def test_manualintegrate_special():
f, F = 4*exp(-x**2/3), 2*sqrt(3)*sqrt(pi)*erf(sqrt(3)*x/3)
assert_is_integral_of(f, F)
f, F = 3*exp(4*x**2), 3*sqrt(pi)*erfi(2*x)/4
assert_is_integral_of(f, F)
f, F = x**Rational(1, 3)*exp(-x/8), -16*uppergamma(Rational(4, 3), x/8)
assert_is_integral_of(f, F)
f, F = exp(2*x)/x, Ei(2*x)
assert_is_integral_of(f, F)
f, F = exp(1 + 2*x - x**2), sqrt(pi)*exp(2)*erf(x - 1)/2
assert_is_integral_of(f, F)
f = sin(x**2 + 4*x + 1)
F = (sqrt(2)*sqrt(pi)*(-sin(3)*fresnelc(sqrt(2)*(2*x + 4)/(2*sqrt(pi))) +
cos(3)*fresnels(sqrt(2)*(2*x + 4)/(2*sqrt(pi))))/2)
assert_is_integral_of(f, F)
f, F = cos(4*x**2), sqrt(2)*sqrt(pi)*fresnelc(2*sqrt(2)*x/sqrt(pi))/4
assert_is_integral_of(f, F)
f, F = sin(3*x + 2)/x, sin(2)*Ci(3*x) + cos(2)*Si(3*x)
assert_is_integral_of(f, F)
f, F = sinh(3*x - 2)/x, -sinh(2)*Chi(3*x) + cosh(2)*Shi(3*x)
assert_is_integral_of(f, F)
f, F = 5*cos(2*x - 3)/x, 5*cos(3)*Ci(2*x) + 5*sin(3)*Si(2*x)
assert_is_integral_of(f, F)
f, F = cosh(x/2)/x, Chi(x/2)
assert_is_integral_of(f, F)
f, F = cos(x**2)/x, Ci(x**2)/2
assert_is_integral_of(f, F)
f, F = 1/log(2*x + 1), li(2*x + 1)/2
assert_is_integral_of(f, F)
f, F = polylog(2, 5*x)/x, polylog(3, 5*x)
assert_is_integral_of(f, F)
f, F = 5/sqrt(3 - 2*sin(x)**2), 5*sqrt(3)*elliptic_f(x, Rational(2, 3))/3
assert_is_integral_of(f, F)
f, F = sqrt(4 + 9*sin(x)**2), 2*elliptic_e(x, Rational(-9, 4))
assert_is_integral_of(f, F)
def test_erf2():
assert erf2(0, 0) is S.Zero
assert erf2(x, x) is S.Zero
assert erf2(nan, 0) is nan
assert erf2(-oo, y) == erf(y) + 1
assert erf2( oo, y) == erf(y) - 1
assert erf2( x, oo) == 1 - erf(x)
assert erf2( x,-oo) == -1 - erf(x)
assert erf2(x, erf2inv(x, y)) == y
assert erf2(-x, -y) == -erf2(x,y)
assert erf2(-x, y) == erf(y) + erf(x)
assert erf2( x, -y) == -erf(y) - erf(x)
assert erf2(x, y).rewrite('fresnels') == erf(y).rewrite(fresnels)-erf(x).rewrite(fresnels)
assert erf2(x, y).rewrite('fresnelc') == erf(y).rewrite(fresnelc)-erf(x).rewrite(fresnelc)
assert erf2(x, y).rewrite('hyper') == erf(y).rewrite(hyper)-erf(x).rewrite(hyper)
assert erf2(x, y).rewrite('meijerg') == erf(y).rewrite(meijerg)-erf(x).rewrite(meijerg)
assert erf2(x, y).rewrite('uppergamma') == erf(y).rewrite(uppergamma) - erf(x).rewrite(uppergamma)
assert erf2(x, y).rewrite('expint') == erf(y).rewrite(expint)-erf(x).rewrite(expint)
assert erf2(I, 0).is_real is False
assert erf2(0, 0, evaluate=False).is_real
assert erf2(0, 0, evaluate=False).is_zero
assert erf2(x, x, evaluate=False).is_zero
assert erf2(x, y).is_zero is None
assert expand_func(erf(x) + erf2(x, y)) == erf(y)
assert conjugate(erf2(x, y)) == erf2(conjugate(x), conjugate(y))
assert erf2(x, y).rewrite('erf') == erf(y) - erf(x)
assert erf2(x, y).rewrite('erfc') == erfc(x) - erfc(y)
assert erf2(x, y).rewrite('erfi') == I*(erfi(I*x) - erfi(I*y))
assert erf2(x, y).diff(x) == erf2(x, y).fdiff(1)
assert erf2(x, y).diff(y) == erf2(x, y).fdiff(2)
assert erf2(x, y).diff(x) == -2*exp(-x**2)/sqrt(pi)
assert erf2(x, y).diff(y) == 2*exp(-y**2)/sqrt(pi)
raises(ArgumentIndexError, lambda: erf2(x, y).fdiff(3))
assert erf2(x, y).is_extended_real is None
xr, yr = symbols('xr yr', extended_real=True)
assert erf2(xr, yr).is_extended_real is True
def test_erfc():
assert erfc(nan) is nan
