Ordinary Differential Equation, Partial Differential Equation, A general linear ordinary differential equation of order n, A nonlinear ordinary differential equation, non-homogeneous

**Classification of
Differential Equations**

If a differential equation contains only ordinary derivatives of
one or more functions with respect to a single independent variable, it is said
to be an **Ordinary Differential Equation (ODE)**.

An equation involving only partial derivatives of one or more
functions of two or more independent variables is called a **Partial Differential Equation (PDE)**.

For
instance, let *y* denote the unknown
function and *x* be independent
variable. Then

are some
examples of ordinary differential equations.

For
instance,

are some
examples of partial differential equations.

In this
chapter, we discuss ordinary differential equations only.

Ordinary
differential equations are classified into two different categories namely **linear **ordinary
differential equations and **nonlinear** ordinary differential equations.

**A general linear
ordinary differential equation of order n**

*a _{n} *(

where the coefficients *a _{n}*
(

**Note**

(1) The
important thing to note about linear differential equations is that there are
no products of the function, *y* ( *x*) , and its derivatives and neither the
function nor its derivatives occur to any power other than the first power.

(2) No transcendental functions – (trigonometric or logarithmic
etc) of *y* or any of its

derivatives occur in differential equation.

(3) Also
note that neither the function nor its derivatives are “inside” another
function, for instance, √*y*′* *or* e ^{y}*

(4) The
coefficients *a*_{0} (
*x* ) , *a*_{1} ( *x*
), . . . ,
*a _{n}*

A **nonlinear
ordinary differential equation** is simply
one that is not linear.

If the coefficients of *y* , *y*′, *y* ′′, , *y*( * ^{n}*)
contain the dependent variable

For instance,

(1) *dy/dx = ax ^{3}
, d^{2}y//dx*

(2) *y *′′
+*
*2*x*^{3}*y *′ =* *7*xy
*+* x*^{2}* *is a second order linear ODE.

(3) *y *′′
+*
y*′ =* *√*x *is a second order
linear ODE.

(4) *y* ^{2} +
*y*′ = √*x* is a first order nonlinear ODE.

(5) *y *′* *=* x *sin(* y*) is a first order nonlinear ODE.

(6) *y *′′* *=* y *sin(* x*) is a second order linear ODE.

If *g* ( *x*) = 0 in (1), then the above equation is said to be **homogeneous**, otherwise it is called **non-homogeneous**.

If *y _{i}*(

* a _{n} *(

then *a _{n}* (

Suppose *u* ( *x*)
=
*c*_{1} *y*_{1} (*x* ) +
*c*_{2} *y* _{2} (*x*) ,
where *c*_{1 }and *c*_{2} are arbitrary constants.
Then, it can be easily_{ }verified that *u* ( *x*) is also a solution
of (2)._{}

Thus, a
first order linear differential equation is written as *y*′* *+* p*(* x *)*
y *=* f *(*
x*)* *. A first order differential
equation that can’t be written like this is nonlinear. Since *y *=* *0*
*is obviously a solution* *of the
homogeneous equation *y*′
+
*p*(*x*
) *y* = 0 , we call it the trivial solution.
Any other solution is nontrivial. In fact this is true for a general linear
homogeneous differential equation as well.

Tags : Mathematics , 12th Maths : UNIT 10 : Ordinary Differential Equations

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12th Maths : UNIT 10 : Ordinary Differential Equations : Classification of Differential Equations | Mathematics

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