def test_rsolve_raises(): x = Function('x') raises(ValueError, lambda: rsolve(y(n) - y(k + 1), y(n))) raises(ValueError, lambda: rsolve(y(n) - y(n + 1), x(n))) raises(ValueError, lambda: rsolve(y(n) - x(n + 1), y(n))) raises(ValueError, lambda: rsolve(y(n) - sqrt(n) * y(n + 1), y(n))) raises(ValueError, lambda: rsolve(y(n) - y(n + 1), y(n), {x(0): 0}))
def test_rsolve_raises(): x = Function("x") raises(ValueError, lambda: rsolve(y(n) - y(k + 1), y(n))) raises(ValueError, lambda: rsolve(y(n) - y(n + 1), x(n))) raises(ValueError, lambda: rsolve(y(n) - x(n + 1), y(n))) raises(ValueError, lambda: rsolve(y(n) - sqrt(n) * y(n + 1), y(n))) raises(ValueError, lambda: rsolve(y(n) - y(n + 1), y(n), {x(0): 0}))
def test_constant_naming(): #issue 8697 assert rsolve(y(n+3) - y(n+2) - y(n+1) + y(n), y(n)) == (-1)**n*C0+C1+C2*n assert rsolve(y(n+3)+3*y(n+2)+3*y(n+1)+y(n), y(n)).expand() == C0*(-1)**n + (-1)**n*C1*n + (-1)**n*C2*n**2 assert rsolve(y(n) - 2*y(n - 3) + 5*y(n - 2) - 4*y(n - 1),y(n),[1,3,8]) == 3*2**n - n - 2 #issue 19630 assert rsolve(y(n+3) - 3*y(n+1) + 2*y(n), y(n), {y(1):0, y(2):8, y(3):-2}) == (-2)**n + 2*n
def test_rsolve(): f = y(n+2) - y(n+1) - y(n) h = sqrt(5)*(S.Half + S.Half*sqrt(5))**n \ - sqrt(5)*(S.Half - S.Half*sqrt(5))**n assert rsolve(f, y(n)) in [ C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n, C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n, ] assert rsolve(f, y(n), [ 0, 5 ]) == h assert rsolve(f, y(n), { 0 :0, 1 :5 }) == h assert rsolve(f, y(n), { y(0):0, y(1):5 }) == h assert rsolve(y(n) - y(n-1) - y(n-2), y(n), [0, 5]) == h f = (n-1)*y(n+2) - (n**2+3*n-2)*y(n+1) + 2*n*(n+1)*y(n) g = C0*factorial(n) + C1*2**n h = -3*factorial(n) + 3*2**n assert rsolve(f, y(n)) == g assert rsolve(f, y(n), [ 0, 3 ]) == h assert rsolve(f, y(n), { 0 :0, 1 :3 }) == h assert rsolve(f, y(n), { y(0):0, y(1):3 }) == h
def test_issue_17990(): f = -10 * y(n) + 4 * y(n + 1) + 6 * y(n + 2) + 46 * y(n + 3) sol = rsolve(f, y(n)) expected = C0 * ( (86 * 18**(S(1) / 3) / 69 + (-12 + (-1 + sqrt(3) * I) * (290412 + 3036 * sqrt(9165))**(S(1) / 3)) * (1 - sqrt(3) * I) * (24201 + 253 * sqrt(9165))**(S(1) / 3) / 276) / ((1 - sqrt(3) * I) * (24201 + 253 * sqrt(9165))**(S(1) / 3)))**n + C1 * ( (86 * 18**(S(1) / 3) / 69 + (-12 + (-1 - sqrt(3) * I) * (290412 + 3036 * sqrt(9165))**(S(1) / 3)) * (1 + sqrt(3) * I) * (24201 + 253 * sqrt(9165))**(S(1) / 3) / 276) / ((1 + sqrt(3) * I) * (24201 + 253 * sqrt(9165))**(S(1) / 3)))**n + C2 * ( -43 * 18**(S(1) / 3) / (69 * (24201 + 253 * sqrt(9165))**(S(1) / 3)) - S(1) / 23 + (290412 + 3036 * sqrt(9165))**(S(1) / 3) / 138)**n assert sol == expected e = sol.subs({C0: 1, C1: 1, C2: 1, n: 1}).evalf() assert abs(e + 0.130434782608696) < 1e-13
