Given a nth order linear ODE for $y\left(x\right)$, the ode_int_y and ode_y1 commands respectively compute the nth order linear ODE satisfied by $\int y\left(x\right)\ⅆx$ and $\frac{\ⅆ}{\ⅆx}y\left(x\right)$.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{diff\_table}\,\mathrm{ode\_int\_y}\,\mathrm{ode\_y1}\right)$

$\left[\mathrm{diff\_table}\,\mathrm{ode\_int\_y}\,\mathrm{ode\_y1}\right]$

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

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

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

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

$Y≔\mathrm{diff\_table}\left(y\left(x\right)\right)\:$

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

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

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

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

$c\left[0\right]\left(x\right)Y\left[\right]+c\left[1\right]\left(x\right)Y\left[x\right]+c\left[2\right]\left(x\right)Y\left[x,x\right]+Y\left[x,x,x,x\right]=0$

$c_{0}y+c_1\mathrm{y\text{'}}+c_2\mathrm{y\text{'}\text{'}}+\mathrm{y\text{'}\text{'}\text{'}\text{'}}=0$

$\mathrm{DEtools}\left[\mathrm{ode\_y1}\right]\left(\right)=0$

$\mathrm{y\text{'}\text{'}\text{'}\text{'}}-\frac{c_0^{\text{'}}\mathrm{y\text{'}\text{'}\text{'}}}{c_0}+c_2\mathrm{y\text{'}\text{'}}-\frac{\left(c_0^{\text{'}}{c}_2-c_1{c}_0-c_2^{\text{'}}{c}_0\right)\mathrm{y\text{'}}}{c_0}-\frac{\left(c_0^{\text{'}}{c}_1-{c_0}^2-c_1^{\text{'}}{c}_0\right)y}{c_0}=0$

$\mathrm{DEtools}\left[\mathrm{ode\_int\_y}\right]\left(\,y\left(x\right)\right)=0$

$c_{0}y+c_1\mathrm{y\text{'}}+c_2\mathrm{y\text{'}\text{'}}+\mathrm{y\text{'}\text{'}\text{'}\text{'}}=0$