Second-order differential equation

Definition

The term second-order differential equation is used for any differential equation whose order is 2. In other words, it is a differential equation of the form:

$F\left(x,y,y',y''\right)=0$

where $F$ is an expression (function) involving the four variables. Note that $F$ must make use of $y''$ though its use of the other variables is optional.

For more on general theory and terminology, see differential equation.

Solution strategies

General idea

In most cases, the solution strategy looks something like this:

1. First, we construct an auxiliary first-order differential equation. How this first-order differential equation is constructed depends on the solution strategy, but in general, we do some type of first-order substitution.
2. For every solution of the auxiliary first-order differential equation, we get a new first-order differential equation. The solutions to the original second-order differential equation are expected to be the union of the solutions to each of these first-order differential equations.

Thus, we need to solve two first-order differential equations, but the second first-order differential equation is not a single equation but a family of first-order differential equations, one for each value of the parameter in the solution for the auxiliary first-order differential equation. Thus, we may need to split the solution strategy for the family of first-order differential equations in Step (2) based on cases on the value of the parameter describing the family. See explicit examples under second-order first-degree autonomous differential equation.

Solution strategies in particular cases

Equation type	Degree (if polynomial in highest order derivative)	Quick summary of solution strategy
second-order homogeneous linear differential equation with constant coefficients	1	Straightforward; find roots of characteristic polynomial using the quadratic formula. See homogeneous linear differential equation with constant coefficients
second-order first-degree autonomous differential equation	1	We do a first-order substitution, solve for the derivative in terms of the function, then plug back and solve a first-order differential equation.