Second-Order Differential Equations
A second-order differential equation is a differential equation in which the highest-order derivative is the second derivative of the unknown function, y'' = f''(x). In general form, it is written as: $$ F(x,y,y',y'')=0 $$ and in its standard form: $$ y''=G(x,y,y') $$
The unknown in the equation is the function y(x).
Solving a second-order differential equation means finding a function y = f(x) whose second derivative satisfies the given relation.
Here’s a basic example of a second-order differential equation:
$$ y'' = y' + y + x $$
The general solution (also called the general integral) of a second-order differential equation is a family of functions that depends on two arbitrary constants, c1 and c2.
$$ y = y(x, c_1, c_2) $$
Assigning specific values to c1 and c2 yields the particular solutions of the differential equation.
Note. A second-order differential equation is said to be homogeneous when the function g(x) is identically zero: $$ y'' + y' + y = g(x) = 0 $$ For example: $$ y'' + 3y' + 4y = 0 $$ It is non-homogeneous (or complete) when g(x) is non-zero at least at one point in the interval of interest: $$ y'' + y' + y = g(x) \ne 0 $$ For instance: $$ y'' + 3y' + 4y = 3x $$
A Practical Example
Consider the following second-order differential equation:
$$ y'' = \sin x $$
This is an elementary differential equation because it can be solved by integrating twice.
We begin by integrating both sides to find the first derivative of the unknown function y':
$$ \int y'' \ dx = \int \sin x \ dx $$
$$ y' = -\cos x + c_1 $$
Next, we integrate again to recover the original function y:
$$ \int y' \ dx = \int (-\cos x + c_1) \ dx $$
$$ y = -\sin x + c_1 x + c_2 $$
This gives the general solution of the differential equation:
$$ y = -\sin x + c_1 x + c_2 $$
Note. Unlike first-order differential equations, the general solution to a second-order differential equation includes two constants, c1 and c2.
Types of Second-Order Differential Equations
Second-order differential equations can be classified into several types:
- Elementary second-order differential equations
These are the simplest cases, typically written in the form y'' = f(x), and are solved by performing two successive integrations: $$ y'' = f(x) $$ - Homogeneous linear equations with constant coefficients
These are linear equations with constant coefficients a, b, and c, and take the form: $$ ay'' + by' + cy = 0 $$ - Non-homogeneous linear differential equations
These are equations of the form: $$ ay'' + by' + cy = g(x) $$ where g(x) ≠ 0 at least at one point in the interval being considered.
And so on.
