Question

Determine whether the equation is exact. If it is, then solve it. 4e+(2y – t)dt + (3 + 8 e") dy = 0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. = C, where is an arbitrary constant. O A. The equation is exact and an implicit solution in the form F(t,y)=C is (Type an expression using t and y as the variables.) O B. The equation is not exact.

Answer 1

Solution Given differential equation is (24-t) at +(3+ Betydy zo -(1) comparing equation (d) with M (t, y) d t + N (t. ,J) dy=0, we get let Mt,y) = 4 et 27-+), N6.4) = 3+8 et Эм = get 2y and aN at det ƏN at Since am therefore the differential ay equation (1) is exact. Hence, there exist function f (t. y) such that af = M (ty) = q et at (2y-t) - (2) and af - Nat, y) = 3+ Bet (3) ay Integrating equation (2) partially with respect to t, we get Saf Sa et (Ryt) at f(ty) = S@yet soy et_ 4t et) at =oyse et = Byet - 4 [tet - 51.etat) dt ot at 4 Stet at

et et + (4) f (t, y) = og et 4 et + V (9) where & (Y) is a function of y only. Rifferentiating equation (4) with respect to y, partially -0+0+ y'a) ay ie. Of - get+4' (y) ay From equation (3), 3f ay af = Bet = 3+ det det + 4 (y) 3+ set = 4' (Y) = 3 =Y y = 3y +C Putting the value of 4 (Y) in equation (7). we get f A, y) = oyet_4t et + 4 et +34 +C solution of the differential equation (2) is given by et_atetta et By et +34 = C where a is an arbitrary constant. Answer: A. The equation is exact and an implicit solution in the form F & 7) = e is oget_qtettqet +3y=c, where c is an arbitrary constant.