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Solving ODEs That Are in Quadrature Format
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Description
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An ODE is said to be in quadrature format when the following conditions are met:
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1) the ODE is of first order and the right hand sides below depend only on x or y(x):
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quadrature_1_x_ode := diff(y(x),x)=F(x);
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quadrature_1_y_ode := diff(y(x),x)=F(y(x));
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2) the ODE is of high order and the right hand side depends only on x. For example:
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quadrature_h_x_ode := diff(y(x),x,x,x,x)=F(x);
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where F is an arbitrary function. These ODEs are just integrals in disguised format, and are solved mainly by integrating both sides.
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Examples
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![[odeadvisor, symgen]](/support/helpjp/helpview.aspx?si=7205/file04382/math46.png)
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![[_quadrature]](/support/helpjp/helpview.aspx?si=7205/file04382/math53.png)
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![[_quadrature]](/support/helpjp/helpview.aspx?si=7205/file04382/math67.png)
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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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![[_xi = 0, _eta = 1]](/support/helpjp/helpview.aspx?si=7205/file04382/math86.png)
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![[_xi = 1, _eta = 0]](/support/helpjp/helpview.aspx?si=7205/file04382/math93.png)
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See Also
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DEtools, odeadvisor, dsolve, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, linear, separable, Bernoulli, exact, homogeneous, homogeneousB, homogeneousC, homogeneousD, homogeneousG, Chini, Riccati, Abel, Abel2A, Abel2C, rational, Clairaut, dAlembert, sym_implicit, patterns; for other differential orders see odeadvisor,types.
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