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Solving Bernoulli's ODEs
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Description
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The general form of Bernoulli's equation is given by:
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Bernoulli_ode := diff(y(x),x)+f(x)*y(x)+g(x)*y(x)^a;
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where f(x) and g(x) are arbitrary functions, and a is a symbolic power. See Differentialgleichungen, by E. Kamke, p. 19. Basically, the method consists of making a change of variables, leading to a linear equation which can be solved in general manner. The transformation is given by the following:
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Examples
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![[odeadvisor]](/support/helpjp/helpview.aspx?si=7206/file04346/math28.png)
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![[_Bernoulli]](/support/helpjp/helpview.aspx?si=7206/file04346/math35.png)
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![[dchange]](/support/helpjp/helpview.aspx?si=7206/file04346/math42.png)
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and the ODE becomes
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![new_ode := dchange(ITR, Bernoulli_ode, [u(t), t])](/support/helpjp/helpview.aspx?si=7206/file04346/math55.png)
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This ODE can then be solved by dsolve. Afterwards, another change of variables will reintroduce the original variables x and y(x).
The present implementation of dsolve can arrive directly at a general solution for Bernoulli's equation:
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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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