Coverage for local/lib/python2.7/site-packages/sage/tests/french_book/recequadiff.py : 0%

## -*- encoding: utf-8 -*- 

""" 

This file (./recequadiff_doctest.sage) was *autogenerated* from ./recequadiff.tex, 

with sagetex.sty version 2011/05/27 v2.3.1. 

It contains the contents of all the sageexample environments from this file. 

You should be able to doctest this file with: 

sage -t ./recequadiff_doctest.sage 

It is always safe to delete this file; it is not used in typesetting your 

document. 

Sage example in ./recequadiff.tex, line 110:: 

sage: x = var('x') 

sage: y = function('y')(x) 

sage: _C = SR.var("_C") 

sage: _K1 = SR.var("_K1") 

sage: _K2 = SR.var("_K2") 

Sage example in ./recequadiff.tex, line 179:: 

sage: x = var('x'); y = function('y')(x) 

Sage example in ./recequadiff.tex, line 182:: 

sage: desolve(diff(y,x) + 3*y == exp(x), y, show_method=True) 

[1/4*(4*_C + e^(4*x))*e^(-3*x), 'linear'] 

Sage example in ./recequadiff.tex, line 194:: 

sage: desolve(y*diff(y,x) == x, y, show_method=True) 

[1/2*y(x)^2 == 1/2*x^2 + _C, 'separable'] 

Sage example in ./recequadiff.tex, line 204:: 

sage: desolve(diff(y,x) == exp(x+y), y, show_method=True) 

[-(e^(x + y(x)) + 1)*e^(-y(x)) == _C, 'exact'] 

Sage example in ./recequadiff.tex, line 215:: 

sage: desolve(diff(y,x)-y == x*y^4, y, show_method=True) 

[e^x/(-1/3*(3*x - 1)*e^(3*x) + _C)^(1/3), 'bernoulli'] 

Sage example in ./recequadiff.tex, line 227:: 

sage: desolve(x^2*diff(y,x) == y^2+x*y+x^2, y, show_method=True) 

[_C*x == e^(arctan(y(x)/x)), 'homogeneous'] 

Sage example in ./recequadiff.tex, line 244:: 

sage: desolve(diff(y,x) == (cos(y)-2*x)/(y+x*sin(y)), y, 

....: show_method=True) 

[x^2 - x*cos(y(x)) + 1/2*y(x)^2 == _C, 'exact'] 

Sage example in ./recequadiff.tex, line 263:: 

sage: desolve(diff(y,x) == x*y^2+y/x-1/x^2, y, 

....: contrib_ode=True, show_method=True)[1] 

'riccati' 

Sage example in ./recequadiff.tex, line 279:: 

sage: desolve(y == x*diff(y,x)-diff(y,x)^2, y, 

....: contrib_ode=True, show_method=True) 

[[y(x) == -_C^2 + _C*x, y(x) == 1/4*x^2], 'clairault'] 

Sage example in ./recequadiff.tex, line 293:: 

sage: x = var('x'); y = function('y')(x) 

Sage example in ./recequadiff.tex, line 297:: 

sage: DE = diff(y,x)+2*y == x**2-2*x+3 

sage: desolve(DE, y) 

1/4*((2*x^2 - 2*x + 1)*e^(2*x) - 2*(2*x - 1)*e^(2*x) + 4*_C 

+ 6*e^(2*x))*e^(-2*x) 

Sage example in ./recequadiff.tex, line 305:: 

sage: desolve(DE, y).expand() 

1/2*x^2 + _C*e^(-2*x) - 3/2*x + 9/4 

Sage example in ./recequadiff.tex, line 321:: 

sage: desolve(DE, y, show_method=True)[1] 

'linear' 

Sage example in ./recequadiff.tex, line 327:: 

sage: desolve(DE, y, ics=[0,1]).expand() 

1/2*x^2 - 3/2*x - 5/4*e^(-2*x) + 9/4 

Sage example in ./recequadiff.tex, line 338:: 

sage: x = var('x'); y = function('y')(x) 

sage: desolve(diff(y,x)*log(y) == y*sin(x), y, show_method=True) 

[1/2*log(y(x))^2 == _C - cos(x), 'separable'] 

Sage example in ./recequadiff.tex, line 348:: 

sage: ed(x) = desolve(diff(y,x)*log(y) == y*sin(x), y); ed(x) 

1/2*log(y(x))^2 == _C - cos(x) 

