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from sympy.core.function import Function |
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from sympy.core.numbers import Rational |
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from sympy.core.relational import Eq |
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from sympy.core.symbol import (Symbol, symbols) |
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from sympy.functions.elementary.exponential import (exp, log) |
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from sympy.functions.elementary.miscellaneous import sqrt |
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from sympy.functions.elementary.trigonometric import (atan, sin, tan) |
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from sympy.solvers.ode import (classify_ode, checkinfsol, dsolve, infinitesimals) |
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from sympy.solvers.ode.subscheck import checkodesol |
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from sympy.testing.pytest import XFAIL |
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C1 = Symbol('C1') |
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x, y = symbols("x y") |
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f = Function('f') |
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xi = Function('xi') |
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eta = Function('eta') |
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def test_heuristic1(): |
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a, b, c, a4, a3, a2, a1, a0 = symbols("a b c a4 a3 a2 a1 a0") |
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df = f(x).diff(x) |
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eq = Eq(df, x**2*f(x)) |
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eq1 = f(x).diff(x) + a*f(x) - c*exp(b*x) |
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eq2 = f(x).diff(x) + 2*x*f(x) - x*exp(-x**2) |
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eq3 = (1 + 2*x)*df + 2 - 4*exp(-f(x)) |
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eq4 = f(x).diff(x) - (a4*x**4 + a3*x**3 + a2*x**2 + a1*x + a0)**Rational(-1, 2) |
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eq5 = x**2*df - f(x) + x**2*exp(x - (1/x)) |
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eqlist = [eq, eq1, eq2, eq3, eq4, eq5] |
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i = infinitesimals(eq, hint='abaco1_simple') |
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assert i == [{eta(x, f(x)): exp(x**3/3), xi(x, f(x)): 0}, |
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{eta(x, f(x)): f(x), xi(x, f(x)): 0}, |
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{eta(x, f(x)): 0, xi(x, f(x)): x**(-2)}] |
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i1 = infinitesimals(eq1, hint='abaco1_simple') |
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assert i1 == [{eta(x, f(x)): exp(-a*x), xi(x, f(x)): 0}] |
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i2 = infinitesimals(eq2, hint='abaco1_simple') |
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assert i2 == [{eta(x, f(x)): exp(-x**2), xi(x, f(x)): 0}] |
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i3 = infinitesimals(eq3, hint='abaco1_simple') |
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assert i3 == [{eta(x, f(x)): 0, xi(x, f(x)): 2*x + 1}, |
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{eta(x, f(x)): 0, xi(x, f(x)): 1/(exp(f(x)) - 2)}] |
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i4 = infinitesimals(eq4, hint='abaco1_simple') |
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assert i4 == [{eta(x, f(x)): 1, xi(x, f(x)): 0}, |
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{eta(x, f(x)): 0, |
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xi(x, f(x)): sqrt(a0 + a1*x + a2*x**2 + a3*x**3 + a4*x**4)}] |
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i5 = infinitesimals(eq5, hint='abaco1_simple') |
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assert i5 == [{xi(x, f(x)): 0, eta(x, f(x)): exp(-1/x)}] |
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ilist = [i, i1, i2, i3, i4, i5] |
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for eq, i in (zip(eqlist, ilist)): |
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check = checkinfsol(eq, i) |
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assert check[0] |
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eq6 = df - (f(x)/x)*(x*log(x**2/f(x)) + 2) |
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i = infinitesimals(eq6, hint='abaco1_product') |
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assert i == [{eta(x, f(x)): f(x)*exp(-x), xi(x, f(x)): 0}] |
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assert checkinfsol(eq6, i)[0] |
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eq7 = x*(f(x).diff(x)) + 1 - f(x)**2 |
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i = infinitesimals(eq7, hint='chi') |
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assert checkinfsol(eq7, i)[0] |
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def test_heuristic3(): |
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a, b = symbols("a b") |
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df = f(x).diff(x) |
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eq = x**2*df + x*f(x) + f(x)**2 + x**2 |
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i = infinitesimals(eq, hint='bivariate') |