assert erfc(oo) is S.Zero
assert erfc(-oo) == 2
assert erfc(0) == 1
assert erfc(I*oo) == -oo*I
assert erfc(-I*oo) == oo*I
assert erfc(-x) == S(2) - erfc(x)
assert erfc(erfcinv(x)) == x
assert erfc(I).is_real is False
assert erfc(0, evaluate=False).is_real
assert erfc(0, evaluate=False).is_zero is False
assert erfc(erfinv(x)) == 1 - x
assert conjugate(erfc(z)) == erfc(conjugate(z))
assert erfc(x).as_leading_term(x) is S.One
assert erfc(1/x).as_leading_term(x) == S.Zero
assert erfc(z).rewrite('erf') == 1 - erf(z)
assert erfc(z).rewrite('erfi') == 1 + I*erfi(I*z)
assert erfc(z).rewrite('fresnels') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erfc(z).rewrite('fresnelc') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erfc(z).rewrite('hyper') == 1 - 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi)
assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi)
assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z
assert erfc(z).rewrite('expint') == S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi)
assert erfc(z).rewrite('tractable') == _erfs(z)*exp(-z**2)
assert expand_func(erf(x) + erfc(x)) is S.One
assert erfc(x).as_real_imag() == \
(erfc(re(x) - I*im(x))/2 + erfc(re(x) + I*im(x))/2,
-I*(-erfc(re(x) - I*im(x)) + erfc(re(x) + I*im(x)))/2)
assert erfc(x).as_real_imag(deep=False) == \
(erfc(re(x) - I*im(x))/2 + erfc(re(x) + I*im(x))/2,
-I*(-erfc(re(x) - I*im(x)) + erfc(re(x) + I*im(x)))/2)
assert erfc(w).as_real_imag() == (erfc(w), 0)
assert erfc(w).as_real_imag(deep=False) == (erfc(w), 0)
raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))
assert erfc(x).inverse() == erfcinv
def test_erf():
assert erf(nan) is nan
assert erf(oo) == 1
assert erf(-oo) == -1
assert erf(0) is S.Zero
assert erf(I*oo) == oo*I
assert erf(-I*oo) == -oo*I
assert erf(-2) == -erf(2)
assert erf(-x*y) == -erf(x*y)
assert erf(-x - y) == -erf(x + y)
assert erf(erfinv(x)) == x
assert erf(erfcinv(x)) == 1 - x
assert erf(erf2inv(0, x)) == x
assert erf(erf2inv(0, x, evaluate=False)) == x # To cover code in erf
assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x
assert erf(I).is_real is False
assert erf(0, evaluate=False).is_real
assert erf(0, evaluate=False).is_zero
assert conjugate(erf(z)) == erf(conjugate(z))
assert erf(x).as_leading_term(x) == 2*x/sqrt(pi)
assert erf(x*y).as_leading_term(y) == 2*x*y/sqrt(pi)
assert (erf(x*y)/erf(y)).as_leading_term(y) == x
assert erf(1/x).as_leading_term(x) == S.One
assert erf(z).rewrite('uppergamma') == sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z
assert erf(z).rewrite('erfc') == S.One - erfc(z)
assert erf(z).rewrite('erfi') == -I*erfi(I*z)
assert erf(z).rewrite('fresnels') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erf(z).rewrite('fresnelc') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erf(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi)
assert erf(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi)
assert erf(z).rewrite('expint') == sqrt(z**2)/z - z*expint(S.Half, z**2)/sqrt(S.Pi)
assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \
2/sqrt(pi)
assert limit((1 - erf(z))*exp(z**2)*z, z, oo) == 1/sqrt(pi)
assert limit((1 - erf(x))*exp(x**2)*sqrt(pi)*x, x, oo) == 1
assert limit(((1 - erf(x))*exp(x**2)*sqrt(pi)*x - 1)*2*x**2, x, oo) == -1
