Wikipedia:Reference desk/Archives/Mathematics/2017 September 28
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September 28
editI had trouble with two problems. The problem asks to solve the differential equations listed below.
- (2x + y + 1) y' = 1
- (2y2sin(x)cos(x) + ycos(x)) dx + (4y + sin(x) - 2ycos2(x)) dy = 0
For the first equation, I tried -ln(4x + 2y + 3) - 4x - 2y - 2/4 = c and ln(4√4x + 2y + 3) - x - 1/2y = c and both were incorrect. For the second equation, I tried 2ysin2(x) + sin(x) - 2y2cos2(x) + 3y2 = c and sin(x)/cos(2x) - 3 = (c(-3 + cos(2x) + sin2(x)/cos(2x) - 3)0.5 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 cos2(x) - sin2(x) instead of cos2(x) + sin2(x) for some reason. Thank you for your help. 147.126.10.129 (talk) 03:10, 29 September 2017 (UTC)