Differential Equations

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June 30th 2010, 07:42 AM
sinichko

Differential Equations

Please help me to solve these equations:

$y'+\frac{y}{x}=(xy)^{2}$
$xy''-6y'=12x^{5}$
June 30th 2010, 09:17 AM
Ackbeet

Well, the second equation is a Cauchy-Euler type if you multiply through by x. Or, since there's no y term, you can reduce the order.
June 30th 2010, 02:15 PM
chisigma

The DE...

$\displaystyle y^{'} = -\frac{y}{x} + x^{2}\ y^{2}$ (1)

... is of the type analysed in XVII° century by the Swiss mathematician Jacob Bernoulli. Deviding both terms of (1) by $y^{2}$ You obtain first...

$\displaystyle \frac{y^{'}}{y^{2}} = -\frac {1}{x\ y} + x^{2}$ (2)

... and then setting $\frac{1}{y}= z$ You obtain...

$\displaystyle z^{'} = \frac{z}{x} - x^{2}$ (3)

... which is linear in z...

Kind regards

$\chi$ $\sigma$

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