assert limit(erf(x)/x, x, 0) == 2/sqrt(pi)
assert limit(x**(-4) - sqrt(pi)*erf(x**2) / (2*x**6), x, 0) == S(1)/3
assert erf(x).as_real_imag() == \
(erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2,
-I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2)
assert erf(x).as_real_imag(deep=False) == \
(erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2,
-I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2)
assert erf(w).as_real_imag() == (erf(w), 0)
assert erf(w).as_real_imag(deep=False) == (erf(w), 0)
# issue 13575
assert erf(I).as_real_imag() == (0, -I*erf(I))
raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
assert erf(x).inverse() == erfinv
def test_erfi_evalf():
assert abs( erfi(Float(2.0)) - 18.5648024145756 ) < 1E-13 # XXX
def test_erfi_series():
assert erfi(x).series(x, 0, 7) == 2*x/sqrt(pi) + \
2*x**3/3/sqrt(pi) + x**5/5/sqrt(pi) + O(x**7)
assert erfi(x).series(x, oo) == \
(3/(4*x**5) + 1/(2*x**3) + 1/x + O(x**(-6), (x, oo)))*exp(x**2)/sqrt(pi) - I
def test_erfi():
assert erfi(nan) is nan
assert erfi(oo) is S.Infinity
assert erfi(-oo) is S.NegativeInfinity
assert erfi(0) is S.Zero
assert erfi(I*oo) == I
assert erfi(-I*oo) == -I
assert erfi(-x) == -erfi(x)
assert erfi(I*erfinv(x)) == I*x
assert erfi(I*erfcinv(x)) == I*(1 - x)
assert erfi(I*erf2inv(0, x)) == I*x
assert erfi(I*erf2inv(0, x, evaluate=False)) == I*x # To cover code in erfi
assert erfi(I).is_real is False
assert erfi(0, evaluate=False).is_real
assert erfi(0, evaluate=False).is_zero
assert conjugate(erfi(z)) == erfi(conjugate(z))
assert erfi(x).as_leading_term(x) == 2*x/sqrt(pi)
assert erfi(x*y).as_leading_term(y) == 2*x*y/sqrt(pi)
assert (erfi(x*y)/erfi(y)).as_leading_term(y) == x
assert erfi(1/x).as_leading_term(x) == erfi(1/x)
assert erfi(z).rewrite('erf') == -I*erf(I*z)
assert erfi(z).rewrite('erfc') == I*erfc(I*z) - I
assert erfi(z).rewrite('fresnels') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
I*fresnels(z*(1 + I)/sqrt(pi)))
assert erfi(z).rewrite('fresnelc') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
I*fresnels(z*(1 + I)/sqrt(pi)))
assert erfi(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], z**2)/sqrt(pi)
assert erfi(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [Rational(-1, 2)], -z**2)/sqrt(pi)
assert erfi(z).rewrite('uppergamma') == (sqrt(-z**2)/z*(uppergamma(S.Half,
-z**2)/sqrt(S.Pi) - S.One))
assert erfi(z).rewrite('expint') == sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(S.Pi)
assert erfi(z).rewrite('tractable') == -I*(-_erfs(I*z)*exp(z**2) + 1)
assert expand_func(erfi(I*z)) == I*erf(z)
assert erfi(x).as_real_imag() == \
(erfi(re(x) - I*im(x))/2 + erfi(re(x) + I*im(x))/2,
-I*(-erfi(re(x) - I*im(x)) + erfi(re(x) + I*im(x)))/2)
assert erfi(x).as_real_imag(deep=False) == \
(erfi(re(x) - I*im(x))/2 + erfi(re(x) + I*im(x))/2,
-I*(-erfi(re(x) - I*im(x)) + erfi(re(x) + I*im(x)))/2)
assert erfi(w).as_real_imag() == (erfi(w), 0)
assert erfi(w).as_real_imag(deep=False) == (erfi(w), 0)
raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
def test_issue_13575():
assert limit(acos(erfi(x)), x, 1) == acos(erfi(S.One))
def test_checksysodesol():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
eq = (Eq(diff(x(t),t), 9*y(t)), Eq(diff(y(t),t), 12*x(t)))
