ok, so here is a problem that I am confused about.....we are learning about substitutions.
xy^2 *dy/dx = x^3 + y^3
I first used v = y/x, and did some algerbraice manipulations.
then I did the switchings and substituions and it ended up being x *dv/dx = 1/v^2 which is
1/x dx = v^2 dv.
integrate both sides and I get ln x = 1/3v^3 + c.
now, usually, after this type of problem, what is my next step? how do I get the v's back to Y.....
and the back of book solution says "y^3 = 3x^3(c + ln x)
how did that come up?
am I wrong?
also, it talks about bernoulli's equation, and I dont know it well. so exactly how do I recognize it and what do I do w/ it??
xy^2 *dy/dx = x^3 + y^3
I first used v = y/x, and did some algerbraice manipulations.
then I did the switchings and substituions and it ended up being x *dv/dx = 1/v^2 which is
1/x dx = v^2 dv.
integrate both sides and I get ln x = 1/3v^3 + c.
now, usually, after this type of problem, what is my next step? how do I get the v's back to Y.....
and the back of book solution says "y^3 = 3x^3(c + ln x)
how did that come up?
am I wrong?
also, it talks about bernoulli's equation, and I dont know it well. so exactly how do I recognize it and what do I do w/ it??
