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I would like to resolve this differential equation:

$xx'-x=f(t)$

any suggestions (or any online texts on similar differential equation) please? Thanks.

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2 Answers

up vote 2 down vote accepted

This is an 'Abel equation of the second kind in the canonical form'.

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Thanks! And is this Abel equation? $xx'-g(t)x=f(t)$ –  Mark Jul 7 '12 at 10:37
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@Mark: More exactly this is a 'special case of the Abel differential equation of the second kind' while the 'Abel differential equation of the second kind' incorporates a x^2 term. –  Raymond Manzoni Jul 7 '12 at 10:46
    
Although the recognition of EqWorld is quite well, it still behaves outdated about Abel differential equation of the second kind, since the article hindawi.com/journals/ijmms/2011/387429/#sec2 has been issued. –  doraemonpaul Jul 7 '12 at 15:58
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Although the recognition of EqWorld is quite well, it still behaves outdated about Abel equation of the second kind.

The following information is really updated indeed:

$xx'-x=f(t)$ itself belongs to an Abel equation of the second kind in the canonical form.

In fact, all Abel equation of the second kind can be transformed into Abel equation of the first kind.

Let $x=\dfrac{1}{y}$,

Then $x'=-\dfrac{y'}{y^2}$

$\therefore-\dfrac{y'}{y^3}-\dfrac{1}{y}=f(t)$

$\dfrac{y'}{y^3}=-f(t)-\dfrac{1}{y}$

$y'=-f(t)y^3-y^2$

Please follow the method in http://www.hindawi.com/journals/ijmms/2011/387429/#sec2

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