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assert i == [{eta(x, f(x)): f(x), xi(x, f(x)): x}] |
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assert checkinfsol(eq, i)[0] |
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eq = x**2*(-f(x)**2 + df)- a*x**2*f(x) + 2 - a*x |
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i = infinitesimals(eq, hint='bivariate') |
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assert checkinfsol(eq, i)[0] |
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def test_heuristic_function_sum(): |
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eq = f(x).diff(x) - (3*(1 + x**2/f(x)**2)*atan(f(x)/x) + (1 - 2*f(x))/x + |
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(1 - 3*f(x))*(x/f(x)**2)) |
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i = infinitesimals(eq, hint='function_sum') |
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assert i == [{eta(x, f(x)): f(x)**(-2) + x**(-2), xi(x, f(x)): 0}] |
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assert checkinfsol(eq, i)[0] |
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def test_heuristic_abaco2_similar(): |
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a, b = symbols("a b") |
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F = Function('F') |
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eq = f(x).diff(x) - F(a*x + b*f(x)) |
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i = infinitesimals(eq, hint='abaco2_similar') |
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assert i == [{eta(x, f(x)): -a/b, xi(x, f(x)): 1}] |
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assert checkinfsol(eq, i)[0] |
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eq = f(x).diff(x) - (f(x)**2 / (sin(f(x) - x) - x**2 + 2*x*f(x))) |
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i = infinitesimals(eq, hint='abaco2_similar') |
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assert i == [{eta(x, f(x)): f(x)**2, xi(x, f(x)): f(x)**2}] |
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assert checkinfsol(eq, i)[0] |
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def test_heuristic_abaco2_unique_unknown(): |
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a, b = symbols("a b") |
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F = Function('F') |
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eq = f(x).diff(x) - x**(a - 1)*(f(x)**(1 - b))*F(x**a/a + f(x)**b/b) |
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i = infinitesimals(eq, hint='abaco2_unique_unknown') |
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assert i == [{eta(x, f(x)): -f(x)*f(x)**(-b), xi(x, f(x)): x*x**(-a)}] |
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assert checkinfsol(eq, i)[0] |
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eq = f(x).diff(x) + tan(F(x**2 + f(x)**2) + atan(x/f(x))) |
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i = infinitesimals(eq, hint='abaco2_unique_unknown') |
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assert i == [{eta(x, f(x)): x, xi(x, f(x)): -f(x)}] |
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assert checkinfsol(eq, i)[0] |
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eq = (x*f(x).diff(x) + f(x) + 2*x)**2 -4*x*f(x) -4*x**2 -4*a |
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i = infinitesimals(eq, hint='abaco2_unique_unknown') |
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assert checkinfsol(eq, i)[0] |
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def test_heuristic_linear(): |
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a, b, m, n = symbols("a b m n") |
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eq = x**(n*(m + 1) - m)*(f(x).diff(x)) - a*f(x)**n -b*x**(n*(m + 1)) |
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i = infinitesimals(eq, hint='linear') |
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assert checkinfsol(eq, i)[0] |
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@XFAIL |
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def test_kamke(): |
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a, b, alpha, c = symbols("a b alpha c") |
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eq = x**2*(a*f(x)**2+(f(x).diff(x))) + b*x**alpha + c |
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i = infinitesimals(eq, hint='sum_function') |
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assert checkinfsol(eq, i)[0] |
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def test_user_infinitesimals(): |
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x = Symbol("x") |
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eq = x*(f(x).diff(x)) + 1 - f(x)**2 |
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sol = Eq(f(x), (C1 + x**2)/(C1 - x**2)) |
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infinitesimals = {'xi':sqrt(f(x) - 1)/sqrt(f(x) + 1), 'eta':0} |
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assert dsolve(eq, hint='lie_group', **infinitesimals) == sol |
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assert checkodesol(eq, sol) == (True, 0) |
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@XFAIL |
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def test_lie_group_issue15219(): |
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eqn = exp(f(x).diff(x)-f(x)) |
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assert 'lie_group' not in classify_ode(eqn, f(x)) |
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