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Solving d'Alembert ODEs
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Description
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The general form of the d'Alembert ODE is given by:
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dAlembert_ode := y(x)=x*f(diff(y(x),x))+g(diff(y(x),x));
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where f and g are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 31. This ODE is actually a generalization of the Clairaut ODE, and is almost always dealt with by looking for a solution in parametric form. For more information, see odeadvisor[patterns].
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Examples
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![[odeadvisor]](/support/helpjp/helpview.aspx?si=7218/file04364/math31.png)
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![[_dAlembert]](/support/helpjp/helpview.aspx?si=7218/file04364/math38.png)
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The general form of the solution for the d'Alembert ODE is returned by dsolve in parametric form, together with a possible singular solution, as follows:
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![y(x) = g(RootOf(_Z-f(_Z)))+x*RootOf(_Z-f(_Z)), [x(_T) = exp(Int((diff(f(_T), _T))/(_T-f(_T)), _T))*(Int((diff(g(_T), _T))*exp(-(Int((diff(f(_T), _T))/(_T-f(_T)), _T)))/(_T-f(_T)), _T)+_C1), y(_T) = exp(Int((diff(f(_T), _T))/(_T-f(_T)), _T))*(Int((diff(g(_T), _T))*exp(-(Int((diff(f(_T), _T))/(_T-f(_T)), _T)))/(_T-f(_T)), _T)+_C1)*f(_T)+g(_T)]](/support/helpjp/helpview.aspx?si=7218/file04364/math50.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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