Solving Bernoulli's ODEs - Maple Help

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Solving Bernoulli's ODEs

Description

• 

The general form of Bernoulli's equation is given by:

Bernoulli_ode := diff(y(x),x)+f(x)*y(x)+g(x)*y(x)^a;

Bernoulli_ode:=ⅆⅆxyx+fxyx+gxyxa

(1)
  

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:

Examples

withDEtools,odeadvisor

odeadvisor

(2)

odeadvisorBernoulli_ode

_Bernoulli

(3)

withPDEtools,dchange

dchange

(4)

ITR:=yx=ut11a,x=t

ITR:=x=t,yx=ut11a

(5)

and the ODE becomes

new_ode:=dchangeITR,Bernoulli_ode,ut,t:

new_ode2:=solvenew_ode,ⅆⅆtut:

opfactorcombineexpandnew_ode2,power

ⅆⅆtut=a1ut1a1agtutaa1+utft

(6)

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:

ans:=dsolveBernoulli_ode

ans:=yx=ⅇ∫fxⅆxa1a∫ⅇ∫fxⅆxgxⅇ∫fxⅆxaⅆx+_C1∫ⅇ∫fxⅆxgxⅇ∫fxⅆxaⅆx1a1ⅇ∫fxⅆxaa1

(7)

See Also

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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