Diff equation: xy' = y + 1

My answer is: y = +/- xe^c - 1

is this correct?

Question

Diff equation: xy' = y + 1

My answer is: y = +/- xe^c - 1

is this correct?

anonymous · Answer

before evaluation the integral i have

dy/(y+1) = dx/x

myininaya · Answer

subtract y on both sides 
$$xy'-y=1 $$
divide both sides by x
$$y'-\frac{1}{x}y=\frac{1}{x}$$
integrating factor is
$$v=e^{\int\limits_{}^{}\frac{-1}{x} dx}=e^{-\ln(x)}  =\frac{1}{x} , x>0$$
$$\frac{1}{x}y'-\frac{1}{x^2}y=\frac{1}{x^2}$$
now we can write
$$(\frac{1}{x}y)'=\frac{1}{x^2}$$
now integrate both sides

amistre64 · Answer

yeah, i was looking for a way to get rid of the x klingon lol

myininaya · Answer

$$\frac{1}{x}y=\frac{-1}{x}+C$$

amistre64 · Answer

seperation of variables would work as well, yes

myininaya · Answer

yes separation of variables works too

amistre64 · Answer

xy' = y + 1

dy/dx  x = (y+1)

dy/(y+1)  x = dx

dy/(y+1) = dx/x ; int the whole thing

ln|y+1|  = ln|x| + C

anonymous · Answer

so far so good @ amistre64

amistre64 · Answer

e^ both sides then

y+1 = e^(ln|x|+C)

y+1 = e^(ln|x|) * e^C

y+1 = x * C

y = Cx - 1

looks about right to me

amistre64 · Answer

e^C is just a constant so we can rewrite it as C

anonymous · Answer

but is it +/- Cx - 1

amistre64 · Answer

I think the generic constant C makes up for any +- business

amistre64 · Answer

the wolf agrees i think

amistre64 · Answer

http://www.wolframalpha.com/input/?i=xy%27+%3D+y%2B1

anonymous · Answer

ahh oke.. but i hadn't simplified it to C so this is correct?

y = +/- xe^c - 1 :)

amistre64 · Answer

thats "correct" yes; but i dont think its simplified as much

myininaya · Answer

e^c gives us positive constant actually

amistre64 · Answer

it could be more generic that is

myininaya · Answer

e^c is never negative

amistre64 · Answer

"e^c gives us a positive constant" he responded in a snitty tone.  lol

myininaya · Answer

he?

amistre64 · Answer

yes, im a he ;)

myininaya · Answer

you're weird

amistre64 · Answer

lol

anonymous · Answer

ofcourse! e^x is always positive so i can leave the negative sign out :)

amistre64 · Answer

yep, leave of the negative sign

anonymous · Answer

okee thank you guys =D

and yeaahhh i got an answer correctly
 ^^

myininaya · Answer

but why would you want to say plus or minus to begin with?

amistre64 · Answer

prolly just to cover the bases

anonymous · Answer

because |y+1| were absolute

myininaya · Answer

oh i did my a different way

anonymous · Answer

and i want to have y+1

anonymous · Answer

thats why i added the plus and minus signs

myininaya · Answer

ok