Question

1.  Solve the following Exact Differential Equation

2. Solve the following Exact Differential Equation with boundary value y(-1) = 2

3. Solve the following higher order differential equation given that y(pi/3 ) = 0, y'(pi/3 ) = 2

Answer 1

o given that (x² + y²) on ar zxy ayo M N +01 Exact D C it has to satisfy - OM OY ON OM UM oy 2 (43+ y²) 342 oy ON OX 11 J (3 x y2) 34 ON ON UN JM ду Solution Jm.art J terms not contains a dy = c J x 3 + y dx = 6 1310 3141 + xy² = 0 4 x y = 0 9 sollnis ntuny² = uc (uy 1st-slat + (6y + ut - 1) dy =0 uy its Neytul-2 014 ay

ON DM 09 EXACT fe mat + hly) = 1 ay tat-sit they) yyt + t² - st they) If = 4x+r(y) = 64 +41-1 sy h'Cy) = 6Y-1 h(9) = J (6 9 - 1) c = 34-94C f = 4ytt test t3ya_y + c y(-1)=2 t=-1 y=2 f = 4(2) (-1) + (-10% 5(-1) + +3 (2) ² - 2 IC - 8+1 +5 + 12-2tc c=8 luyt + test +34 - y = 8 alaty=0 mo + m = II m I

enerdi Solution y = e** [C, cos (Bx) + C2 sin (3x)] - e" [c, cos(x) +Cg sin(sx)) CiCosi + casin (2)