1. Sec 2.4: Exact Differential Equations
Find the differential

2. 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 1

3. How?? : check exact or not Given a DE: (1) exact If Theorem 2.1
Sec 2.4
How?? : check exact or not
Theorem 2.1
Given a DE:
----- (1)
(1) exact If 1 2 3

5. How to Solve ? Step 1 Step 2 Step 3 Method of Solution: Given a DE:
Sec 2.4
How to Solve ?
Given a DE:
----- (1)
Method of Solution:
Step 1
Check if (1) is an exact
Step 2
Find
Step 3
The family of solutions is: 1

6. How to find f(x,y) ? Step 1 Step 2 Step 3 Step 3 Given an exact DE:
Sec 2.4
How to find f(x,y) ?
Given an exact DE:
----- (1)
Step 1
Integrate wrt x:
----- (2)
Step 2
Differentiate (2) wrt y and equate to N
Check point: function of y only
Step 3
Find:
----- (3)
Step 3
Use (2) and (3) to write: 1 2

7. How to find f(x,y) ? Step 1 Step 2 Step 3 Step 3 Given an exact DE:
Sec 2.4
How to find f(x,y) ?
Given an exact DE:
----- (1)
Step 1
Integrate wrt x:
----- (2)
Step 2
Differentiate (2) wrt y and equate to N
Check point: function of y only
Step 3
Find:
----- (3)
Step 3
Use (2) and (3) to write: 1

8. Made Exact case 1 case 2 Given an non-exact DE: ----- (1)
Find u so that :
Exact
This u is called integrating factor
case 1
case 2