Solve the following: 1.    (siny – y sinx) dx + (cosx + xcosy – y ) dy = 0 2.    ( yex – 2x ) dx + ex dy = 0 3.    (2x siny + y) dx + (x2 cosy + x ) dy = 0 4.    (2x3 – y) dx + x dy = 0

To solve the given differential equations step by step:

1. (siny – y sinx) dx + (cosx + xcosy – y ) dy = 0: Rearrange the equation to isolate dy/dx: (siny – y sinx) dx = (y - cosx - xcosy) dy Divide both sides by (y - cosx - xcosy): (siny – y sinx)/(y - cosx - xcosy) dx = dy Integrate both sides with respect to x: ∫(siny – y sinx)/(y - cosx - xcosy) dx = ∫dy Solve the integral on the left side to find the solution.

2. (yex – 2x) dx + ex dy = 0: Rearrange the equation to isolate dy/dx: (yex – 2x) dx = -ex dy Divide both sides by -ex: (yex – 2x)/ex dx = dy Integrate both sides with respect to x: ∫(yex – 2x)/ex dx = ∫dy Solve the integral on the left side to find the solution.

3. (2x siny + y) dx + (x2 cosy + x ) dy = 0: Rearrange the equation to isolate dy/dx: (2x siny + y) dx = -(x2 cosy + x) dy Divide both sides by -(x2 cosy + x): (2x siny + y)/-(x2 cosy + x) dx = dy Integrate both sides with respect to x: ∫(2x siny + y)/-(x2 cosy + x) dx = ∫dy Solve the integral on the left side to find the solution.

4. (2x3 – y) dx + x dy = 0: Rearrange the equation to isolate dy/dx: (2x3 – y) dx = -x dy Divide both sides by -x: (2x3 – y)/-x dx = dy Integrate both sides with respect to x: ∫(2x3 – y)/-x dx = ∫dy Solve the integral on the left side to find the solution.

Please note that the specific solutions to these differential equations will depend on the integration results, which can vary depending on the complexity of the integrals involved.