Question

solve 5c

5. (24 points) Find the solution of each of the following initial value problems: a) xy' + 3y = x V),y (1) = 0 (Bernoulli equation) 18 b) y" – 4y' – 12y = 3e5, y (0) =- (Hint: use the method of undetermined 7 coefficients) c) (2xy - 9x?) dx + (2y + x2 + 1) dy = 0,y (0) = - 3 (Hint: first show this is an exact DE)

Answer 1

Solution 5 (c) =0 Griven differential equation is (any -92)else + (24+ +P+1) dy -0 e glo)= -3, comparing equation with. redy M=any - goce N = 2y+*+ Maret we have & IM = 22 4 ЭМ - 22 an Hence IM =2N ose = C sy Griven differential ry egucetion is exact Now general solution is Smare + gray Y-constant scang- gx²) dze + J cay +1) dy 9 Suéctre efydytsay=c 24 x 2 - 9 x3 +24% ty zc does not contain any ferm en re e C + 2 y=constant ay fredac Yaconstant azy - 3x3 + y 2 ty=0 =C y cityt a2 - 3x3

400)=-3 => -3(1-3 +0%) -3x03=0 → -3(-2+0) -0 =0 6=0 c=6 above ein C geneesal use value of a solletion y city them) - 3x3 = 6 dies is required ed Soletion