# Classification of Differential Equations

Once we have written a differential equation in the form

We can talk about whether a differential equation is
linear or not .

We say that the differential equation above is a linear differential equation if

for all i and j. Any linear ordinary differential equation of degree n can be written as

**Examples**

3x^{2}y'' + 2ln(x)y' + e^{x} y = 3xcos x

is a second order linear ordinary differential equation.

4yy''' - x^{3}y' + cos y = e^{2x}

is not a linear differential equation because of the 4yy''' and the cos y terms .

**MyPhysicsLab – Classifying Differential Equations
**When you study differential equations, it is kind of like botany. You learn to
look at an

equation and classify it into a certain group . The reason is that the techniques for solving

differential equations are common to these various classification groups. And sometimes

you can transform an equation of one type into an equivalent equation of another type, so

that you can use easier solution techniques . Here then are some of the major

classifications of differential equations:

**First Order, Second Order, etc.
**The order of a differential equation is equal to the highest derivative in the
equation. The

single-quote indicates differention. So x' is a first derivative, while x'' is a second

derivative. x' = 1/x is first-order x'' = −x is second-order x'' + 2 x' + x = 0 is second-order.

**Linear vs. Non-linear**

Linear just means that the variable in an equation appears only with a power of
one . So x

is linear but x^{2} is non-linear. Also any function like cos(x) is non-linear.

In math and physics , linear generally means "simple" and non-linear means

"complicated". The theory for solving linear equations is very well developed
because

linear equations are simple enough to be solveable. Non-linear equations can
usually not

be solved exactly and are the subject of much on-going research. Here is a brief

description of how to recognize a linear equation.

Recall that the equation for a line is y = m x + b where m, b are constants ( m
is the slope ,

and b is the y- intercept ). In a differential equation, when the variables and
their

derivatives are only multiplied by constants , then the equation is linear. The
variables

and their derivatives must always appear as a simple first power. Here are some
examples.

x'' + x = 0 is linear x'' + 2x' + x = 0 is linear x' + 1/x = 0 is non-linear
because 1/x is not a

first power x' + x^{2} = 0 is non-linear because x^{2} is not a first power x'' + sin(x) = 0 is nonlinear

because sin(x) is not a first power x x' = 1 is non-linear because x' is not
multiplied by a

constant Similar rules apply to multiple variable problems. x' + y' = 0 is
linear x y' = 1 is

non-linear because y' is not multiplied by a constant Note, however, that an
exception is made

for the time variable t (ie. the variable that we are differentiating by). We
can have any

function of t appear in the equation, but still have an equation that is linear
in x. x'' + 2 x'

+ x = sin(t) is linear in x x' + t^{2}x = 0 is linear in x

**Linear vs. non-linear
**Linear differential equations do not contain any higher powers of any
differentials, nonlinear

differential equations do. In fully linear differential equations, higher powers of the

independent variable(s) are also excluded.

Examples:

and are ODE, but and are

PDE.

All of the examples above are linear, but isn't. Note that

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