Differential Equations: Finding the General Solution

amsscorpio

Hello, this is the first time I post here, I'm really stumped and tried everything, even my TI-89 calculator won't give me something nice XD

1. The problem statement, all variables and given/known data

Find the general solution of

dy/dt= 1/(ty+t+y+1)


2. Relevant equations
No relevant equations.


3. The attempt at a solution

The first step I did was,

(ty+t+y+1)dy = 1dt

by cross multiplying proportions.
I can't figure out anyway to seperate my terms to the proper places...I then tried this:

integral( ty+t+y+1 dy ) = integral( 1dt )
and resulted in:

ty^2/2 + ty + y^/2 + y + c = t

But this leads me nowhere...Any ideas? Or should I come into conclusion that this differential equation is not valid and unable to do?

amsscorpio

Ahh I THOUGHT I tried everything XD simple algebra error...

I found that (ty + t + y + 1) = (t + 1)(y + 1).

when factored...
then my next step would be

(y + 1) dy = dt/(t + 1).
which then i can integrate both sides to give me

y^2/2 + y + c = ln|t+1| + c

solving for y ultimately gives me

y = + or - (sqrt( 4ln|t+1| +1 ) +1 )/2

All over 2.

Am I right?

rock.freak667

I replied in your other thread that same hint but it looks correct.

But just know that when you have things like y²+y³=ln(x²+x+1)+e^87x+x³+C

you don't always need to make 'y' the subject of the formula, you can leave it as is.

Mark44

You don't need constants on both sides in the equation above.Now you don't have any constants. The constant that should have been on the right side in your previous equation will show up in this one inside the radical. You can check by differenting and seeing whether it is a solution of the original differential equation. You might want to make life a bit easier on yourself by working with y² rather than y, differentiating implicitly.

amsscorpio

Ahh I don't know how to choose best answer + feedback on yahoo answers lol, I did it to have more chances of geting help, thank you and yes I will remember that.