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September 28

Differential equations

I had trouble with two problems. The problem asks to solve the differential equations listed below.

$(2 x + y + 1) y' = 1$
$(2 y 2 sin(x)cos(x) + y cos(x)) dx + (4 y + sin(x) - 2 y cos 2 (x)) dy = 0$

For the first equation, I tried $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠-ln(4x + 2y + 3) - 4x - 2y - 2/4⁠ = c$ and $ln(4 \sqrt 4 x + 2 y + 3) - x - ⁠ 1 / 2 ⁠ y = c$ and both were incorrect. For the second equation, I tried $2 y sin 2 (x) + sin(x) - 2 y 2 cos 2 (x) + 3 y 2 = c$ and $⁠ sin(x) / cos(2 x) - 3 ⁠ = (⁠ c (-3 + cos(2 x) + sin 2 (x) / cos(2 x) - 3 ⁠) 0.5$ 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)[reply]

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)[reply]

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)[reply]

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

cos 2 (x) - sin 2 (x)

instead of

cos 2 (x) + sin 2 (x)

for some reason. Thank you for your help. 147.126.10.129 (talk) 03:10, 29 September 2017 (UTC)[reply]