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04/18/2023 Differential Equation 1
EXACT & NON EXACT DIFFERENTIAL EQUATION
EXACT DIFFERENTIAL EQUATION
04/18/2023 Differential Equation 3
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
04/18/2023 Differential Equation 4
SOLUTION OF EXACT D.E.
• The solution is given by :
04/18/2023 Differential Equation 5
Example : 1
Find the solution of differential equation
.Solution: Let M(x, y)= and N(x, y)= Now, = , =
04/18/2023 Differential Equation 6
Example : 1 (cont.)
The given differential equation is exact ,⇒
⇒
⇒𝒚𝒆𝒙+𝒚𝟐=c
NON EXACT DIFFERENTIAL
EQUATION
04/18/2023 Differential Equation 8
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.
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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
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CASE I :
If ( i.e. function of x only
Then I.F. =
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Example : 2
Solve :
Solution: Let M(x, y)= and N(x, y)= Now, = , =-
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(
Now, I.F. = = = = =
Example : 2 (cont.)
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Multiply both side by I.F. (i.e. ), we get
Example : 2 (cont.)
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Example : 2 (cont.)
Let M(x, y)= and N(x, y)= Now, = , =
04/18/2023 Differential Equation 16
Example : 2 (cont.)
,which is exact differential equation.It’s solution is :
x −𝑦2
𝑥−3𝑥
=𝑐
04/18/2023 Differential Equation 17
CASE II :
If ) is a function of y only , say g(y), then is an I.F.(Integrating Factor).
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Example : 3
Solve =0Solution: Here M=and so +2 N=and so Thus, and so the given differential equation is non exact.
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Example : 3 (cont.)
Now, =- which is a function of y only . Therefore I.F.==
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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
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Example : 3 (cont.)
Therefore , General Solution is=cwhere c is an arbitrary constant.
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CASE III :
If the given differential equation is homogeneous with then is an I.F.
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Example : 4
Solve Solution: Here M= and so N=and so
The given differential equation is non exact.
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Example : 4 (cont.)
The given differential equation is homogeneous function of same degree=3.[ = = = =
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Example : 4 (cont.)
Now, = =Thus, I.F.=Now, multiplying given differential equation by we have
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Example : 4 (cont.)
Here, M= and so N= and so Thus, and hence (i) is an exact differential equation.
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Example : 4 (cont.)
Therefore , General Solution is
where c is an arbitrary constant.
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CASE IV :
If the given differential equation is of the form is an I.F.
04/18/2023 Differential Equation 29
Example : 4 (cont.)
Solve (Solution: Here, M=( and so N=( and so The given differential equation is non exact.
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Example : 4 (cont.)
Now, = =So, I.F.=Multiplying the given equation by , we have
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Example : 4 (cont.)
----------(i)Here, M=and so N= and so Thus, and hence (i) is an exact differential equation.
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Example : 4 (cont.)
Therefore , General Solution is
where c is an arbitrary constant.
04/18/2023 Differential Equation 33