as I understand, we need to solve the system
Integrating first equation we obtain . Plugging to the second and integrating, we get , and then plugging that to the third equation and integrating, we get . Hence for any constant c
have you done differential equations before? this is done in much the same way as the method of exact equations. essentially, we are looking for a function so that .
so we have,
.............(1)
...............(2)
...............(3)
integrating (1) with respect to , we get
...........we think of our constant as a function of and , for (hopefully) obvious reasons
now, differentiate this with respect to
comparing this to (2), we see that
so that , by integrating with respect to , leaving a constant in terms of a funciton of .
so now, go back to , we see that,
differentiate this with respect to and compare to (3), we see that:
, a constant, so finally,
it is trivial to verify that