# A first-order ODE solvable in a case

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?

## 3 Answers

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.

According to Maple, the solutions can be expressed with the help of Whittaker functions: • I think that the solution is much simpler. – Claude Leibovici Jul 20 at 14:12
• Maybe, but Maple cannot simplify it. – Héhéhé Jul 20 at 14:14

Since this equation can be put in the form $$y'+a(x)y=b(x)$$, you can solve by using the integrating factor $$e^{\int a(x)\operatorname dx}$$.