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Solving Abel's ODEs of the Second Kind, Class C
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Description
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The general form of Abel's equation, second kind, class C is given by:
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Abel_ode2C := (g1(x)*y(x)+g0(x))*diff(y(x),x)
-f3(x)*y(x)^3-f2(x)*y(x)^2+f1(x)*y(x)+f0(x);
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where f3(x), f2(x), f1(x), f0(x), g1(x) and g0(x) are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 28. There is as yet no general solution for this ODE.
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Examples
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![[odeadvisor]](/support/helpjp/helpview.aspx?si=7127/file04345/math29.png)
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All ODEs of type Abel, second kind, can be rewritten as ODEs of type Abel, first kind, using the following transformation:
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![[dchange]](/support/helpjp/helpview.aspx?si=7127/file04345/math38.png)
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![new_ode := dchange(ITR, Abel_ode2C, [u(t), t])](/support/helpjp/helpview.aspx?si=7127/file04345/math49.png)
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![odeadvisor(new_ode, u(t), [Abel])](/support/helpjp/helpview.aspx?si=7127/file04345/math60.png)
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![[_Abel]](/support/helpjp/helpview.aspx?si=7127/file04345/math63.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, rational, Clairaut, dAlembert, sym_implicit, patterns; for other differential orders see odeadvisor,types.
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