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The ODE

$$y'+\left(-1+\frac{a-1}{x}\right)y+b x -c=0$$

is solved by $y=bx$ in the particular case $c=ab$.

How would you proceed to find a more general solution?

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The equation $$y'+\left(-1+\frac{a-1}{x}\right)y+b x -c=0$$ is linear.

You have $$y= y_h + y_p$$ where $y_h$ is the solution to the homogeneous equation, $$y'+\left(-1+\frac{a-1}{x}\right)y=0$$ and $:y_p $. is a particular solution to the in-homogeneous equation.

Both solutions are found easily because for the first one the equation is separable and for the second one you may use variation of constant.

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According to Maple, the solutions can be expressed with the help of Whittaker functions:

enter image description here

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  • $\begingroup$ I think that the solution is much simpler. $\endgroup$ – Claude Leibovici Jul 20 at 14:12
  • $\begingroup$ Maybe, but Maple cannot simplify it. $\endgroup$ – Héhéhé Jul 20 at 14:14
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Since this equation can be put in the form $y'+a(x)y=b(x)$, you can solve by using the integrating factor $e^{\int a(x)\operatorname dx}$.

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