exact & non differential equation

04/18/2023 Differential Equation 1

EXACT & NON EXACT DIFFERENTIAL EQUATION

EXACT DIFFERENTIAL EQUATION


EXACT DIFFERENTIAL EQUATION

A differential equation of the form M(x, y)dx + N(x, y)dy = 0

is called an exact differential equation if and only if x

NyM


SOLUTION OF EXACT D.E.

• The solution is given by :


Example : 1

Find the solution of differential equation

.Solution: Let M(x, y)= and N(x, y)= Now, = , =


Example : 1 (cont.)

The given differential equation is exact ,⇒

⇒

⇒𝒚𝒆𝒙+𝒚𝟐=c

NON EXACT DIFFERENTIAL

EQUATION


NON EXACT DIFFERENTIAL EQUATION

• For the differential equation IF then,

• If the given differential equation is not exact then make that equation exact by finding INTEGRATING FACTOR.


INTEGRATING FACTOR

• In general, for differential equationM(x, y)dx + N(x, y)dy = 0

is not exact.In such situation, we find a function such that by multiplying to the equation, it becomes an exact equation.So,M(x, y)dx +N(x, y)dy = 0 becomes exact equation

Here the function is then called an Integrating Factor

Methods to find an INTEGRATING FACTOR (I.F.) for given non exact

equation:

M(x, y)dx + N(x, y)dy = 0

CASES:CASE ICASE IICASE IIICASE IV


CASE I :

If ( i.e. function of x only

Then I.F. =


Example : 2

Solve :

Solution: Let M(x, y)= and N(x, y)= Now, = , =-


(

Now, I.F. = = = = =

Example : 2 (cont.)


Multiply both side by I.F. (i.e. ), we get

Example : 2 (cont.)


Example : 2 (cont.)

Let M(x, y)= and N(x, y)= Now, = , =


Example : 2 (cont.)

,which is exact differential equation.It’s solution is :

x −𝑦2

𝑥−3𝑥

=𝑐


CASE II :

If ) is a function of y only , say g(y), then is an I.F.(Integrating Factor).


Example : 3

Solve =0Solution: Here M=and so +2 N=and so Thus, and so the given differential equation is non exact.


Example : 3 (cont.)

Now, =- which is a function of y only . Therefore I.F.==


Example : 3 (cont.)

Multiplying the given differential equation by ,we have ----------------(i)Now here, M= and so N= and so Thus, and hence (i) is an exact differential equatio


Example : 3 (cont.)

Therefore , General Solution is=cwhere c is an arbitrary constant.


CASE III :

If the given differential equation is homogeneous with then is an I.F.


Example : 4

Solve Solution: Here M= and so N=and so

The given differential equation is non exact.


Example : 4 (cont.)

The given differential equation is homogeneous function of same degree=3.[ = = = =


Example : 4 (cont.)

Now, = =Thus, I.F.=Now, multiplying given differential equation by we have


Example : 4 (cont.)

Here, M= and so N= and so Thus, and hence (i) is an exact differential equation.


Example : 4 (cont.)

Therefore , General Solution is

where c is an arbitrary constant.


CASE IV :

If the given differential equation is of the form is an I.F.


Example : 4 (cont.)

Solve (Solution: Here, M=( and so N=( and so The given differential equation is non exact.


Example : 4 (cont.)

Now, = =So, I.F.=Multiplying the given equation by , we have


Example : 4 (cont.)

----------(i)Here, M=and so N= and so Thus, and hence (i) is an exact differential equation.


Example : 4 (cont.)

Therefore , General Solution is

where c is an arbitrary constant.

exact & non differential equation

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