the integrating factor method takes care of all these. see post #21 here
I have a few questions about finding the general solution of first-order linear differential equations..
please help, I have no clue!! =/
http://i28.tinypic.com/2yv3343.gif
http://i31.tinypic.com/2818w02.gif
http://i28.tinypic.com/n81ee.gif
it is just something you multiply through by to make the left hand side the derivative given by the product and the integrating factor. for the first one:
(with practice, you will be able to recognize the integrating factor immediately, but let's go through the method to see it)
the integrating factor is
multiply through by the integrating factor, we get:
now the left hand side is the result of differentiating by the product rule, thus
integrate both sides, we get:
the others are done similarly