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Does anybody suggest how to face the inhomogeneous Bernoulli differential equation $y'+P(x)y=Q(x)y^n+f(x)$ for the simple case $f=const.$ and for the generic case. I would like to know about techniques of approximation, bounds, asymptotic limit, numerical techniques etc. Thank you Roberto

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I fear you should move your question to – Jon Aug 5 '12 at 14:27
up vote 1 down vote accepted

I presume your $f(t)$ should be $f(x)$. This is Chini's equation. See e.g.

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Thank you, of course $f(t)$ is $f(x)$. Thanks for the answer. Roberto – Roberto D'Autilia Aug 6 '12 at 18:06

Solution: let u=y^(1-n) du/dx = (1-n) y^(-n) dy/dx = (1-n) y^(-n) [ -Py+Qy^n+f] =-(1-n) Pu +(1-n)Q +(1-n)f u^(n/n-1)

Separation of variables [1/(1-n)]Int[du/ { f u^(n/n-1)+Pu-Q}]= Int[dx]= x+ C Factor out the division and integrates the LHS. We get F(u,x)=0 then transform into F(y,x)=0.

For n=2, it is very easy to solve. You can try it.

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You can't separate the variables since $P,~Q,~f$ depend on $x$. Have you tried it? In general these equations are not solvable by quadrature when $PQf\neq0$. – Loïc Teyssier Nov 9 '15 at 18:42
Yes, it can be done for the cases that PQf are constants. If they are functions of x, cannot solved by separation. Thks a lot for your reminding. I walked too quickly in simplified form. – Li-Jeng Huang Nov 9 '15 at 19:13
When PQf are constants and n=2, LHS becomes (1/1-n) (1/f)(1/(u1-u2) ln |(u-u1)/(u-u2)|= x+C where u1 and u2 are two roots of u^2 +P/f u - Q/f =0. – Li-Jeng Huang Nov 9 '15 at 19:29
When n=2, PQf depend on x. This is Ricatti's Eq. It can also be transformed to 2nd-order linear ODE. See – Li-Jeng Huang Nov 9 '15 at 19:45
Well, yes, I know that (and a lot of other people here also do…) so that referring to Wikipedia is besides the point. Just check your maths before posting, please. And also, please use LaTeX typesetting… – Loïc Teyssier Nov 9 '15 at 20:09

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