def test_rsolve(): f = y(n + 2) - y(n + 1) - y(n) h = sqrt(5)*(S.Half + S.Half*sqrt(5))**n \ - sqrt(5)*(S.Half - S.Half*sqrt(5))**n assert rsolve(f, y(n)) in [ C0 * (S.Half - S.Half * sqrt(5))**n + C1 * (S.Half + S.Half * sqrt(5))**n, C1 * (S.Half - S.Half * sqrt(5))**n + C0 * (S.Half + S.Half * sqrt(5))**n, ] assert rsolve(f, y(n), [0, 5]) == h assert rsolve(f, y(n), {0: 0, 1: 5}) == h assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = (n - 1) * y(n + 2) - (n**2 + 3 * n - 2) * y(n + 1) + 2 * n * (n + 1) * y(n) g = C1 * factorial(n) + C0 * 2**n h = -3 * factorial(n) + 3 * 2**n assert rsolve(f, y(n)) == g assert rsolve(f, y(n), []) == g assert rsolve(f, y(n), {}) == g assert rsolve(f, y(n), [0, 3]) == h assert rsolve(f, y(n), {0: 0, 1: 3}) == h assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - y(n - 1) - 2 assert rsolve(f, y(n), {y(0): 0}) == 2 * n assert rsolve(f, y(n), {y(0): 1}) == 2 * n + 1 assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = 3 * y(n - 1) - y(n) - 1 assert rsolve(f, y(n), {y(0): 0}) == -3**n / 2 + S.Half assert rsolve(f, y(n), {y(0): 1}) == 3**n / 2 + S.Half assert rsolve(f, y(n), {y(0): 2}) == 3 * 3**n / 2 + S.Half assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - 1 / n * y(n - 1) assert rsolve(f, y(n)) == C0 / factorial(n) assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - 1 / n * y(n - 1) - 1 assert rsolve(f, y(n)) is None f = 2 * y(n - 1) + (1 - n) * y(n) / n assert rsolve(f, y(n), {y(1): 1}) == 2**(n - 1) * n assert rsolve(f, y(n), {y(1): 2}) == 2**(n - 1) * n * 2 assert rsolve(f, y(n), {y(1): 3}) == 2**(n - 1) * n * 3 assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = (n - 1) * (n - 2) * y(n + 2) - (n + 1) * (n + 2) * y(n) assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n * (n - 1) * (n - 2) assert rsolve(f, y(n), { y(3): 6, y(4): -24 }) == -n * (n - 1) * (n - 2) * (-1)**(n) assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 assert rsolve(Eq(y(n + 1), a * y(n)), y(n), {y(1): a}).simplify() == a**n assert rsolve(y(n) - a*y(n-2),y(n), \ {y(1): sqrt(a)*(a + b), y(2): a*(a - b)}).simplify() == \ a**(n/2)*(-(-1)**n*b + a) f = (-16 * n**2 + 32 * n - 12) * y(n - 1) + (4 * n**2 - 12 * n + 9) * y(n) assert expand_func(rsolve(f, y(n), \ {y(1): binomial(2*n + 1, 3)}).rewrite(gamma)).simplify() == \ 2**(2*n)*n*(2*n - 1)*(4*n**2 - 1)/12 assert (rsolve(y(n) + a * (y(n + 1) + y(n - 1)) / 2, y(n)) - (C0 * ((sqrt(-a**2 + 1) - 1) / a)**n + C1 * ((-sqrt(-a**2 + 1) - 1) / a)**n)).simplify() == 0
def test_issue_6844(): f = y(n + 2) - y(n + 1) + y(n) / 4 assert rsolve(f, y(n)) == 2**(-n) * (C0 + C1 * n) assert rsolve(f, y(n), {y(0): 0, y(1): 1}) == 2 * 2**(-n) * n
def test_issue_15751(): f = y(n) + 21 * y(n + 1) - 273 * y(n + 2) - 1092 * y(n + 3) + 1820 * y( n + 4) + 1092 * y(n + 5) - 273 * y(n + 6) - 21 * y(n + 7) + y(n + 8) assert rsolve(f, y(n)) is not None
def test_issue_18751(): r = Symbol('r', real=True, positive=True) theta = Symbol('theta', real=True) f = y(n) - 2 * r * cos(theta) * y(n - 1) + r**2 * y(n - 2) assert rsolve(f, y(n)) == \ C0*(r*(cos(theta) - I*Abs(sin(theta))))**n + C1*(r*(cos(theta) + I*Abs(sin(theta))))**n
