I with simple Differential Equation.

[JDStupi]

Homework Statement

Hello, I'm trying to go back to school and haven't done any math in awhile, as such, my skills are terribly out of practice. I am unable to arrive at the book's solution and suspect I am forgetting a simple algebraic trick and would like somebody to show me it and explain the details so I can learn it and apply it in the future.

Homework Equations

dy/dt=1/(2y+1)

The Attempt at a Solution

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So, I do the Ol' separation trick and get ∫(2y+1)dy=∫dt
Which then leads me to y²/2+y+c=t+c. I combine the constants to get y²/2+y=t+k
This is basically as far as I get, algebraic fiddling notwithstanding.

The actual solution is y= (-1 ± √(4t+c))/2

I see the form of the quadratic in the solution, but I'm used to applying the quadratic when the right hand side is zero. So, if anybody could show me how to arrive at that solution and where I go wrong or what I'm missing I would highly appreciate it.

Also, if anybody can let me in on how to make the equations look nice in the text editor that would be awesome.

 

[fresh_42]

I see the form of the quadratic in the solution, but I'm used to applying the quadratic when the right hand side is zero. So, if anybody could show me how to arrive at that solution and where I go wrong or what I'm missing I would highly appreciate it.

But the RHS is zero, if you like. You got $\frac{1}{2}y^2+y=t+k$ which is wrong by a factor $2$: $\frac{d}{dy}y^2=2y$
Corrected you have $y^2+y-t-k=0$ which resolves to $y_{1,2}=\dfrac{1}{2}\left(-1 \pm \sqrt{1+4t+c}\right)\,.$

Also, if anybody can let me in on how to make the equations look nice in the text editor that would be awesome.

https://www.physicsforums.com/help/latexhelp/

 

[JDStupi]

Ah! You're the man! I knew I was being stupid and overlooking something simple. Thank you very much for the help, and for the latexhelp link.