Sec 2.4: Exact Differential Equations

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Sec 2.4 Exact Differential Equations
Find the differential
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Sec 2.4
Sec 2.4 Exact Differential Equations
Definition 2.3 (part 1)
An expression
is an exact if it corresponds to the
differential of some function f(x,y)
Definition 2.3 (part 2)
A 1st order DE
is an exact if LHS is an exact differential
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Sec 2.4
How?? check exact or not
Theorem 2.1
----- (1)
Given a DE
(1) exact
If
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Sec 2.4
How to Solve ?
----- (1)
Given a DE
Method of Solution
Step 1
Check if (1) is an exact
Step 2
Find
Step 3
The family of solutions is
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Sec 2.4
How to find f(x,y) ?
----- (1)
Given an exact DE
Step 1
Integrate wrt x
----- (2)
Step 2
Differentiate (2) wrt y and equate to N
Step 3
----- (3)
Find
Step 3
Use (2) and (3) to write
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Sec 2.4
How to find f(x,y) ?
----- (1)
Given an exact DE
Step 1
Integrate wrt x
----- (2)
Step 2
Differentiate (2) wrt y and equate to N
Step 3
----- (3)
Find
Step 3
Use (2) and (3) to write
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Sec 2.4
Made Exact
----- (1)
Given an non-exact DE
Find u so that
Exact
This u is called integrating factor
case 1
case 2