Sage example in ./recequadiff.tex, line 356:: 

sage: solve(ed, y) 

[y(x) == e^(-sqrt(2*_C - 2*cos(x))), y(x) == e^(sqrt(2*_C - 2*cos(x)))] 

Sage example in ./recequadiff.tex, line 367:: 

sage: solve(ed, y)[0].subs(_C == 5).rhs() 

e^(-sqrt(-2*cos(x) + 10)) 

Sage example in ./recequadiff.tex, line 377:: 

sage: ed.variables() 

(_C, x) 

Sage example in ./recequadiff.tex, line 384:: 

sage: c = ed.variables()[0] 

sage: solve(ed, y)[0].subs(c == 5).rhs() 

e^(-sqrt(-2*cos(x) + 10)) 

Sage example in ./recequadiff.tex, line 396:: 

sage: plot(solve(ed, y)[0].subs(c == 2).rhs(), x, -3, 3) 

Graphics object consisting of 1 graphics primitive 

Sage example in ./recequadiff.tex, line 408:: 

sage: P = Graphics() 

sage: for k in range(1,20,2): 

....: P += plot(solve(ed, y)[0].subs(c == 1+k/4).rhs(), x, -3, 3) 

sage: P 

Graphics object consisting of 1... graphics primitives 

Sage example in ./recequadiff.tex, line 426:: 

sage: P = Graphics() 

sage: for j in [0,1]: 

....: for k in range(1,10,2): 

....: f = solve(ed,y)[j].subs(c == 2+0.25*k).rhs() 

....: P += plot(f, x, -3, 3) 

sage: P 

Graphics object consisting of 10 graphics primitives 

Sage example in ./recequadiff.tex, line 472:: 

sage: u = function('u')(x) 

sage: y = x*u 

sage: DE = x*diff(y,x) == y + sqrt(x**2 + y**2) 

Sage example in ./recequadiff.tex, line 484:: 

sage: forget() 

Sage example in ./recequadiff.tex, line 488:: 

sage: assume(x>0) 

sage: desolve(DE, u) 

x == _C*e^arcsinh(u(x)) 

Sage example in ./recequadiff.tex, line 505:: 

sage: S = desolve(DE,u)._maxima_().ev(logarc=True).sage().solve(u); S 

[u(x) == -(sqrt(u(x)^2 + 1)*_C - x)/_C] 

Sage example in ./recequadiff.tex, line 519:: 

sage: solu = (x-S[0]*c)^2; solu 

(_C*u(x) - x)^2 == (u(x)^2 + 1)*_C^2 

sage: sol = solu.solve(u); sol 

[u(x) == -1/2*(_C^2 - x^2)/(_C*x)] 

Sage example in ./recequadiff.tex, line 526:: 

sage: y(x) = x*sol[0].rhs(); y(x) 

-1/2*(_C^2 - x^2)/_C 

Sage example in ./recequadiff.tex, line 535:: 

sage: P = Graphics() 

sage: for k in range(-19,19,2): 

....: P += plot(y(x).subs(c == 1/k), x, 0, 3) 

sage: P 

Graphics object consisting of 19 graphics primitives 

Sage example in ./recequadiff.tex, line 567:: 

sage: x = var('x'); y = function('y')(x); a, b = var('a, b') 

sage: DE = diff(y,x) - a*y == -b*y**2 

sage: sol(x) = desolve(DE,[y,x]); sol(x) 

-(log(b*y(x) - a) - log(y(x)))/a == _C + x 

Sage example in ./recequadiff.tex, line 575:: 

sage: Sol(x) = solve(sol, y)[0]; Sol(x) 

log(y(x)) == _C*a + a*x + log(b*y(x) - a) 

Sage example in ./recequadiff.tex, line 582:: 

sage: Sol(x) = Sol(x).lhs()-Sol(x).rhs(); Sol(x) 

-_C*a - a*x - log(b*y(x) - a) + log(y(x)) 

sage: Sol = Sol.simplify_log(); Sol(x) 

-_C*a - a*x + log(y(x)/(b*y(x) - a)) 

sage: solve(Sol, y)[0].simplify() 

y(x) == a*e^(_C*a + a*x)/(b*e^(_C*a + a*x) - 1) 

Sage example in ./recequadiff.tex, line 602:: 

sage: x = var('x'); y = function('y')(x) 

sage: DE = diff(y,x,2)+3*y == x^2-7*x+31 

sage: desolve(DE, y).expand() 

