Table of Contents
A second-order homogeneous differential equation is a second-order differential equation in which every term involves the unknown function or its derivatives, and there is no “free” term depending only on the independent variable. In symbols, a linear second-order equation has the general form
$$
a_2(x)\,y'' + a_1(x)\,y' + a_0(x)\,y = g(x),
$$
and it is called homogeneous precisely when $g(x) = 0$ for all $x$:
$$
a_2(x)\,y'' + a_1(x)\,y' + a_0(x)\,y = 0.
$$
Here we focus on the most important and most manageable case: linear second-order homogeneous equations with constant coefficients.
Linear homogeneous equations with constant coefficients
A very common and useful class of equations is
$$
y'' + ay' + by = 0,
$$
where $a$ and $b$ are constants. This is called:
- second-order (because of $y''$),
- linear (because $y, y', y''$ appear only to the first power and are not multiplied by each other),
- homogeneous (right-hand side equals $0$),
- with constant coefficients (the numbers in front of $y'', y', y$ are constants).
The key idea is to guess solutions of a special form and see what conditions they must satisfy.
Exponential trial solution and characteristic equation
We look for solutions of the form
$$
y = e^{rx},
$$
where $r$ is a number to be determined. Then
$$
y' = re^{rx}, \quad y'' = r^2 e^{rx}.
$$
Substitute into
$$
y'' + ay' + by = 0:
$$
$$
r^2 e^{rx} + a\,r e^{rx} + b\,e^{rx} = 0.
$$
Factor out $e^{rx}$:
$$
e^{rx}(r^2 + ar + b) = 0.
$$
Since $e^{rx} \neq 0$ for all $x$, we must have
$$
r^2 + ar + b = 0.
$$
This quadratic equation in $r$ is called the characteristic equation (or auxiliary equation) of the differential equation.
So the differential equation
$$
y'' + ay' + by = 0
$$
is reduced to studying the roots of
$$
r^2 + ar + b = 0.
$$
Three cases for the characteristic roots
The discriminant of the quadratic equation $r^2 + ar + b = 0$ is
$$
\Delta = a^2 - 4b.
$$
The nature of the solutions to the differential equation depends on whether $\Delta$ is positive, zero, or negative.
We will describe the form of the general solution in each case. The details of why the forms look this way are part of general theory of linear differential equations and exponentials; here we just focus on the results specific to homogeneous equations.
Case 1: Two distinct real roots ($\Delta > 0$)
If $a^2 - 4b > 0$, the characteristic equation has two distinct real roots $r_1$ and $r_2$:
$$
r_1 \neq r_2, \quad r_1, r_2 \in \mathbb{R}.
$$
Then the corresponding exponential functions $e^{r_1 x}$ and $e^{r_2 x}$ are both solutions to the differential equation, and the general solution is
$$
y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x},
$$
where $C_1$ and $C_2$ are arbitrary constants.
Example pattern: For
$$
y'' - 3y' + 2y = 0,
$$
the characteristic equation is
$$
r^2 - 3r + 2 = 0 \quad \Rightarrow \quad (r - 1)(r - 2) = 0,
$$
so $r_1 = 1,\ r_2 = 2$, and the general solution has the form
$$
y = C_1 e^{x} + C_2 e^{2x}.
$$
Case 2: Repeated real root ($\Delta = 0$)
If $a^2 - 4b = 0$, the characteristic equation has a single real root $r$ of multiplicity two:
$$
r_1 = r_2 = r.
$$
In this situation, $y_1 = e^{rx}$ is one solution, but $y_2 = e^{rx}$ is not new—we need a second, linearly independent solution. For this case, it turns out that $x e^{rx}$ is also a solution, and independent from $e^{rx}$.
Therefore, the general solution is
$$
y(x) = C_1 e^{rx} + C_2 x e^{rx}.
$$
Example pattern: For
$$
y'' - 4y' + 4y = 0,
$$
the characteristic equation is
$$
r^2 - 4r + 4 = 0 \quad \Rightarrow \quad (r - 2)^2 = 0,
$$
so $r = 2$, and
$$
y = C_1 e^{2x} + C_2 x e^{2x}.
$$
Case 3: Complex conjugate roots ($\Delta < 0$)
If $a^2 - 4b < 0$, the characteristic equation has complex conjugate roots
$$
r_{1,2} = \alpha \pm i\beta,
$$
with $\alpha, \beta \in \mathbb{R}$ and $\beta \neq 0$.
