Table of Contents
Overview
In this chapter we focus on second-order differential equations: equations that involve a function $y(x)$ and its derivatives up to $y''(x)$, but not higher. The simplest and most important class for a first course are linear second-order equations, especially those with constant coefficients. These already cover many classical models in physics and engineering (springs, circuits, vibrations, and more).
We assume you are already familiar with:
- What a differential equation is.
- Basic ideas for first-order equations (from the parent chapter).
Here we concentrate on the new features that appear with order $2$.
A general second-order differential equation has the form
$$
F\bigl(x,\,y(x),\,y'(x),\,y''(x)\bigr)=0.
$$
A linear second-order equation has the special form
$$
a_2(x)\,y'' + a_1(x)\,y' + a_0(x)\,y = g(x),
$$
where $a_2(x)\neq 0$ and $a_0, a_1, a_2, g$ are given functions of $x$. The function $g(x)$ is called the forcing term or nonhomogeneous term.
The two most important subclasses we study are:
- Homogeneous linear equations: $g(x)=0$.
- Nonhomogeneous linear equations: $g(x)\neq 0$.
Here we introduce the main ideas and solution methods for these equations, with emphasis on constant coefficients (where $a_2, a_1, a_0$ are constants).
Homogeneous linear equations with constant coefficients
A homogeneous linear second-order equation with constant coefficients has the form
$$
a\,y'' + b\,y' + c\,y = 0,
$$
where $a, b, c$ are constants and $a\neq 0$.
The key idea is to look for solutions of the form
$$
y = e^{rx},
$$
where $r$ is a constant to be determined. Substituting this into the equation leads to an algebraic equation for $r$ called the characteristic equation.
Characteristic equation
If $y = e^{rx}$, then $y' = r e^{rx}$ and $y'' = r^2 e^{rx}$. Substitute into
$$
a\,y'' + b\,y' + c\,y = 0:
$$
$$
a(r^2 e^{rx}) + b(r e^{rx}) + c(e^{rx}) = 0.
$$
Factor out $e^{rx}$ (which is never zero):
$$
e^{rx}\bigl(a r^2 + b r + c\bigr) = 0.
$$
Thus we must have
$$
a r^2 + b r + c = 0.
$$
This quadratic equation in $r$ is the characteristic equation.
Solving it gives roots $r_1$ and $r_2$, which determine the shape of the general solution. There are three cases, depending on the discriminant $D = b^2 - 4ac$.
Case 1: Two distinct real roots
If $D>0$, the characteristic equation has two distinct real roots $r_1\neq r_2$. Then
$$
y_1(x) = e^{r_1 x}, \qquad y_2(x) = e^{r_2 x}
$$
are two independent solutions, and the general solution of the differential equation 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 shape:
- If both $r_1$ and $r_2$ are negative, solutions decay to $0$ as $x\to\infty$.
- If one root is positive and the other negative, solutions grow in one direction and decay in the other.
Case 2: Repeated real root
If $D=0$, the characteristic equation has a repeated real root $r_1 = r_2 = r$. Then $e^{rx}$ is one solution. To get a second, independent solution, we use the fact (provable by deeper theory) that
$$
y_1(x) = e^{r x}, \qquad y_2(x) = x e^{r x}
$$
form a pair of independent solutions.
The general solution is
$$
y(x) = \bigl(C_1 + C_2 x\bigr)e^{r x}.
$$
This form is specific to repeated roots: the extra factor $x$ supplies a second independent solution.
Case 3: Complex conjugate roots
If $D<0$, the characteristic equation has complex roots
$$
r = \alpha \pm i\beta, \quad \beta\neq 0.
$$
We still get exponential solutions $e^{(\alpha \pm i\beta)x}$, but it is more useful to rewrite these in terms of real functions using Euler’s formula:
$$
e^{(\alpha + i\beta)x} = e^{\alpha x}(\cos \beta x + i \sin \beta x).
$$
From this, one can show that
$$
y_1(x) = e^{\alpha x}\cos(\beta x), \qquad
y_2(x) = e^{\alpha x}\sin(\beta x)
$$
are two real independent solutions.
