Wikipedia:Reference desk/Archives/Mathematics/2017 September 28

= September 28 =

Differential equations
I had trouble with two problems. The problem asks to solve the differential equations listed below. For the first equation, I tried $(2x + y + 1) y ' = 1$ and $(2y^{2}sin(x)cos(x) + ycos(x)) dx + (4y + sin(x) - 2ycos^{2}(x)) dy = 0$ and both were incorrect. For the second equation, I tried $-ln(4x + 2y + 3) - 4x - 2y - 2⁄4 = c$ and $4}) - x - 1⁄2y = c$ and both were incorrect. I have attempted to use Wolfram Alpha and Wolfram Mathematica, but I cannot use the product log or the error function in my answer. 147.126.10.148 (talk) 23:38, 28 September 2017 (UTC)


 * On the first one: Not sure where you went wrong but a method of solution is to put v=y+1 to get (2x+v)dv = dx, then put w=2x+v and eliminate x to get (2w+1)dv=dw. This is easily solved by separation of variables to get 2v = log(2w+1)+c or 2y = log(4x+2y+3)+c. This is similar to what you got so I think you were on the right track; maybe you lost a factor of 2 somewhere. Note that you could do a lot more to absorb parts of the solution equation into the constant in order to simplify. Full disclosure, I'm pretty rusty at this kind of thing so I had to consult my generic ODE text, but apparently this method is well known. Haven't looked at the second one yet so no promises. --RDBury (talk) 01:49, 29 September 2017 (UTC)
 * On the second one, it looks like you're going in the wrong direction there. The equation is already exact and can be written
 * d(y sin x + 2 y2 - y2 cos2 x)=0.
 * Not sure why Wolfram is giving non-elementary solutions; maybe it's trying to give an explicit formula for y rather than an implicit equation. --RDBury (talk) 02:32, 29 September 2017 (UTC)
 * They both worked out. I suppose I lost a "2" somewhere for the first problem. I went down the "exact" path for the second problem, but ended up getting $2ysin^{2}(x) + sin(x) - 2y^{2}cos^{2}(x) + 3y^{2} = c$ instead of $sin(x)⁄cos(2x) - 3 = (c(-3 + cos(2x) + sin^{2}(x)⁄cos(2x) - 3)^{0.5}$ for some reason. Thank you for your help. 147.126.10.129 (talk) 03:10, 29 September 2017 (UTC)