An ODE is said to be in quadrature format when the following conditions are met:

1) the ODE is of first order and the right hand sides below depend only on x or y(x):

quadrature_1_x_ode := diff(y(x),x)=F(x);

$\mathrm{quadrature\_1\_x\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=F\left(x\right)$

quadrature_1_y_ode := diff(y(x),x)=F(y(x));

$\mathrm{quadrature\_1\_y\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=F\left(y\left(x\right)\right)$

2) the ODE is of high order and the right hand side depends only on x. For example:

quadrature_h_x_ode := diff(y(x),x,x,x,x)=F(x);

$\mathrm{quadrature\_h\_x\_ode}≔\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)=F\left(x\right)$

where F is an arbitrary function. These ODEs are just integrals in disguised format, and are solved mainly by integrating both sides.

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

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

$\mathrm{odeadvisor}\left(\mathrm{quadrature\_1\_x\_ode}\right)$

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

$\mathrm{dsolve}\left(\mathrm{quadrature\_1\_x\_ode}\right)$

$y\left(x\right)=\int F\left(x\right)\ⅆx+\mathrm{\_C1}$

$\mathrm{odeadvisor}\left(\mathrm{quadrature\_1\_y\_ode}\right)$

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

$\mathrm{dsolve}\left(\mathrm{quadrature\_1\_y\_ode}\right)$

$x-\left({\int }_{}^{y\left(x\right)}\frac{1}{F\left(\mathrm{\_a}\right)}\ⅆ\mathrm{\_a}\right)+\mathrm{\_C1}=0$

$\mathrm{symgen}\left(\mathrm{quadrature\_1\_x\_ode}\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=1\right]$

$\mathrm{symgen}\left(\mathrm{quadrature\_1\_y\_ode}\right)$

$\left[\mathrm{\_\ξ}=1\,\mathrm{\_\η}=0\right]$