# 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?

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