solve the problem using Bernoulli....

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February 10th 2011, 06:42 PM
slapmaxwell1

solve the problem using Bernoulli....

ok i dont see how the book came up with the substitution of u= y^-1

so here is the problem..

(t^2)dy/dt + y^2 = ty

my question is this, if u = y^(1-n) wouldnt n = 0?

i think the book may have made a mistake on this problem....can someone please check the problem and see if the substitution is correct? thanks in advance..
February 10th 2011, 06:52 PM
Krizalid

to make it clearer rewrite the equation as $\dfrac{t^2}{y^2}y'+1=\dfrac ty$ and put $u=\dfrac1y.$
February 10th 2011, 06:56 PM
slapmaxwell1

well i did rewrite it...

i got dy/dt + (y^2/t^2) = y/t but the way i have it written it seems like n would be 1?
February 10th 2011, 07:13 PM
slapmaxwell1

ok i got it..thanks for your help...
February 10th 2011, 08:49 PM
Prove It

To use Bernoulli...

$\displaystyle t^2\,\frac{dy}{dt} + y^2 = t\,y$

$\displaystyle t^2\,\frac{dy}{dt} - t\,y = -y^2$ .

Make the substitution $\displaystyle v = y^{1-2} = y^{-1} \implies y = v^{-1}$ .

Then $\displaystyle \frac{dy}{dt} = \frac{d}{dt}(v^{-1}) = \frac{d}{dv}(v^{-1})\,\frac{dv}{dt} = -v^{-2}\,\frac{dv}{dt}$ .

Substituting into the DE gives

$\displaystyle t^2\left(-v^{-2}\,\frac{dv}{dt}\right) - t\,v^{-1} = -v^{-2}$

$\displaystyle t^2\,\frac{dv}{dt} + t\,v = 1$

$\displaystyle \frac{dv}{dt} + t^{-1}v = t^{-2}$ .

This is now first order linear, so multiplying both sides of the DE by the Integrating Factor $\displaystyle e^{\int{t^{-1}\,dt}} = e^{\ln{t}} = t$ gives

$\displaystyle t\,\frac{dv}{dt} + v = t^{-1}$

$\displaystyle \frac{d}{dt}(t\,v) = t^{-1}$

$\displaystyle t\,v = \int{t^{-1}\,dt}$

$\displaystyle t\,v = \ln{|t|} + C$

$\displaystyle v = \frac{\ln{|t|} + C}{t}$

$\displaystyle y^{-1} = \frac{\ln{|t|} + C}{t}$

$\displaystyle y = \frac{t}{\ln{|t|} + C}$ .