def test_rsolve(): f = y(n + 2) - y(n + 1) - y(n) h = sqrt(5) * (S.Half + S.Half * sqrt(5)) ** n - sqrt(5) * (S.Half - S.Half * sqrt(5)) ** n assert rsolve(f, y(n)) in [ C0 * (S.Half - S.Half * sqrt(5)) ** n + C1 * (S.Half + S.Half * sqrt(5)) ** n, C1 * (S.Half - S.Half * sqrt(5)) ** n + C0 * (S.Half + S.Half * sqrt(5)) ** n, ] assert rsolve(f, y(n), [0, 5]) == h assert rsolve(f, y(n), {0: 0, 1: 5}) == h assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = (n - 1) * y(n + 2) - (n ** 2 + 3 * n - 2) * y(n + 1) + 2 * n * (n + 1) * y(n) g = C1 * factorial(n) + C0 * 2 ** n h = -3 * factorial(n) + 3 * 2 ** n assert rsolve(f, y(n)) == g assert rsolve(f, y(n), []) == g assert rsolve(f, y(n), {}) == g assert rsolve(f, y(n), [0, 3]) == h assert rsolve(f, y(n), {0: 0, 1: 3}) == h assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - y(n - 1) - 2 assert rsolve(f, y(n), {y(0): 0}) == 2 * n assert rsolve(f, y(n), {y(0): 1}) == 2 * n + 1 assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = 3 * y(n - 1) - y(n) - 1 assert rsolve(f, y(n), {y(0): 0}) == -3 ** n / 2 + S.Half assert rsolve(f, y(n), {y(0): 1}) == 3 ** n / 2 + S.Half assert rsolve(f, y(n), {y(0): 2}) == 3 * 3 ** n / 2 + S.Half assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - 1 / n * y(n - 1) assert rsolve(f, y(n)) == C0 / factorial(n) assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - 1 / n * y(n - 1) - 1 assert rsolve(f, y(n)) is None f = 2 * y(n - 1) + (1 - n) * y(n) / n assert rsolve(f, y(n), {y(1): 1}) == 2 ** (n - 1) * n assert rsolve(f, y(n), {y(1): 2}) == 2 ** (n - 1) * n * 2 assert rsolve(f, y(n), {y(1): 3}) == 2 ** (n - 1) * n * 3 assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = (n - 1) * (n - 2) * y(n + 2) - (n + 1) * (n + 2) * y(n) assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n * (n - 1) * (n - 2) assert rsolve(f, y(n), {y(3): 6, y(4): -24}) == n * (n - 1) * (n - 2) * (-1) ** (3 - n) assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 a, b, c = symbols("a,b,c") assert rsolve(Eq(y(n + 1), a * y(n)), y(n)).simplify() == C0 * a ** n assert rsolve(Eq(y(n + 1), a * y(n)), y(n), {y(1): a}).simplify() == a ** n f = y(n) - a * y(n - 2) assert rsolve(f, y(n)) == C0 * a ** (n / 2) + C1 * (-sqrt(a)) ** n assert rsolve(f, y(n), {y(1): sqrt(a) * (a + b), y(2): a * (a - b)}).simplify() == a ** (n / 2) * ( (-1) ** (n + 1) * b + a )
def test_rsolve(): f = y(n + 2) - y(n + 1) - y(n) h = sqrt(5)*(S.Half + S.Half*sqrt(5))**n \ - sqrt(5)*(S.Half - S.Half*sqrt(5))**n assert rsolve(f, y(n)) in [ C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n, C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n, ] assert rsolve(f, y(n), [0, 5]) == h assert rsolve(f, y(n), {0: 0, 1: 5}) == h assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n) g = C1*factorial(n) + C0*2**n h = -3*factorial(n) + 3*2**n assert rsolve(f, y(n)) == g assert rsolve(f, y(n), []) == g assert rsolve(f, y(n), {}) == g assert rsolve(f, y(n), [0, 3]) == h assert rsolve(f, y(n), {0: 0, 1: 3}) == h assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - y(n - 1) - 2 assert rsolve(f, y(n), {y(0): 0}) == 2*n assert rsolve(f, y(n), {y(0): 1}) == 2*n + 1 assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = 3*y(n - 1) - y(n) - 1 assert rsolve(f, y(n), {y(0): 0}) == -3**n/2 + S.Half assert rsolve(f, y(n), {y(0): 1}) == 3**n/2 + S.Half assert