Solving ODEs That Are in Quadrature Format
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);
quadrature_1_x_ode ≔ ⅆⅆx⁢y⁡x=F⁡x
quadrature_1_y_ode := diff(y(x),x)=F(y(x));
quadrature_1_y_ode ≔ ⅆⅆx⁢y⁡x=F⁡y⁡x
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);
quadrature_h_x_ode ≔ ⅆ4ⅆx4⁢y⁡x=F⁡x
where F is an arbitrary function. These ODEs are just integrals in disguised format, and are solved mainly by integrating both sides.
From the point of view of their symmetries, all ODEs "missing y" have the symmetry [xi = 0, eta = 1], and all ODEs "missing x" have the symmetry [xi = 1, eta = 0] (see symgen);
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