Solve the questions

Homogeneous equations 5. Verify that each of the following equations is homogeneous and then solve it. (a) 2x= =x + yz Ans: y = x -= 2x Inx+ 4 (b) (x] + yz ) "/ dx = xy Ans: y = Agx*/zy (c) xyy' =x + yz Ans: y' = 2x* (In]x] + () Exact equations 6. Determine whether each of the following DE are exact equations or not. If there are exact equations, find their solutions. (a) (2x + 3) + (2y - 2)y' = 0 Ans: yz - 2y + x3 + 3x = C (b) (x+ 912 - 1) - (4x -t)x = 0 Ane: 3t' + tx - t - 2x= = C (c) (sinxy + xy cos xy + 2x)dx + (x'cosxy + 2y)dy =0 Ans: x sinxy + x] + y' = C Linear equations Find the general solution of the following linear equations. If initial condition is given, find the corresponding particular solution. (@) x +2y =5x3 Ans: y = x3 - : (b) y' - ytanx = 8 sin] x Ans: y = 2 sin* */ cos x + /cos x Ans y = e* (e* + ()