rsolve(f, y(n), {y(0): 2}) == 3*3**n/2 + S.Half assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - 1/n*y(n - 1) assert rsolve(f, y(n)) == C0/factorial(n) assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - 1/n*y(n - 1) - 1 assert rsolve(f, y(n)) is None f = 2*y(n - 1) + (1 - n)*y(n)/n assert rsolve(f, y(n), {y(1): 1}) == 2**(n - 1)*n assert rsolve(f, y(n), {y(1): 2}) == 2**(n - 1)*n*2 assert rsolve(f, y(n), {y(1): 3}) == 2**(n - 1)*n*3 assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = (n - 1)*(n - 2)*y(n + 2) - (n + 1)*(n + 2)*y(n) assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n*(n - 1)*(n - 2) assert rsolve( f, y(n), {y(3): 6, y(4): -24}) == n*(n - 1)*(n - 2)*(-1)**(3 - n) assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
def test_rsolve(): f = y(n + 2) - y(n + 1) - y(n) h = sqrt(5)*(S.Half + S.Half*sqrt(5))**n \ - sqrt(5)*(S.Half - S.Half*sqrt(5))**n assert rsolve(f, y(n)) in [ C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n, C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n, ] assert rsolve(f, y(n), [0, 5]) == h assert rsolve(f, y(n), {0: 0, 1: 5}) == h assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n) g = C1*factorial(n) + C0*2**n h = -3*factorial(n) + 3*2**n assert rsolve(f, y(n)) == g assert rsolve(f, y(n), []) == g assert rsolve(f, y(n), {}) == g assert rsolve(f, y(n), [0, 3]) == h assert rsolve(f, y(n), {0: 0, 1: 3}) == h assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - y(n - 1) - 2 assert rsolve(f, y(n), {y(0): 0}) == 2*n assert rsolve(f, y(n), {y(0): 1}) == 2*n + 1 assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = 3*y(n - 1) - y(n) - 1 assert rsolve(f, y(n), {y(0): 0}) == -3**n/2 + S.Half assert rsolve(f, y(n), {y(0): 1}) == 3**n/2 + S.Half assert rsolve(f, y(n), {y(0): 2}) == 3*3**n/2 + S.Half assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - 1/n*y(n - 1) assert rsolve(f, y(n)) == C0/factorial(n) assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - 1/n*y(n - 1) - 1 assert rsolve(f, y(n)) is None f = 2*y(n - 1) + (1 - n)*y(n)/n assert rsolve(f, y(n), {y(1): 1}) == 2**(n - 1)*n assert rsolve(f, y(n), {y(1): 2}) == 2**(n - 1)*n*2 assert rsolve(f, y(n), {y(1): 3}) == 2**(n - 1)*n*3 assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = (n - 1)*(n - 2)*y(n + 2) - (n + 1)*(n + 2)*y(n) assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n*(n - 1)*(n - 2) assert rsolve( f, y(n), {y(3): 6, y(4): -24}) == -n*(n - 1)*(n - 2)*(-1)**(-n) assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 assert rsolve(Eq(y(n + 1), a*y(n)), y(n), {y(1): a}).simplify() == a**n assert rsolve(y(n) - a*y(n-2),y(n), \ {y(1): sqrt(a)*(a + b), y(2): a*(a - b)}).simplify() == \ a**(n/2)*((-1)**(n + 1)*b + a) f = (-16*n**2 + 32*n - 12)*y(n - 1) + (4*n**2 - 12*n + 9)*y(n) assert expand_func(rsolve(f, y(n), \ {y(1): binomial(2*n + 1, 3)}).rewrite(gamma)).simplify() == \ 4**n*n*(8*n**3 - 4*n**2 - 2*n + 1)/12 assert (rsolve(y(n) + a*(y(n + 1) + y(n - 1))/2, y(n)) - (C0*((sqrt(-a**2 + 1) - 1)/a)**n + C1*((-sqrt(-a**2 + 1) - 1)/a)**n)).simplify() == 0
