Please help me to solve these equations:

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

$\displaystyle xy''-6y'=12x^{5}$

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- Jun 30th 2010, 07:42 AMsinichkoDifferential Equations
Please help me to solve these equations:

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

$\displaystyle xy''-6y'=12x^{5}$ - Jun 30th 2010, 09:17 AMAckbeet
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.

- Jun 30th 2010, 02:15 PMchisigma
The DE...

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

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

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

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

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

... which is*linear*in z...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$