1/3*x^2 + _K2*cos(sqrt(3)*x) + _K1*sin(sqrt(3)*x) - 7/3*x + 91/9 

Sage example in ./recequadiff.tex, line 611:: 

sage: desolve(DE, y, ics=[0,1,2]).expand() 

1/3*x^2 + 13/9*sqrt(3)*sin(sqrt(3)*x) - 7/3*x 

- 82/9*cos(sqrt(3)*x) + 91/9 

Sage example in ./recequadiff.tex, line 621:: 

sage: desolve(DE, y, ics=[0,1,-1,0]).expand() 

1/3*x^2 - 7/3*x - 82/9*cos(sqrt(3))*sin(sqrt(3)*x)/sin(sqrt(3)) 

+ 115/9*sin(sqrt(3)*x)/sin(sqrt(3)) - 82/9*cos(sqrt(3)*x) + 91/9 

Sage example in ./recequadiff.tex, line 674:: 

sage: x, t = var('x, t'); f = function('f')(x); g = function('g')(t) 

sage: z = f*g 

sage: eq(x,t) = diff(z,x,2) == diff(z,t); eq(x,t) 

g(t)*diff(f(x), x, x) == f(x)*diff(g(t), t) 

Sage example in ./recequadiff.tex, line 688:: 

sage: eqn = eq/z; eqn(x,t) 

diff(f(x), x, x)/f(x) == diff(g(t), t)/g(t) 

Sage example in ./recequadiff.tex, line 702:: 

sage: k = var('k') 

sage: eq1(x,t) = eqn(x,t).lhs() == k; eq2(x,t) = eqn(x,t).rhs() == k 

Sage example in ./recequadiff.tex, line 709:: 

sage: g(t) = desolve(eq2(x,t),[g,t]); g(t) 

_C*e^(k*t) 

Sage example in ./recequadiff.tex, line 717:: 

sage: desolve(eq1,[f,x]) 

Traceback (most recent call last): 

... 

TypeError: Computation failed ... 

Is k positive, negative or zero? 

Sage example in ./recequadiff.tex, line 728:: 

sage: assume(k>0); desolve(eq1,[f,x]) 

_K1*e^(sqrt(k)*x) + _K2*e^(-sqrt(k)*x) 

Sage example in ./recequadiff.tex, line 782:: 

sage: x, s = var('x, s'); f = function('f')(x) 

sage: f(x) = sin(x); f.laplace(x,s) 

x |--> 1/(s^2 + 1) 

Sage example in ./recequadiff.tex, line 795:: 

sage: X(s) = 1/(s^2-3*s-4)/(s^2+1) + (s-4)/(s^2-3*s-4) 

sage: X(s).inverse_laplace(s, x) 

3/34*cos(x) + 1/85*e^(4*x) + 9/10*e^(-x) - 5/34*sin(x) 

Sage example in ./recequadiff.tex, line 807:: 

sage: X(s).partial_fraction() 

1/34*(3*s - 5)/(s^2 + 1) + 9/10/(s + 1) + 1/85/(s - 4) 

Sage example in ./recequadiff.tex, line 818:: 

sage: x = var('x'); y = function('y')(x) 

sage: eq = diff(y,x,x) - 3*diff(y,x) - 4*y - sin(x) == 0 

sage: desolve_laplace(eq, y) 

1/85*(17*y(0) + 17*D[0](y)(0) + 1)*e^(4*x) + 1/10*(8*y(0) 

- 2*D[0](y)(0) - 1)*e^(-x) + 3/34*cos(x) - 5/34*sin(x) 

sage: desolve_laplace(eq, y, ics=[0,1,-1]) 

3/34*cos(x) + 1/85*e^(4*x) + 9/10*e^(-x) - 5/34*sin(x) 

Sage example in ./recequadiff.tex, line 869:: 

sage: x = var('x'); y1 = function('y1')(x) 

sage: y2 = function('y2')(x); y3 = function('y3')(x) 

sage: y = vector([y1, y2, y3]) 

sage: A = matrix([[2,-2,0],[-2,0,2],[0,2,2]]) 

sage: system = [diff(y[i], x) - (A * y)[i] for i in range(3)] 

sage: desolve_system(system, [y1, y2, y3], ics=[0,2,1,-2]) 

[y1(x) == e^(4*x) + e^(-2*x), 

y2(x) == -e^(4*x) + 2*e^(-2*x), 

y3(x) == -e^(4*x) - e^(-2*x)] 

