The general form of the exact, linear ODE is given by the following:

exact_linear_ode := diff(linear_ODE(x),x) = 0;

$\mathrm{exact\_linear\_ode}≔\frac{\ⅆ}{\ⅆx}\mathrm{linear\_ODE}\left(x\right)=0$

where linearODE(x) is a linear ODE of any differential order; see Murphy, "Ordinary Differential Equations and their Solutions", p. 221. The order of these exact linear ODEs can be reduced since they are the total derivative of an ODE of one order lower. The reduced ODE is:

linear_ODE(x) + _C1;

$\mathrm{linear\_ODE}\left(x\right)+\mathrm{\_C1}$

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

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

$\mathrm{ODE}≔\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)=A\left(x\right)y\left(x\right)+B\left(x\right)\,x\right)$

$\mathrm{ODE}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\left(\frac{\ⅆ}{\ⅆx}A\left(x\right)\right)y\left(x\right)+A\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\frac{\ⅆ}{\ⅆx}B\left(x\right)$

$\mathrm{odeadvisor}\left(\mathrm{ODE}\,y\left(x\right)\right)$

$\left[\left[\mathrm{\_2nd\_order}\,\mathrm{\_exact}\,\mathrm{\_linear}\,\mathrm{\_nonhomogeneous}\right]\right]$

$\mathrm{dsolve}\left(\mathrm{ODE}\,y\left(x\right)\right)$

$y\left(x\right)=\left(\mathrm{c\_\_2}+\int \left(\mathrm{c\_\_1}+B\left(x\right)\right){\ⅇ}^{\int -A\left(x\right)\ⅆx}\ⅆx\right){\ⅇ}^{-\int -A\left(x\right)\ⅆx}$

$\mathrm{ODE}≔\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=A\left(x\right)y\left(x\right)+B\left(x\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+F\left(x\right)\,x\,x\right)$

$\mathrm{ODE}≔\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)=\left(\frac{{\ⅆ}^2}{\ⅆx^2}A\left(x\right)\right)y\left(x\right)+2\left(\frac{\ⅆ}{\ⅆx}A\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+A\left(x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(\frac{{\ⅆ}^2}{\ⅆx^2}B\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+2\left(\frac{\ⅆ}{\ⅆx}B\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+B\left(x\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+\frac{{\ⅆ}^2}{\ⅆx^2}F\left(x\right)$

$\mathrm{odeadvisor}\left(\mathrm{ODE}\,y\left(x\right)\right)$

$\left[\left[\mathrm{\_high\_order}\,\mathrm{\_exact}\,\mathrm{\_linear}\,\mathrm{\_nonhomogeneous}\right]\right]$

$\mathrm{dsolve}\left(\mathrm{ODE}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{DESol}\left(\left\{-A\left(x\right)\mathrm{\_Y}\left(x\right)-B\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{\_Y}\left(x\right)\right)+\frac{{\ⅆ}^2}{\ⅆx^2}\mathrm{\_Y}\left(x\right)-\mathrm{c\_\_2}-\mathrm{c\_\_1}x-F\left(x\right)\right\}\,\left\{\mathrm{\_Y}\left(x\right)\right\}\right)$

$\mathrm{ODE}≔\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)=A\left(x\right)y\left(x\right)+B\left(x\right)\,x\,x\,x\,x\right)$

$\mathrm{ODE}≔\frac{{\ⅆ}^5}{\ⅆx^5}y\left(x\right)=\left(\frac{{\ⅆ}^4}{\ⅆx^4}A\left(x\right)\right)y\left(x\right)+4\left(\frac{{\ⅆ}^3}{\ⅆx^3}A\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+6\left(\frac{{\ⅆ}^2}{\ⅆx^2}A\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+4\left(\frac{\ⅆ}{\ⅆx}A\left(x\right)\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+A\left(x\right)\left(\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)\right)+\frac{{\ⅆ}^4}{\ⅆx^4}B\left(x\right)$

$\mathrm{odeadvisor}\left(\mathrm{ODE}\,y\left(x\right)\right)$

$\left[\left[\mathrm{\_high\_order}\,\mathrm{\_fully}\,\mathrm{\_exact}\,\mathrm{\_linear}\right]\right]$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{ODE}\,y\left(x\right)\right)$

$\mathrm{ans}≔y\left(x\right)=\left(\mathrm{c\_\_5}+\int \left(4\mathrm{c\_\_1}{x}^3+3\mathrm{c\_\_2}{x}^2+2\mathrm{c\_\_3}x+\mathrm{c\_\_4}+B\left(x\right)\right){\ⅇ}^{\int -A\left(x\right)\ⅆx}\ⅆx\right){\ⅇ}^{-\int -A\left(x\right)\ⅆx}$

$\mathrm{odetest}\left(\mathrm{ans}\,\mathrm{ODE}\right)$

$0$