homogeneous_second_order

Homogeneous Second-Order ODE

Definition

The homogeneous second-order ODE with constant coefficients:

$$a\ddot{x}+b\dot{x}+cx=0,$$

where $a,b,c$ are constants.

Intuition

The equation models a damped harmonic oscillator: $a$ is inertia, $b$ is damping, and $c$ is restoring force. Trying a solution of the form $e^{rt}$ is natural because exponentials are eigenfunctions of the derivative operator. The discriminant $b^2-4ac$ then determines whether the system oscillates, decays monotonically, or sits at the boundary.

Formal Description

The exponential ansatz $x=e^{rt}$ converts the ODE into an algebraic characteristic equation:

$$a{r}^2+br+c=0,$$

with roots

$$r_{\pm }=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

The general solution is a linear superposition of two linearly independent solutions (verified by the Wronskian); the two free constants are determined from initial values on $x$ and $\dot{x}$.

Key Results

Three cases arise depending on the discriminant $b^2-4ac$:

Discriminant	Root type	General solution
$b^2-4ac>0$	Distinct real $r_1,r_2$	$x(t)=c_1{e}^{r_{1}t}+c_2{e}^{r_{2}t}$
$b^2-4ac<0$	Complex conjugates $\lambda \pm i\mu$	$x(t)=e^{\lambda t}(A\cos \mu t+B\sin \mu t)$
$b^2-4ac=0$	Repeated root $r$	$x(t)=(c_1+c_{2}t)e^{rt}$

See Characteristic Roots for derivations of all three cases.

Applications

• Mechanical spring-mass-damper systems.
• Electrical RLC circuits ($L\ddot{q}+R\dot{q}+q/C=0$).
• Foundation for the inhomogeneous solution and linear ODE systems.

Trade-offs

• Restricted to constant coefficients; variable-coefficient second-order ODEs generally require series solutions or numerical methods.
• The exponential ansatz always works for constant-coefficient homogeneous linear ODEs; no guessing about the solution form is needed.
• Linear independence of the two solutions must be confirmed via the Wronskian before forming the general solution.

Links

• Characteristic Roots
• Principle of Superposition
• Wronskian
• Inhomogeneous Second-Order ODE