The general real solution is
$$
y(x) = e^{\alpha x}\bigl(C_1\cos(\beta x) + C_2\sin(\beta x)\bigr).
$$
Interpretation:
- $\alpha$ controls exponential growth or decay.
- $\beta$ controls oscillation frequency.
This case is fundamental for modeling oscillations such as mass–spring systems and electrical circuits.
Initial value problems and uniqueness
A second-order initial value problem typically specifies $y$ and $y'$ at some initial point $x_0$:
$$
\begin{cases}
a\,y'' + b\,y' + c\,y = 0,\\
y(x_0)=y_0,\\
y'(x_0)=v_0.
\end{cases}
$$
Once the general solution is written in terms of $C_1$ and $C_2$, these two conditions allow you to solve for $C_1, C_2$ uniquely (under mild assumptions on the coefficients, satisfied in the constant coefficient case). Thus a second-order equation typically needs two initial conditions to determine one particular solution.
Nonhomogeneous linear equations with constant coefficients
A nonhomogeneous linear second-order equation with constant coefficients has the form
$$
a\,y'' + b\,y' + c\,y = g(x),
$$
where $g(x)$ is some given function (the forcing term).
The solution method is based on two pieces:
- The homogeneous solution $y_h(x)$: solve
$$
a\,y'' + b\,y' + c\,y = 0
$$
as above. - A particular solution $y_p(x)$: find one specific function $y_p$ that satisfies the full equation
$$
a\,y_p'' + b\,y_p' + c\,y_p = g(x).
$$
Once you have these, the general solution of the nonhomogeneous equation is
$$
y(x) = y_h(x) + y_p(x).
$$
The most common systematic method for $y_p$ in constant-coefficient problems is the method of undetermined coefficients.
Method of undetermined coefficients (idea)
This method applies when $g(x)$ is of a “simple” type: polynomials, exponentials, sines and cosines, or finite sums/products of these.
The basic strategy:
- Guess a form for $y_p(x)$ that has the same general shape as $g(x)$, with unknown constants (“undetermined coefficients”).
- Substitute your guess into the differential equation.
- Solve for the unknown coefficients so that the equation holds for all $x$.
Some typical choices:
- If $g(x)$ is a polynomial of degree $n$, guess a general polynomial of degree $n$ for $y_p(x)$.
- If $g(x)=e^{kx}$, guess $y_p(x)=Ae^{kx}$.
- If $g(x)=\cos(\omega x)$ or $\sin(\omega x)$, guess $y_p(x)=A\cos(\omega x)+B\sin(\omega x)$.
- If $g(x)$ is a sum, use a sum of guesses.
There is an important adjustment: if your guessed form for $y_p$ overlaps with any part of the homogeneous solution $y_h$, you multiply the guessed form by a suitable power of $x$ to make it linearly independent. (The details of this adjustment are often treated step by step in examples.)
Structure of solutions: two-dimensional solution space
For a second-order linear homogeneous equation
$$
a_2(x) y'' + a_1(x) y' + a_0(x) y = 0
$$
(with mild conditions on $a_0,a_1,a_2$), the set of all solutions behaves like a two-dimensional vector space:
- Any linear combination of solutions is again a solution.
- There exist two independent solutions $y_1, y_2$ such that every solution can be written uniquely as
$$
y(x) = C_1 y_1(x) + C_2 y_2(x).
$$
The pair $\{y_1, y_2\}$ is called a fundamental set of solutions. The fact that a second-order equation needs two initial conditions (for $y$ and $y'$) corresponds to this two-dimensional structure.
For nonhomogeneous equations, adding any solution of the homogeneous equation to one particular solution yields another solution. This is why the general solution is always $y = y_h + y_p$.
Beyond constant coefficients (brief outlook)
This chapter has focused on second-order linear equations with constant coefficients because they are solvable by simple algebraic methods and arise in many applications.
For equations with variable coefficients,
$$
a_2(x) y'' + a_1(x) y' + a_0(x) y = g(x),
$$
the structure (two independent solutions, a homogeneous part plus a particular part) is still valid for linear equations, but:
- We usually cannot use the characteristic equation.
- Methods such as reduction of order and variation of parameters are often employed.
- Many equations cannot be solved in closed form and require numerical methods.
These more advanced techniques and examples are treated in the more specialized subsections and in later applications.