A solution of a differential equation is an expression for the dependent variable in terms of the independent variable(s) which satisfies the differential equation.

**Solution of
Ordinary Differential Equations**

A **solution** of a differential
equation is an expression for the dependent variable in terms of the
independent variable(s) which satisfies the differential equation.

(i) There
is no guarantee that a differential equation has a solution.

For
instance, ( *y* '(*x*))^{2} + *y*^{2}
+1
=
0 has no solution, since ( *y*
'( *x*))^{2} = − ( *y*^{2}
+1)
and so *y* '(*x*) cannot be real.

(ii) Also,
a solution of a differential equation, if exists, is not unique.

For
instance, the functions *y* =
*e*^{2x} , *y* =
2*e*^{2x} , *y *=
√8*e*^{2x} are solutions of same equation *dy/dx* − 2 *y* = 0. In
fact, *y* = *ce*^{2x} , *c* ∈ ℝ, are all solutions of the differential equation *dy/dx* − 2*y* = 0.

Thus, to
represent all possible solutions of a differential equation, we introduce the
notion of the general solution of a differential equation.

The solution which contains as many arbitrary constants as the
order of the differential equation is called the **general solution**

The **general
solution** includes all possible solutions and typically includes
arbitrary constants (in the case of an ODE) or arbitrary functions (in the case
of a PDE.)

If we
give particular values to the arbitrary constants in the general solution of
differential equation, the resulting solution is called a Particular Solution.

**Remark**

(i) Often
we find a particular solution to a differential equation by giving extra
conditions.

(ii) The
general solution of a first order differential equation *y* ' = *f*
( *x* , *y*) represents a one parameter family of curves in *xy* -plane.

For
instance, *y *=* ce*^{2x}* *,*
c *∈* *ℝ,*
*is the general solution of the differential equation *dy*/*dx* − 2*y *= 0.

For instance,
we have already seen that *y* =
*a* cos *x* + *b* sin *x* satisfies the second order differential
equation *d*^{2}*y/dx*^{2} +
*y* = 0 . Since it contains two arbitrary
constants, it is the general solution of *d*^{2}*y/dx*^{2} +
*y* = 0 . If we put *a* = 1, *b* =
0 in the general solution, then we get *y*
=
cos *x* is a particular solution of the
differential equation *d*^{2}*y/dx*^{2} +
*y* = 0.

In
application, differential equations do not arise by eliminating the arbitrary constants.
They frequently arise while investigating many physical problems in all fields
of engineering, science and even in social sciences. Mostly these differential
equations are also accompanied by certain conditions on the variables to obtain
unique solution satisfying the given conditions.

** **

**Example 10.7**

Show
that *x*^{2}* *+* y*^{2 }= *r*^{2} ,
where *r* is a constant, is a solution of the
differential equation* dy/dx * = − *x/ y*.

**Solution**

Given
that *x*^{2}* *+* y*^{2 }= *r*^{2} , *r* ∈ ℝ ... (1)

The
given equation contains exactly one arbitrary constant.

So, we
have to differentiate the given equation once. Differentiate (1) with respect
to *x* , we get

Thus,* x*^{2} + *y*^{2} = r^{2} satisfies the differential equation* dy/dx* = −* x/y*.

Hence,* x*^{2} + y^{2} = *r*^{2} is a solution of the
differential equation* dy/dx* = −* x/y*.

**Example 10.8 **

Show
that* y *=* mx *+ *7/m*,* m *≠ 0 is a solution of the differential
equation *xy*'+ 7(*1/y*'*)* –* y *=
0.

**Solution**

The
given function is* y *=* mx *+ *7/m*, where* m *is an
arbitrary constant. ... (1)

Differentiating
both sides of equation (1) with respect to*
x *, we get* y *' =* m *.

Substituting
the values of* y *' and* y *in the given differential equation,

we get *xy*′ + 7/*y*′ −* y *= *xm* + 7/*m* −* mx *– 7/*m* = 0 .

Therefore,
the given function is a solution of the differential equation *xy*'+ 7(1/y’) −* y *= 0.

**Example 10.9**

Show
that* y *= 2(x^{2} – 1) + Ce^{−
x2} is a solution of the differential equation *dy*/*dx* + 2xy − 4x^{3}
= 0 .

**Solution**

The
given function is y = 2 ( *x*^{2}
−1) + Ce^{− }^{x}^{2} , where C is an arbitrary constant.
...(1)

Differentiating
both sides of equation (1) with respect to*
x *, we get* dy/dx* = 4*x *− 2xCe^{− }^{x}^{2} .

Substituting
the values of *dy*/*dx* and *y* in the given
differential equation, we get

Therefore,
the given function is a solution of the differential equation *dy/dx* + 2*xy* − 4*x*^{3} = 0 .

Show that* y *= a cos(log* x *) +* b *sin ( log* x *) ,* x *> 0 is a solution of the differential equation x^{2}
y′′ + xy′ +* y *= 0 .

The
given function is* y *= a cos(log* x *) +* b *sin ( log x) ...(1)

where a,* b *are two arbitrary constants. In order
to eliminate the two arbitrary constants, we have to differentiate the given
function two times successively.

Differentiating
equation (1) with respect to* x *, we
get

Again
differentiating this with respect to x, we get

Therefore,* y *= a cos(log* x *) +* b *sin ( log x) is
a solution of the given differential equation.

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

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