CONSTANT COEFFICIENT HOMOGENEOUS
SECOND ORDER EQUATIONS

General Second Order Equations

are in the form

$$\frac{d^2y}{dt^2}= f(t,y,\frac{dy}{dt}).$$

Linear Second Order Equations

are in the form

$$\frac{d^2y}{dt^2}+ p(t)\frac{dy}{dt}+ q(t)y = g(t).$$

• Homogeneous equations have g(t) = 0.
• Nonhomogeneous equations have $g(t) \neq 0$.
• Constant Coefficient homogeneous equations:
  
  ay'' + by' + cy = 0.
  
  

Constant Coefficient Homogeneous Equations

• Try solutions of the form y(t) = e^rt.
• Then ( ar² + br + c )e^rt = 0.
• So r values are roots of the characteristic equation
  ar² + br + c = 0; $r_{1,2} = -\frac{b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a}$.
• Three cases:
  a) both roots real and distinct: y(t) = c₁e^r₁t + c₂e^r₂t;
  b) both roots complex: $r_{1,2} = \lambda \pm i\mu$, so
  $y(t) = (c_1\sin(\mu t)+c_2\cos(\mu t))e^{\lambda t}$;
  c) one double root: y(t) = (c₁ + c₂t)e^rt.

FUNDAMENTAL SOLUTIONS for
LINEAR HOMOGENEOUS EQUATIONS

Consider equations of the form

$$L[y]=\frac{d^2y}{dt^2}+ p(t)\frac{dy}{dt}+ q(t)y = g(t),$$

given y(t₀)=y₀, y'(t₀)=y'₀.

Existence and Uniqueness:

if p(t), q(t), and g(t) are
continuous in some open interval I containing t₀, there exists exactly one solution for all $t\in I$.

Solutions of Homogeneous Equations

L[y] = 0.

• Superposition Principle: if y₁ and y₂ are solutions, then any combination c₁y₁+c₂y₂ is a solution.
• The Wronskian for two solutions y₁ and y₂ is
  
  $$W(t) = \left\vert\begin{array}{ll} y_1(t) & y_2(t)\\ y'_1(t)... ...'_2(t) \end{array}\right\vert = y_1(t)y'_2(t)- y_2(t)y'_1(t).$$
  
  

• If y₁ and y₂ are solutions, and $W(t_0)\neq 0$, then
  
  $$c_1 = \left\vert\begin{array}{ll}y_0&y_2(t_0)\\ y'_0&y'_2(t_... ...l}y_1(t_0)&y_0\\ y'_1(t_0)&y'_0\end{array}\right\vert/W(t_0).$$
  
  

• If y₁ and y₂ are solutions, and $W(t_0)\neq 0$, for some t₀ then any solution has the form c₁y₁+c₂y₂; $\{y_1,y_2\}$ is a fundamental set of solutions.

• If y₁ and y₂ are solutions with
  
  $$y_1(t_0)=1,\ y'_1(t_0)=0,\ \ y_2(t_0)=0,\ y'_2(t_0)=1,$$
  
  
  then $\{y_1,y_2\}$ is a fundamental set.