Differential Equations:Linear first Order Equations

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July 13th 2007, 11:40 AM
googoogaga

Differential Equations:Linear first Order Equations

How would you solve these? (I use your explanations and answers as reference to my homework, because the summer teacher is out of reach oustide of class).

1) find an explicit general solution of the differential equation:
y' - 2xy= e^(x^(2))

2) Find an explicit general solution of the differential equation:
xy'= 2y + x^(3) cosx

3) Find an explicit general and particular solution of the initial value problem:
xy' - y= x, y(1)= 7

4) Find an explicit general and particular solution of the initial value problem:
y' + y= e^(x), y(0)= 1

5) A 400-gal tank initially contains 100 gal of brine (salt water) containing
50 lb of salt. Brine containing 1 lb of salt per gallon enters the tank at
the rate 5 gal/s, and the well-mixed brine in the tank flows out at the rate
of 3 gal/s. How much salt will the tank contain when it is full of brine?
(Solve questions using linear first order equation method primarily.
If you have any other methodic approaches to this problem, please
feel free to demonstrate).
July 13th 2007, 12:03 PM
Krizalid

Quote:

Originally Posted by googoogaga

4) Find an explicit general and particular solution of the initial value problem:
y' + y= e^(x), y(0)= 1

Linear ODE $y'+P(x)y=Q(x)$

Integrating factor defined by $\mu(x)=\exp\int{P(x)}~dx$

In this problem, you can see that the integrating factor is $\mu(x)=e^x$ , then multiplying the equation by this I.F. yields $e^xy'+e^xy=e^{2x}\iff(e^xy)'=e^{2x}$

The rest, for you.
July 13th 2007, 01:12 PM
galactus

Quote:

5) A 400-gal tank initially contains 100 gal of brine (salt water) containing
50 lb of salt. Brine containing 1 lb of salt per gallon enters the tank at
the rate 5 gal/s, and the well-mixed brine in the tank flows out at the rate
of 3 gal/s. How much salt will the tank contain when it is full of brine?
(Solve questions using linear first order equation method primarily.
If you have any other methodic approaches to this problem, please
feel free to demonstrate).

At time t there's 100+2t gallons of brine in the tank. So

$\frac{dy}{dt}=\underbrace{5}_{\text{in}}-\overbrace{\frac{3y}{100+2t}}^{\text{out}}$

$\frac{dy}{dt}+\frac{3y}{100+2t}=5$

Integrating factor:

$e^{\int\frac{3}{100+2t}dt}=(t+50)^{\frac{3}{2}}$

$\frac{d}{dt}((t+50)^{\frac{3}{2}}y)=5(t+50)^{\frac {3}{2}}$

Integrate and get:

$(t+50)^{\frac{3}{2}}y=2(t+50)^{\frac{5}{2}}+C$

$y=2t+100+C(t+50)^{\frac{-3}{2}}$

Using the IC that t(0)=50, we get $C=-12500\sqrt{2}$

$\boxed{y=2t+100-12500\sqrt{2}(t+50)^{\frac{-3}{2}}}$

The tank is full when 100+2t=400, t=150

$y=2(150)+100-12500\sqrt{2}(150+50)^{\frac{-3}{2}}\approx{393.75}~lbs~of~brine$

Please check my calculations. Easy to go astray.
July 16th 2007, 04:52 AM
googoogaga

Problem # 4: Linear First Order Equations( I still did not understand)

:(I still did not understand it your explanation. This is what I did to Find the general equation... Is this correct? If not could someone please help me with this?

y'+ p(x) y= Q(x) p(x)= -2x
y'- 2xy = e^(x*x) Q(x)= e^(x*x)

v(x)= e^(Int p(x) dx) = e^ (Int -2x dx)= e^(-x^(2))

y(x)= Int v(x)Q(x)+c
v(x)

y(x)= Int e^(-x^(2)) * e^(x*x) + c
(e^(-x^(2)
= e^(-x^(2)) * 1/2 e ^(x*x) +c
e^(-x^(2))

G.S. ==> Y(x)= 1/2 e^(x*x) + c
e^(-x^(2))
July 16th 2007, 04:54 AM
googoogaga

Sorry! Wrong Numbering

:oI meant problem #1
July 16th 2007, 04:55 AM
googoogaga

To Mr. Galactus

Thanks! That's the Answer I got for the Brine Problem you've helped me with.:)