Sage example in ./recequadiff.tex, line 913:: 

sage: x = var('x'); y1 = function('y1')(x); y2 = function('y2')(x) 

sage: y = vector([y1,y2]) 

sage: A = matrix([[3,-4],[1,3]]) 

sage: system = [diff(y[i], x) - (A * y)[i] for i in range(2)] 

sage: desolve_system(system, [y1, y2], ics=[0,2,0]) 

[y1(x) == 2*cos(2*x)*e^(3*x), y2(x) == e^(3*x)*sin(2*x)] 

Sage example in ./recequadiff.tex, line 966:: 

sage: x = var('x'); u1 = function('u1')(x); u2 = function('u2')(x) 

sage: u3 = function('u3')(x); u4 = function('u4')(x) 

sage: u = vector([u1,u2,u3,u4]) 

sage: A = matrix([[0,0,1,0],[0,0,0,1],[2,-6,1,3],[-2,6,1,-1]]) 

sage: system = [diff(u[i], x) - (A*u)[i] for i in range(4)] 

sage: sol = desolve_system(system, [u1, u2, u3, u4]) 

Sage example in ./recequadiff.tex, line 977:: 

sage: sol[0] 

u1(x) == 1/12*(2*u1(0) - 6*u2(0) + 5*u3(0) + 3*u4(0))*e^(2*x) 

+ 1/24*(2*u1(0) - 6*u2(0) - u3(0) + 3*u4(0))*e^(-4*x) 

+ 3/4*u1(0) + 3/4*u2(0) - 3/8*u3(0) - 3/8*u4(0) 

sage: sol[1] 

u2(x) == -1/12*(2*u1(0) - 6*u2(0) - u3(0) - 3*u4(0))*e^(2*x) 

- 1/24*(2*u1(0) - 6*u2(0) - u3(0) + 3*u4(0))*e^(-4*x) 

+ 1/4*u1(0) + 1/4*u2(0) - 1/8*u3(0) - 1/8*u4(0) 

Sage example in ./recequadiff.tex, line 1095:: 

sage: x = var('x'); f = function('f')(x) 

sage: f(x) = 3.83*x*(1 - x/100000) 

sage: def u(n): 

....: if n==0: return(20000) 

....: else: return f(u(n-1)) 

Sage example in ./recequadiff.tex, line 1105:: 

sage: def v(n): 

....: V = 20000; 

....: for k in [1..n]: 

....: V = f(V) 

....: return V 

Sage example in ./recequadiff.tex, line 1118:: 

sage: def nuage(u,n): 

....: L = [[0,u(0)]]; 

....: for k in [1..n]: 

....: L += [[k,u(k)]] 

....: points(L).show() 

Sage example in ./recequadiff.tex, line 1128:: 

sage: nuage(u,50) 

Sage example in ./recequadiff.tex, line 1144:: 

sage: def escargot(f,x,u0,n,xmin,xmax): 

....: u = u0 

....: P = plot(x, x, xmin, xmax, color='gray') 

....: for i in range(n): 

....: P += line([[u,u],[u,f(u)],[f(u),f(u)]], color = 'red') 

....: u = f(u) 

....: P += f.plot(x, xmin, xmax, color='blue') # Courbe de f 

....: P.show() 

Sage example in ./recequadiff.tex, line 1157:: 

sage: f(x) = 3.83*x*(1 - x/100000) 

sage: escargot(f,x,20000,100,0,100000) 

Sage example in ./recequadiff.tex, line 1197:: 

sage: from sympy import Function, Symbol 

sage: u = Function('u'); n = Symbol('n', integer=True) 

Sage example in ./recequadiff.tex, line 1208 (WARNING: the order of factors is 

inverted, see :trac:`23496` ):: 

sage: f = u(n+2)-u(n+1)*(3/2)+u(n)*(1/2) 

Sage example in ./recequadiff.tex, line 1214:: 

sage: from sympy import rsolve 

sage: rsolve(f, u(n), {u(0):-1,u(1):1}) 

3 - 4*2**(-n) 

Sage example in ./recequadiff.tex, line 1265:: 

sage: from sympy import rsolve_hyper 

sage: n = Symbol('n', integer=True) 

sage: rsolve_hyper([-2,1],2**(n+2),n) # known bug (Trac #24334) 

2**n*C0 + 2**(n + 2)*(C0 + n/2) 

"""