A fully exact nth-order linear ODE is one which can be obtained by directly differentiating a first order linear ODE n-1 times; for example, a fourth order ODE of this type is of the form:

fully_exact_linear_4th_order_ODE := Diff( A(x)*diff(y(x),x) + B(x)*y(x) + C(x) = 0, x$3);

$\mathrm{fully\_exact\_linear\_4th\_order\_ODE}≔\frac{{\ⅆ}^3}{\ⅆx^3}\left(A\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+B\left(x\right)y\left(x\right)+C\left(x\right)=0\right)$

Linear ODEs (LODEs) of this form can be recognized by the fact that the adjoint of their homogeneous part is always a first order ODE in disguised form (see "How it works" below). This type of LODE is systematically solved by dsolve.

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

$\left[\mathrm{adjoint}\,\mathrm{firint}\,\mathrm{odeadvisor}\right]$

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(y\left(x\right)\,\mathrm{prime}=x\right)$

$y\left(x\right)\mathrm{will\; now\; be\; displayed\; as}y$

$\mathrm{derivatives\; with\; respect\; to}x\mathrm{of\; functions\; of\; one\; variable\; will\; now\; be\; displayed\; with\; \text{'}}$

$\mathrm{ODE4}≔\left(x^2+\exp \left(x\right)x\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\,x\right)+\left(4\exp \left(x\right)+6x+3\exp \left(x\right)x\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+\left(6+3\exp \left(x\right)x+9\exp \left(x\right)\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\left(\exp \left(x\right)x+6\exp \left(x\right)\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+\exp \left(x\right)y\left(x\right)=0$

$\mathrm{ODE4}≔\left(x^2+{\ⅇ}^{x}x\right)\mathrm{y\text{'}\text{'}\text{'}\text{'}}+\left(4{\ⅇ}^x+6x+3{\ⅇ}^{x}x\right)\mathrm{y\text{'}\text{'}\text{'}}+\left(6+3{\ⅇ}^{x}x+9{\ⅇ}^x\right)\mathrm{y\text{'}\text{'}}+\left({\ⅇ}^{x}x+6{\ⅇ}^x\right)\mathrm{y\text{'}}+{\ⅇ}^{x}y=0$

$\mathrm{odeadvisor}\left(\mathrm{ODE4}\,y\left(x\right)\right)$

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

$\mathrm{ODE3}≔\mathrm{firint}\left(\mathrm{ODE4}\,y\left(x\right)\right)$

$\mathrm{ODE3}≔{\ⅇ}^{x}y+\left({\ⅇ}^{x}x+4{\ⅇ}^x+2\right)\mathrm{y\text{'}}+\left(2{\ⅇ}^{x}x+3{\ⅇ}^x+4x\right)\mathrm{y\text{'}\text{'}}+\left(x^2+{\ⅇ}^{x}x\right)\mathrm{y\text{'}\text{'}\text{'}}+\mathrm{\_C1}=0$

$\mathrm{ODE2}≔\mathrm{firint}\left(\mathrm{ODE3}\,y\left(x\right)\right)$

$\mathrm{ODE2}≔{\ⅇ}^{x}y+\left({\ⅇ}^{x}x+2{\ⅇ}^x+2x\right)\mathrm{y\text{'}}+\left(x^2+{\ⅇ}^{x}x\right)\mathrm{y\text{'}\text{'}}+\mathrm{\_C1}x+\mathrm{\_C2}=0$

$\mathrm{ODE1}≔\mathrm{firint}\left(\mathrm{ODE2}\,y\left(x\right)\right)$

$\mathrm{ODE1}≔{\ⅇ}^{x}y+\left(x^2+{\ⅇ}^{x}x\right)\mathrm{y\text{'}}+\frac{\mathrm{\_C1}{x}^2}{2}+\mathrm{\_C2}x+\mathrm{\_C3}=0$

$\mathrm{sol1}≔\mathrm{dsolve}\left(\mathrm{ODE1}\,y\left(x\right)\right)$

$\mathrm{sol1}≔y=\left(\int -\frac{{\ⅇ}^{\int \frac{{\ⅇ}^x}{x\left({\ⅇ}^x+x\right)}\ⅆx}\left(\mathrm{\_C1}{x}^2+2\mathrm{\_C2}x+2\mathrm{\_C3}\right)}{2x\left({\ⅇ}^x+x\right)}\ⅆx+\mathrm{c\_\_4}\right){\ⅇ}^{\int -\frac{{\ⅇ}^x}{x\left({\ⅇ}^x+x\right)}\ⅆx}$

$\mathrm{odetest}\left(\mathrm{sol1}\,\mathrm{ODE4}\right)$

$0$

$\mathrm{dsolve}\left(\mathrm{ODE4}\,y\left(x\right)\right)$

$y=\left(\mathrm{c\_\_4}+\int \frac{\left(3\mathrm{c\_\_5}{x}^2+2\mathrm{c\_\_6}x+\mathrm{c\_\_7}\right){\ⅇ}^{\int \frac{{\ⅇ}^x}{x^2+{\ⅇ}^{x}x}\ⅆx}}{x^2+{\ⅇ}^{x}x}\ⅆx\right){\ⅇ}^{-\left(\int \frac{{\ⅇ}^x}{x^2+{\ⅇ}^{x}x}\ⅆx\right)}$

$\mathrm{ADJ}≔\mathrm{adjoint}\left(\mathrm{ODE4}\,y\left(x\right)\right)$

$\mathrm{ADJ}≔\left({\ⅇ}^{x}x+2x\right)\mathrm{y\text{'}\text{'}\text{'}}+\left(x^2+{\ⅇ}^{x}x\right)\mathrm{y\text{'}\text{'}\text{'}\text{'}}$

$\mathrm{subs}\left(\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,3\right)\right)=y\left(x\right)\,\mathrm{ADJ}\right)$

$\left({\ⅇ}^{x}x+2x\right)y+\left(x^2+{\ⅇ}^{x}x\right)\mathrm{y\text{'}}$

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

${\ⅇ}^{x}y+\left(x^2+{\ⅇ}^{x}x\right)\mathrm{y\text{'}}$

$\mathrm{diff}\left(\,\mathrm{`\$`}\left(x\,3\right)\right)=0$

${\ⅇ}^{x}y+3\mathrm{y\text{'}}{\ⅇ}^x+3\mathrm{y\text{'}\text{'}}{\ⅇ}^x+\mathrm{y\text{'}\text{'}\text{'}}{\ⅇ}^x+\left({\ⅇ}^{x}x+3{\ⅇ}^x\right)\mathrm{y\text{'}}+3\left(2+{\ⅇ}^{x}x+2{\ⅇ}^x\right)\mathrm{y\text{'}\text{'}}+3\left(2x+{\ⅇ}^{x}x+{\ⅇ}^x\right)\mathrm{y\text{'}\text{'}\text{'}}+\left(x^2+{\ⅇ}^{x}x\right)\mathrm{y\text{'}\text{'}\text{'}\text{'}}=0$

$\mathrm{normal}\left(-\mathrm{ODE4}\right)$

$0=0$