Differential equations with linear coefficients - First implementation

    
    Solves a differential equation with linear-coefficients.

    The general form of a differential equation with linear-coefficients is
    y' + F((a1*x + b1*y + c1)/(a2*x + b2*y + c2)) = 0,
    where a1, b1, c1, a2, b2, c2 are constants and (a1*b2 - a2*b1) != 0

    This can be solved by substituting x = x' + ((b2*c1 - b1*c2)/(a2*b1 - a1*b2))
    and y = y' + ((a1*c2 - a2*c1)/(a2*b1 - a1*b2)). Making the given
    substitution reduces it to a homogeneous differential equation
    (see the docstring of ode_1st_homogeneous_coeff_best(), ode_1st_homogeneous_coeff_subs_indep_div_dep()
    and 1st_homogeneous_coeff_subs_dep_div_indep())

    Examples
    ========

    >>> from sympy import Function, Derivative, pprint
    >>> from sympy.solvers.ode import dsolve, classify_ode
    >>> from sympy.abc import x
    >>> f = Function('f')
    >>> df = f(x).diff(x)
    >>> eq = (x + f(x) + 1)*df + (f(x) -6*x + 1)
    >>> dsolve(eq, hint = 'linear_coefficients')
    [f(x) == -x - sqrt(C1 + 7*x**2) - 1, f(x) == -x + sqrt(C1 + 7*x**2) - 1]
    >>> pprint(dsolve(eq, hint = 'linear_coefficients'))
                      ___________                     ___________
                   /         2                     /         2
    [f(x) = -x - \/  C1 + 7*x   - 1, f(x) = -x + \/  C1 + 7*x   - 1]


    References
    ==========

    - Joel Moses, "Symbolic Integration - The Stormy Decade",
        Communications of the ACM, Volume 14, Number 8, August 1971, pp. 558