2,2 (0,2) ( ) II A (2,0) (0,0) Exercise 1 Let 6f = (2x3; x2y) dx+ (x2 xzy) dy (a) Show that 5f is an inexact di'erential. (b) Consider two path in the xy-plane from (0, 0) to (2,2): Path 1: (0,0) ) (2,0) at constant 3; then from (2,0) > (2,2) at constant x. Path 11: (0,0) > (0,2) at constant x then from (0, 2) > (2,2) at constant 3;. Show the line integral along paths I and II yield different values demonstrating again the differential to be inexact. (0) An integrating factor, (x,y), is a function which multiplies an inexact differential to make it exact. right!) = (x:y)5f If (x,y) = xny'". Determine the form of the integration factor (what are m and 11?). Hint: When (1065 63(1); (2):}; x230) = 810}: (x2 xzy) )? (d) Now with the integrating factor determined in (c). Repeat the calculations in (a) and verify dg is now an exact differential