def test_issue_3745(): f = y(n + 2) - y(n + 1) + y(n)/4 assert rsolve(f, y(n)) == 2**(-n)*(C0 + C1*n) assert rsolve(f, y(n), {y(0): 0, y(1): 1}) == 2*2**(-n)*n
def test_issue_15751(): f = y(n) + 21*y(n + 1) - 273*y(n + 2) - 1092*y(n + 3) + 1820*y(n + 4) + 1092*y(n + 5) - 273*y(n + 6) - 21*y(n + 7) + y(n + 8) assert rsolve(f, y(n)) is not None
def test_rsolve(): f = y(n + 2) - y(n + 1) - y(n) h = sqrt(5)*(S.Half + S.Half*sqrt(5))**n \ - sqrt(5)*(S.Half - S.Half*sqrt(5))**n assert rsolve(f, y(n)) in [ C0 * (S.Half - S.Half * sqrt(5))**n + C1 * (S.Half + S.Half * sqrt(5))**n, C1 * (S.Half - S.Half * sqrt(5))**n + C0 * (S.Half + S.Half * sqrt(5))**n, ] assert rsolve(f, y(n), [0, 5]) == h assert rsolve(f, y(n), {0: 0, 1: 5}) == h assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = (n - 1) * y(n + 2) - (n**2 + 3 * n - 2) * y(n + 1) + 2 * n * (n + 1) * y(n) g = C1 * factorial(n) + C0 * 2**n h = -3 * factorial(n) + 3 * 2**n assert rsolve(f, y(n)) == g assert rsolve(f, y(n), []) == g assert rsolve(f, y(n), {}) == g assert rsolve(f, y(n), [0, 3]) == h assert rsolve(f, y(n), {0: 0, 1: 3}) == h assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - y(n - 1) - 2 assert rsolve(f, y(n), {y(0): 0}) == 2 * n assert rsolve(f, y(n), {y(0): 1}) == 2 * n + 1 assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = 3 * y(n - 1) - y(n) - 1 assert rsolve(f, y(n), {y(0): 0}) == -3**n / 2 + S.Half assert rsolve(f, y(n), {y(0): 1}) == 3**n / 2 + S.Half assert rsolve(f, y(n), {y(0): 2}) == 3 * 3**n / 2 + S.Half assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - 1 / n * y(n - 1) assert rsolve(f, y(n)) == C0 / factorial(n) assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = y(n) - 1 / n * y(n - 1) - 1 assert rsolve(f, y(n)) is None f = 2 * y(n - 1) + (1 - n) * y(n) / n assert rsolve(f, y(n), {y(1): 1}) == 2**(n - 1) * n assert rsolve(f, y(n), {y(1): 2}) == 2**(n - 1) * n * 2 assert rsolve(f, y(n), {y(1): 3}) == 2**(n - 1) * n * 3 assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 f = (n - 1) * (n - 2) * y(n + 2) - (n + 1) * (n + 2) * y(n) assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n * (n - 1) * (n - 2) assert rsolve(f, y(n), { y(3): 6, y(4): -24 }) == -n * (n - 1) * (n - 2) * (-1)**(n) assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 assert rsolve(Eq(y(n + 1), a * y(n)), y(n), {y(1): a}).simplify() == a**n assert rsolve(y(n) - a*y(n-2),y(n), \ {y(1): sqrt(a)*(a + b), y(2): a*(a - b)}).simplify() == \ a**(n/2)*(-(-1)**n*b + a) f = (-16 * n**2 + 32 * n - 12) * y(n - 1) + (4 * n**2 - 12 * n + 9) * y(n) yn = rsolve(f, y(n), {y(1): binomial(2 * n + 1, 3)}) sol = 2**(2 * n) * n * (2 * n - 1)**2 * (2 * n + 1) / 12 assert factor(expand(yn, func=True)) == sol sol = rsolve(y(n) + a * (y(n + 1) + y(n - 1)) / 2, y(n)) Y = lambda i: sol.subs(n, i) assert (Y(n) + a * (Y(n + 1) + Y(n - 1)) / 2).expand().cancel() == 0 assert rsolve((k + 1) * y(k), y(k)) is None assert (rsolve((k + 1) * y(k) + (k + 3) * y(k + 1) + (k + 5) * y(k + 2), y(k)) is None) assert rsolve(y(n) + y(n + 1) + 2**n + 3**n, y(n)) == (-1)**n * C0 - 2**n / 3 - 3**n / 4