A differential equation is any equation which contains at least one derivative of an unknown function, either ordinary derivative or partial derivative.

**Differential
Equation, Order, and Degree**

A **differential equation** is any equation which
contains at least one derivative of an unknown function, either ordinary
derivative or partial derivative.

For
instance, let *y* =
*f* ( *x*) where *y* is a dependent
variable ( *f* is an unknown function)
and *x* is an independent variable.

(1) The
equation *dy/dx* =
0 is a differential equation.

(2) The
equation *dy/dx* =
sin *x* is a differential equation.

(3) The
equation *dy/dx* +
*y* = 7*x* + 5 is a differential equation.

(4) The
equation *d*^{2}*y/dx*^{2} +
*dy/dx* + *y*
=
sin *x* is a differential equation.

(5) The
equation *e ^{dy/dx}* =
ln

(6) The
equation is a differential equation.

The **order** of a differential
equation is the highest order derivative present in the differential equation.

Thus, if
the highest order derivative of the unknown function* y *in the equation is *k*^{th}
derivative, then the order of the differential equation is *k*. Clearly *k* must be a
positive integer.

For
example, is a differential equation of order three.

If a differential equation is expressible in a polynomial form,
then the integral power of the highest order derivative appears is called the **degree** of the differential equation

In other
words, the **degree**
of a differential equation is the power of the highest order derivative
involved in the differential equation when the differential equation (after
expressing in polynomial form) satisfies the following conditions :

(i) All
of the derivatives in the equation are free from fractional powers, if any.

(ii) Highest
order derivative should not be an argument of a transcendental function,
trigonometric or exponential, etc. The coefficient of any term containing the
highest order derivative should just be a function of *x, y*, or some lower order derivative but not as transcendental,
trigonometric, exponential, logarithmic function of derivatives.

If one
or more of the aforementioned conditions are not satisfied by the differential
equation, it should be first reduced to the polynomial form in which it
satisfies all of the above conditions.

**If a differential
equation is not expressible to polynomial equation form having the highest
order derivative as the leading term then that the degree of the differential
equation is not defined.**

The
determination of the degree of a given differential equation can be tricky if
you are not well versed with the conditions under which the degree of the
differential equation is defined. So go through the given solved examples
carefully and master the technique of calculating the degree of the given
differential equation just by sheer inspection!

(1) Consider the differential equation

The
highest order derivative involved here is 2, and its power is 1 in the
equation. Thus, the order of the differential equation is 2 and degree is 1.

(2) Consider
the differential equation

Since
this equation involves fractional powers, we must first get rid of them. On
squaring the equation, we get

Now, we
can clearly make out that the highest order derivative is 3. Therefore order of
the differential equation is 3 and since its power is 2 in the equation, the
degree of the differential equation is 2.

(3) Consider
the differential equation

Here,
the highest order derivative is 2. Because of sine of first derivative, the
given differential equation can not be expressed as polynominal equation. So,
the order of the differential equation is 2, and, it is not in polynomial
equation in derivatives and so degree is not defined.

(4) Consider the equation

Here,
the highest order derivative (order is 2) has involvement in an exponential
function. This cannot be expressed as polynomial equation with *d*^{2}*y / dx*^{2} as the leading term So, the degree of the
equation is not defined. The order of the equation is 2.

(5) Further,
the following differential equations do not have degrees.

(6) The
differential equation 10(*y*′′′)^{4}
+
7 ( *y*′′)^{5} +
sin( *y*′) + 5 = 0 has order 3 but degree is not defined.

(7) The
differential equation cos( *y*′)
*y*′′′ + 5 *y* ′′ + 7 *y*′ = sin *x* has order 3 and degree is not defined.

Observe
that the degree of a differential equation is always a positive integer.

** **

Determine
the order and degree (if exists) of the following differential equations:

(i) In
this equation, the highest order derivative is *dy/dx* whose power is 1 Therefore, the given differential equation
is of order 1 and degree 1.

(ii) Here,
the highest order derivative is *d*^{4}*y/ dx ^{2}* whose power is 3.

Therefore,
the given differential equation is of order 4 and degree 3.

(iii) In
the given differential equation, the highest order derivative is *d*^{2}*y/dx*^{2} whose power is 1.

Therefore,
the given differential equation is of order 2.

The
given differential equation is not a polynomial equation in its derivatives and
so its degree is not defined.

(iv) The
given differential equation is

Squaring
both sides, we get

In this
equation, the highest order derivative is *d*^{2}*y/ dx*^{2} whose power is 2.

Therefore,
the given differential equation is of order 2 and degree 2.

(v) *dy* + ( *xy*
−
cos *x* ) *dx*
=
0 is a first order differential equation with degree 1, since the equation can
be rewritten as *dy*/*dx* + *xy* − cos *x* = 0 .

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

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12th Maths : UNIT 10 : Ordinary Differential Equations : Differential Equation, Order, and Degree | Ordinary Differential Equations | Mathematics

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