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Lagerstrom ODEs
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Description
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The general form of the Lagerstrom ODE is given by the following:
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Lagerstrom_ode := diff(y(x),x,x)= -k*diff(y(x),x)/x-epsilon*y(x)*diff(y(x),x);
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See Rosenblat and Shepherd, "On the Asymptotic Solution of the Lagerstrom Model Equation".
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Examples
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The second order Lagerstrom ODE can be reduced to a first order ODE of Abel type once the system succeeds in finding one polynomial symmetry for it (see ?symgen):
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![[_Lagerstrom, [_2nd_order, _with_linear_symmetries]]](/support/helpjp/helpview.aspx?si=7240/file04356/math37.png)
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![[_xi = -x, _eta = y]](/support/helpjp/helpview.aspx?si=7240/file04356/math44.png)
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From which, giving the same indication directly to dsolve, you obtain the reduction of order
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![ans := y(x) = `&where`(_a*exp(Int(_b(_a), _a)+_C1), [{diff(_b(_a), _a) = (2*_a-k*_a-epsilon*_a^2)*_b(_a)^3+(3-k-epsilon*_a)*_b(_a)^2}, {_a = y(x)*x, _b(_a) = -1/(((diff(y(x), x))*x+y(x))*x)}, {x = 1/exp(Int(_b(_a), _a)+_C1), y(x) = _a*exp(Int(_b(_a), _a)+_C1)}])](/support/helpjp/helpview.aspx?si=7240/file04356/math56.png)
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For the structure of the solution above see ?ODESolStruc. Reductions of order can also be tested with odetest
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The reduced ODE is of Abel type and can be selected using the mouse, or as follows
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![reduced_ode := op([2, 2, 1, 1], ans)](/support/helpjp/helpview.aspx?si=7240/file04356/math77.png)
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![[_Abel]](/support/helpjp/helpview.aspx?si=7240/file04356/math87.png)
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See Also
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DEtools, odeadvisor, dsolve, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, missing, reducible, linear_ODEs, exact_linear, exact_nonlinear, sym_Fx, linear_sym, Bessel, Painleve, Halm, Gegenbauer, Duffing, ellipsoidal, elliptic, erf, Emden, Jacobi, Hermite, Lagerstrom, Laguerre, Liouville, Lienard, Van_der_Pol, Titchmarsh; for other differential orders see odeadvisor,types.
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