Differential Equations

A differential equation describes the relationship between an independent variable (on the bottom of the derivative), and a dependent variable (on the top of the derivative) and its derivatives. Differential equations can be categorized by:

The order of a differential equation is the highest order derivative of the dependent variable.
A linear differential equation is where the dependent variable appears to a maximum power of 1. Any differential equation with other functions of the dependent variable, e.g. , , is not linear.
A homogenous differential equation is where every term involves the dependent variable. All non-homogeneous differential equations have an associated homogeneous differential equation by removing terms that don't involve the dependent variable.

For a differential equation with independent variable and dependent variable , solving a differential equation involves finding in terms of . The solution will contain arbitrary constants of integration, where the general solution to an -th order differential equation has arbitrary constants.

To find the arbitrary constants, initial conditions or boundary conditions are needed. Each piece of information, e.g. or for a value of gives another arbitrary constant. The particular solution is where all the arbitrary constants have been found so the solution fits the conditions.

Separable differential equations

Some differential equations can be solved by separation of variables. This is where the equation can be rearranged so one side only has terms and the other side only has terms. Then:

can be separated as a fraction.
Integrate both sides.

A proof for this result is given below. A separable differential equation can be written in the following form:

By considering the integrals of and :

Consider the following equation:

Where going from the first line to the second involves differentiating by and applying the chain rule on the left. Thus, our original equation can be solved by multiplying across by and integrating both sides.

Non-homogeneous differential equations

For linear differential equations, the general solution to a non-homogeneous differential equation is the sum of:

The general solution to the associated homogeneous equation, called the complementary function.
One solution to the full non-homogeneous equation, called the particular integral.
For a linear differential equation, the general solution is: Where is the complementary function and is a particular integral.

The integrating factor

Consider a linear first-order differential equation:

We define , the integrating factor, to be:

By multiplying both sides of the original equation by :

Now, the left-hand side is a result of applying the product rule:

Giving a solution to the original differential equation.

Second-order differential equations with constant coefficients

A homogeneous second-order differential equation with constant coefficients is of the form:

To solve this, an auxiliary equation is formed:

The general solution depends on the solutions to the auxiliary equation:

If there are two real roots and , the general solution is .
If there is a repeated real root , the general solution is .
If there are two complex roots , the general solution is .

A non-homogeneous second-order differential equation with constant coefficients is of the form:

To solve equations of this form, first solve the homogeneous equation to obtain the complementary function. Then, find a particular integral using a trial function, by substituting in a trial function into the differential equation to find the coefficients. The trial function is of the form:

for a linear polynomial
A general polynomial of the same order for any polynomial
for
for either or .
If the trial function is already a part of the complementary function, then multiply the trial function by .