The general form of a first order linear ODE is given by the following:

linear_ode := diff(y(x),x)+f(x)*y(x)-g(x);

$\mathrm{linear\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)+f\left(x\right)y\left(x\right)-g\left(x\right)$

where f(x) and g(x) are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 16. This type of ODE can be solved in a general manner by dsolve as follows:

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{odeadvisor}\left(\mathrm{linear\_ode}\right)$

$\left[\mathrm{\_linear}\right]$

$\mathrm{dsolve}\left(\mathrm{linear\_ode}\right)$

$y\left(x\right)=\left(\int g\left(x\right){\ⅇ}^{\int f\left(x\right)\ⅆx}\ⅆx+\mathrm{c\_\_1}\right){\ⅇ}^{\int -f\left(x\right)\ⅆx}$