Sec 2.4: Exact Differential Equations

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Presentation transcript:

Sec 2.4: Exact Differential Equations Find the differential

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

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

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

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

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

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