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

Solve the IVP and find the interval in which the solution exists

y'=2ty^{2}, y(0)=y_{0}>0

2. Relevant equations

3. The attempt at a solution

y'=2ty^{2}

y'/y^{2}=2t

[tex]\int[/tex]y^{-2}y'=[tex]\int[/tex]2t

-1/y=t^{2}+c[tex]\Rightarrow[/tex] c = -1/y_{0}

and therefore [tex]y= -1/(t^2-1/y_0)[/tex]

So it appears that the interval is all real excluding when t^{2}=1/y_{0}. Is this correct?

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

A tank contains 100 gallons of water and 50 oz of salt. Water containing a salt concentration of .25(1+.5sint) oz/gal flows into the tank at a rate of 2 gal/min, and the mixture in the tank flows out at the same rate. Find the amount of salt in the tank at any given time.

2. Relevant equations

I set up the eq. like so:

[tex]dQ/dt=r_1a+Qr_2/100[/tex] where r_{1}is rate in, r_{2}is rate out, and a is the concentration of salt coming in.

3. The attempt at a solution

Q'-2/100Q = .5(1+.5sin(t)) ; u = e^{-t/50}

(uQ)' = u(.5+.25sin(t))

[tex]uQ=\int.5+.25sin(t)[/tex]

uQ = .5t+1/8sin(t^{2})+c

Q = e^{t/50}(.5t+1/8sin(t^{2})+c) ; c = 50 (from IC)

and therefore

Q = e^{t/50}(.5t+1/8sin(t^{2})+50)

Sorry guys, one more if you will.

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

Find the solution of the IVP:

y''+y'+y=0, y(o)=1, y'(0)=-1

2. Relevant equations

3. The attempt at a solution

[tex]-1/2 \pm (\sqrt{1^2-4(1)(1)})/2[/tex]

[tex]-.5 \pm \sqrt{3/2}i[/tex] ; let [tex]\mu = \sqrt{3/2}[/tex]

[tex]y=c_1 e^{(-t/2)}cos(\mu t)+c_2 e^{(-t/2)}sin(\mu t)[/tex]

[tex]y(0)=1=c_1cos(0)+c_2sin(0)=1\Rightarrow c_1=1[/tex]

[tex]y'=-\mu sin(\mu t)e^{(-t/2)}-.5e^{(-t/2)}cos(\mu t)+c_2 \mu cos(\mut)e^{(-t/2)}-.4e^{(-t/2)}sin(\mu t)[/tex]

[tex]y'(0)=-1=-1/2+\mu c_2 \Rightarrow c_2=-.5/\mu[/tex]

[tex]y=e^{(-t/2)}cos(\mu t)+c_2e^{(-t/2)}sin(\mu t)[/tex] where c_{2}is defined above

Thanks a lot for your time. I would really appreciate any replies as to the correctness of the solution, or of I made some mistakes where they might be and how I might go about fixing them. Hopefully I didn't butcher the formatting :).

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# Homework Help: Several Differential Equation problems

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