sol = [Eq(x(t), 9*C1*exp(-6*sqrt(3)*t) + 9*C2*exp(6*sqrt(3)*t)), \
Eq(y(t), -6*sqrt(3)*C1*exp(-6*sqrt(3)*t) + 6*sqrt(3)*C2*exp(6*sqrt(3)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 2*x(t) + 4*y(t)), Eq(diff(y(t),t), 12*x(t) + 41*y(t)))
sol = [Eq(x(t), 4*C1*exp(t*(-sqrt(1713)/2 + Rational(43, 2))) + 4*C2*exp(t*(sqrt(1713)/2 + \
Rational(43, 2)))), Eq(y(t), C1*(-sqrt(1713)/2 + Rational(39, 2))*exp(t*(-sqrt(1713)/2 + \
Rational(43, 2))) + C2*(Rational(39, 2) + sqrt(1713)/2)*exp(t*(sqrt(1713)/2 + Rational(43, 2))))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t)), Eq(diff(y(t),t), -2*x(t) + 2*y(t)))
sol = [Eq(x(t), (C1*sin(sqrt(7)*t/2) + C2*cos(sqrt(7)*t/2))*exp(t*Rational(3, 2))), \
Eq(y(t), ((C1/2 - sqrt(7)*C2/2)*sin(sqrt(7)*t/2) + (sqrt(7)*C1/2 + \
C2/2)*cos(sqrt(7)*t/2))*exp(t*Rational(3, 2)))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
sol = [Eq(x(t), C1*exp(t*(-sqrt(6) + 3)) + C2*exp(t*(sqrt(6) + 3)) - \
Rational(22, 3)), Eq(y(t), C1*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) + C2*(2 + \
sqrt(6))*exp(t*(sqrt(6) + 3)) - Rational(5, 3))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23))
sol = [Eq(x(t), (C1*sin(sqrt(2)*t) + C2*cos(sqrt(2)*t))*exp(t) - Rational(58, 3)), \
Eq(y(t), (sqrt(2)*C1*cos(sqrt(2)*t) - sqrt(2)*C2*sin(sqrt(2)*t))*exp(t) - Rational(185, 3))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t)))
sol = [Eq(x(t), (C1*exp(Integral(2, t).doit()) + C2*exp(-(Integral(2, t)).doit()))*\
exp((Integral(5*t, t)).doit())), Eq(y(t), (C1*exp((Integral(2, t)).doit()) - \
C2*exp(-(Integral(2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t)))
sol = [Eq(x(t), (C1*cos((Integral(t**2, t)).doit()) + C2*sin((Integral(t**2, t)).doit()))*\
exp((Integral(5*t, t)).doit())), Eq(y(t), (-C1*sin((Integral(t**2, t)).doit()) + \
C2*cos((Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + (5*t+9*t**2)*y(t)))
sol = [Eq(x(t), (C1*exp((-sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()) + \
C2*exp((sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit())), \
Eq(y(t), (C1*(-sqrt(77)/2 + Rational(9, 2))*exp((-sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()) + \
C2*(sqrt(77)/2 + Rational(9, 2))*exp((sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 5*x(t) + 43*y(t)), Eq(diff(y(t),t,t), x(t) + 9*y(t)))
root0 = -sqrt(-sqrt(47) + 7)
root1 = sqrt(-sqrt(47) + 7)
root2 = -sqrt(sqrt(47) + 7)
root3 = sqrt(sqrt(47) + 7)
sol = [Eq(x(t), 43*C1*exp(t*root0) + 43*C2*exp(t*root1) + 43*C3*exp(t*root2) + 43*C4*exp(t*root3)), \
Eq(y(t), C1*(root0**2 - 5)*exp(t*root0) + C2*(root1**2 - 5)*exp(t*root1) + \
C3*(root2**2 - 5)*exp(t*root2) + C4*(root3**2 - 5)*exp(t*root3))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 8*x(t)+3*y(t)+31), Eq(diff(y(t),t,t), 9*x(t)+7*y(t)+12))
root0 = -sqrt(-sqrt(109)/2 + Rational(15, 2))
root1 = sqrt(-sqrt(109)/2 + Rational(15, 2))
root2 = -sqrt(sqrt(109)/2 + Rational(15, 2))
root3 = sqrt(sqrt(109)/2 + Rational(15, 2))
sol = [Eq(x(t), 3*C1*exp(t*root0) + 3*C2*exp(t*root1) + 3*C3*exp(t*root2) + 3*C4*exp(t*root3) - Rational(181, 29)), \
Eq(y(t), C1*(root0**2 - 8)*exp(t*root0) + C2*(root1**2 - 8)*exp(t*root1) + \
