operator then

Homogeneous

$$L[y]=0$$

Inhomogeneous

$$L[y]=f\not\equiv 0$$

Boundary Value Problem of a Second-Order ODE

Generally to have two boundary conditions for a unique solution

Boundary Value Problem refers to the way in how the boundary conditions are imposed so
ODE is posed on an interval, say $a<x<b$ and boundary conditions involve

$$y\text{ and/or }{y}^{\prime }\text{ at both ends of domain }x=a\text{ and }x=b$$

provided coefficients $P_i(x)$ and $f(x)$ are sufficiently well behaved and $P_2(x)\ne 0$

Space of Solutions to an ODE

Ignoring the Boundary Conditions

Let $(H)$ denote the Homogenous Equation ($Ly=0$)
Let $(N)$ denote the Inhomogeneous Equation ($Ly=f$)

The following properties of the solutions of $(H)$ and $(N)$ are

1. Solutions of $(H)$ form a vector space since
   If $L[y_1]+L[y_2]=0$ then

$$L[\alpha y_1+\beta y_2]=0$$

2. If $y_1$ and $y_2$ satisfy $(N)$ then $y_1-y_2$ satisfy $(H)$

3. It follows that the general solution of $(N)$ may be written in the form

$$y(x)=\underset{\text{any solution of }(N)}{\underbrace{y_{\text{PI}}(x)}}+\underset{\text{general solution of }(H)}{\underbrace{y_{\text{CF}}(x)}}$$

where $y_{\text{ PI }}(x)$ is called the particular integral and $y_{\text{CF}}(x)$ the complementary function

4. For a Second-Order ODE , then the vector space of solutions to $(H)$ has dimension $2$
   Complementary Function then takes form

$$y_{\text{CF}}(x)=c_1{y}_1(x)+c_2{y}_2(x)$$

where $c_1$ and $c_2$ are arbitrary constants, and $y_1,y_2$ are two linearly independent solutions to $(H)$