The beauty of DE's is it's often much easier to check your own solution than to find the solution in the first place. Just plug your solution in to the DE and see if you get equality! You got the same answer both ways, which is a good sign.
Solve in two different ways.
Could someone please check the finer details of my solution, such as signs of integration constants and modulus signs?
Separable:
Let .
, where is a positive constant.
, where .
Bernoulli:
Let .
The integrating factor is .
The beauty of DE's is it's often much easier to check your own solution than to find the solution in the first place. Just plug your solution in to the DE and see if you get equality! You got the same answer both ways, which is a good sign.
Note that
the signum function.
You can use the chain rule multiple times to get the final result.