C3*(root2**2 - 8)*exp(t*root2) + C4*(root3**2 - 8)*exp(t*root3) + Rational(183, 29))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t) - 9*diff(y(t),t) + 7*x(t),0), Eq(diff(y(t),t,t) + 9*diff(x(t),t) + 7*y(t),0))
sol = [Eq(x(t), C1*cos(t*(Rational(9, 2) + sqrt(109)/2)) + C2*sin(t*(Rational(9, 2) + sqrt(109)/2)) + \
C3*cos(t*(-sqrt(109)/2 + Rational(9, 2))) + C4*sin(t*(-sqrt(109)/2 + Rational(9, 2)))), Eq(y(t), -C1*sin(t*(Rational(9, 2) + sqrt(109)/2)) \
+ C2*cos(t*(Rational(9, 2) + sqrt(109)/2)) - C3*sin(t*(-sqrt(109)/2 + Rational(9, 2))) + C4*cos(t*(-sqrt(109)/2 + Rational(9, 2))))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 9*t*diff(y(t),t)-9*y(t)), Eq(diff(y(t),t,t),7*t*diff(x(t),t)-7*x(t)))
I1 = sqrt(6)*7**Rational(1, 4)*sqrt(pi)*erfi(sqrt(6)*7**Rational(1, 4)*t/2)/2 - exp(3*sqrt(7)*t**2/2)/t
I2 = -sqrt(6)*7**Rational(1, 4)*sqrt(pi)*erf(sqrt(6)*7**Rational(1, 4)*t/2)/2 - exp(-3*sqrt(7)*t**2/2)/t
sol = [Eq(x(t), C3*t + t*(9*C1*I1 + 9*C2*I2)), Eq(y(t), C4*t + t*(3*sqrt(7)*C1*I1 - 3*sqrt(7)*C2*I2))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 21*x(t)), Eq(diff(y(t),t), 17*x(t)+3*y(t)), Eq(diff(z(t),t), 5*x(t)+7*y(t)+9*z(t)))
sol = [Eq(x(t), C1*exp(21*t)), Eq(y(t), 17*C1*exp(21*t)/18 + C2*exp(3*t)), \
Eq(z(t), 209*C1*exp(21*t)/216 - 7*C2*exp(3*t)/6 + C3*exp(9*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),3*y(t)-11*z(t)),Eq(diff(y(t),t),7*z(t)-3*x(t)),Eq(diff(z(t),t),11*x(t)-7*y(t)))
sol = [Eq(x(t), 7*C0 + sqrt(179)*C1*cos(sqrt(179)*t) + (77*C1/3 + 130*C2/3)*sin(sqrt(179)*t)), \
Eq(y(t), 11*C0 + sqrt(179)*C2*cos(sqrt(179)*t) + (-58*C1/3 - 77*C2/3)*sin(sqrt(179)*t)), \
Eq(z(t), 3*C0 + sqrt(179)*(-7*C1/3 - 11*C2/3)*cos(sqrt(179)*t) + (11*C1 - 7*C2)*sin(sqrt(179)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(3*diff(x(t),t),4*5*(y(t)-z(t))),Eq(4*diff(y(t),t),3*5*(z(t)-x(t))),Eq(5*diff(z(t),t),3*4*(x(t)-y(t))))
sol = [Eq(x(t), C0 + 5*sqrt(2)*C1*cos(5*sqrt(2)*t) + (12*C1/5 + 164*C2/15)*sin(5*sqrt(2)*t)), \
Eq(y(t), C0 + 5*sqrt(2)*C2*cos(5*sqrt(2)*t) + (-51*C1/10 - 12*C2/5)*sin(5*sqrt(2)*t)), \
Eq(z(t), C0 + 5*sqrt(2)*(-9*C1/25 - 16*C2/25)*cos(5*sqrt(2)*t) + (12*C1/5 - 12*C2/5)*sin(5*sqrt(2)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),4*x(t) - z(t)),Eq(diff(y(t),t),2*x(t)+2*y(t)-z(t)),Eq(diff(z(t),t),3*x(t)+y(t)))
sol = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t) + C3*exp(2*t)), \
Eq(y(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t)), \
Eq(z(t), 2*C1*exp(2*t) + 2*C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t) + C3*t*exp(2*t) + C3*exp(2*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),4*x(t) - y(t) - 2*z(t)),Eq(diff(y(t),t),2*x(t) + y(t)- 2*z(t)),Eq(diff(z(t),t),5*x(t)-3*z(t)))
sol = [Eq(x(t), C1*exp(2*t) + C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), \
Eq(y(t), C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), Eq(z(t), C1*exp(2*t) + 5*C2*cos(t) + 5*C3*sin(t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol = [Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(Rational(-1, 4)))), Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5))
sol = [Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(x(t),t*diff(x(t),t)+diff(x(t),t)*diff(y(t),t)), Eq(y(t),t*diff(y(t),t)+diff(y(t),t)**2))
sol = {Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)}
assert checksysodesol(eq, sol) == (True, [0, 0])