From the exponential form $e^{(\alpha \pm i\beta)x}$ and the relationship between complex exponentials and sines and cosines, it follows that a real-valued general solution can be written as
$$
y(x) = e^{\alpha x}\bigl(C_1 \cos(\beta x) + C_2 \sin(\beta x)\bigr).
$$
Example pattern: For
$$
y'' + y = 0,
$$
the characteristic equation is
$$
r^2 + 1 = 0 \quad \Rightarrow \quad r = \pm i.
$$
Here $\alpha = 0$ and $\beta = 1$, so the general solution is
$$
y = C_1 \cos x + C_2 \sin x.
$$
As another pattern, for
$$
y'' + 4y' + 13y = 0,
$$
the characteristic equation is
$$
r^2 + 4r + 13 = 0,
$$
with discriminant
$$
\Delta = 4^2 - 4\cdot 13 = 16 - 52 = -36 < 0.
$$
The roots are
$$
r = \frac{-4 \pm \sqrt{-36}}{2} = -2 \pm 3i.
$$
Thus $\alpha = -2$, $\beta = 3$, and the general solution has the form
$$
y = e^{-2x}\bigl(C_1 \cos(3x) + C_2 \sin(3x)\bigr).
$$
Superposition principle for homogeneous linear equations
A key feature of homogeneous linear equations such as
$$
y'' + ay' + by = 0
$$
is that sums of solutions are also solutions, and multiplying a solution by a constant gives another solution. This is sometimes called the superposition principle.
More precisely, if $y_1$ and $y_2$ are both solutions of the homogeneous equation, then
$$
y(x) = C_1 y_1(x) + C_2 y_2(x)
$$
is also a solution, for any constants $C_1, C_2$.
This principle is exactly why the general solutions above are written as combinations of two basic solutions; they describe all possible solutions to the homogeneous equation.
Initial value problems for homogeneous second-order equations
A typical initial value problem (IVP) for a second-order homogeneous equation with constant coefficients looks like
$$
y'' + ay' + by = 0, \quad y(x_0) = y_0, \quad y'(x_0) = v_0,
$$
where $x_0, y_0, v_0$ are given numbers.
The homogeneous equation has a general solution of one of the forms described above (depending on the characteristic roots). Plugging in $x = x_0$ into $y(x)$ and $y'(x)$ gives two equations in the unknown constants $C_1$ and $C_2$. Solving those two equations uniquely determines $C_1$ and $C_2$, and thus gives the specific solution satisfying the initial conditions.
The general theory of existence and uniqueness guarantees (under mild assumptions) that this homogeneous IVP has exactly one solution.
Homogeneous equations and oscillations
Many physical systems at rest, or in equilibrium, lead to homogeneous equations when no external forcing is present. For example:
- A simple mass–spring system without damping or driving forces leads to an equation like
$$
y'' + \omega^2 y = 0,
$$
whose solutions are combinations of sines and cosines. - Adding damping (resistance) leads to an equation like
$$
y'' + c y' + k y = 0,
$$
where the characteristic roots determine whether the motion is oscillatory or not, and whether it decays or persists.
The presence of $\cos$ and $\sin$ (or exponentials with complex roots) in the solutions of homogeneous equations with constant coefficients is closely tied to oscillatory behavior, but the detailed physical interpretation belongs to applications rather than the abstract theory itself.
Summary of solution forms
For a homogeneous linear second-order differential equation with constant coefficients
$$
y'' + ay' + by = 0,
$$
the characteristic equation is
$$
r^2 + ar + b = 0.
$$
Let the roots be $r_1$ and $r_2$.
- If $r_1$ and $r_2$ are distinct real numbers:
$$
y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x}.
$$ - If there is a repeated real root $r$:
$$
y(x) = C_1 e^{rx} + C_2 x e^{rx}.
$$ - If the roots are complex conjugates $r = \alpha \pm i\beta$ with $\beta \neq 0$:
$$
y(x) = e^{\alpha x}\bigl(C_1 \cos(\beta x) + C_2 \sin(\beta x)\bigr).
$$
These patterns are central to understanding homogeneous second-order differential equations and form the foundation for studying more general and more complicated equations, including non-homogeneous ones and equations with variable coefficients.