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Tutorial

Voici un petit tuto qui contient:



%typeset_mode True

2+3
5\displaystyle 5
sin(x^2)
sin⁡(x2)\displaystyle \sin\left(x^{2}\right)

diff(1 + x + x^3, x)
3 x2+1\displaystyle 3 \, x^{2} + 1
x
x\displaystyle x
type(x)
<type 'sage.symbolic.expression.Expression'> 
isinstance(2, Expression)
False\displaystyle \mathrm{False}


SR(2)
2\displaystyle 2
SR(2).derivative(x)
0\displaystyle 0
type(x)
<type 'sage.symbolic.expression.Expression'> 
print(x.parent())
Symbolic Ring 
e^(I*pi)+1
0\displaystyle 0
e^(i*pi)+1
0\displaystyle 0
pi
π\displaystyle \pi
pi.numerical_approx(digits=1000)
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420199\displaystyle 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420199
n(pi)
3.14159265358979\displaystyle 3.14159265358979
N(pi)
3.14159265358979\displaystyle 3.14159265358979
a = [1, 3, -5]

a
[1\displaystyle 1, 3\displaystyle 3, −5\displaystyle -5]
a[1]
3\displaystyle 3
a = RR(pi)

type(a)
<type 'sage.rings.real_mpfr.RealNumber'> 
a
3.14159265358979\displaystyle 3.14159265358979
n(pi, digits=1000)
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420199\displaystyle 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420199
f = diff(sin(x^2), x)

f
2 xcos⁡(x2)\displaystyle 2 \, x \cos\left(x^{2}\right)
x = 1

f
2 xcos⁡(x2)\displaystyle 2 \, x \cos\left(x^{2}\right)
x
1\displaystyle 1
f
2 xcos⁡(x2)\displaystyle 2 \, x \cos\left(x^{2}\right)
type(x)
<type 'sage.rings.integer.Integer'> 
x = var("x")

x
x\displaystyle x
type(x)
<type 'sage.symbolic.expression.Expression'> 
f(x) = diff(sin(x^2), x)

f
x ↦ 2 xcos⁡(x2)\displaystyle x \ {\mapsto}\ 2 \, x \cos\left(x^{2}\right)
f(1)
2 cos⁡(1)\displaystyle 2 \, \cos\left(1\right)
g(x) = e^(x^2) ; g
x ↦ e(x2)\displaystyle x \ {\mapsto}\ e^{\left(x^{2}\right)}
def G(x, a=0):
    """
    Cette fonction calcule G
    """
    return e^(x^a)

G(x)
e\displaystyle e
G(x,43)
e(x43)\displaystyle e^{\left(x^{43}\right)}
diff(G(x,43), x)
43 x42e(x43)\displaystyle 43 \, x^{42} e^{\left(x^{43}\right)}
G?

File: /projects/d1f616f5-b5fa-499d-aa93-4337149d096f
Signature : G(x, a=0)
Docstring :
Cette fonction calcule G

g
x ↦ e(x2)\displaystyle x \ {\mapsto}\ e^{\left(x^{2}\right)}
integrate(g(x), x)
−12i πerf(i x)\displaystyle -\frac{1}{2} i \, \sqrt{\pi} \text{erf}\left(i \, x\right)
var('y')
y\displaystyle y
y
y\displaystyle y
x+y^2
y2+x\displaystyle y^{2} + x
s = x+y^2

s
y2+x\displaystyle y^{2} + x
diff(s, x, y)
0\displaystyle 0
xi = var('xi', latex_name=r'\xi')

xi
ξ\displaystyle {{\xi}}
s = sin(x*xi)*y^3

s
y3sin⁡(xξ)\displaystyle y^{3} \sin\left(x {{\xi}}\right)
s = 2

s
2\displaystyle 2
s = sin(x*xi)*y^3

s.subs(xi=3, y=2)
8 sin⁡(3 x)\displaystyle 8 \, \sin\left(3 \, x\right)
s
y3sin⁡(xξ)\displaystyle y^{3} \sin\left(x {{\xi}}\right)
a = s.subs({xi: 3, y: 2})

a
8 sin⁡(3 x)\displaystyle 8 \, \sin\left(3 \, x\right)
f(x,xi,y) = s

f
(x,ξ,y) ↦ y3sin⁡(xξ)\displaystyle \left( x, {{\xi}}, y \right) \ {\mapsto} \ y^{3} \sin\left(x {{\xi}}\right)
f(x,3,2)
8 sin⁡(3 x)\displaystyle 8 \, \sin\left(3 \, x\right)
f(4,3,2)
8 sin⁡(12)\displaystyle 8 \, \sin\left(12\right)
n(f(4,3,2), digits=50)
−4.2925833440034797733229938259392143385259082262243\displaystyle -4.2925833440034797733229938259392143385259082262243
s
y3sin⁡(xξ)\displaystyle y^{3} \sin\left(x {{\xi}}\right)
var('z')
z\displaystyle z
a = s.subs(y=1+z) ; a
(z+1)3sin⁡(xξ)\displaystyle {\left(z + 1\right)}^{3} \sin\left(x {{\xi}}\right)
a.expand()
z3sin⁡(xξ)+3 z2sin⁡(xξ)+3 zsin⁡(xξ)+sin⁡(xξ)\displaystyle z^{3} \sin\left(x {{\xi}}\right) + 3 \, z^{2} \sin\left(x {{\xi}}\right) + 3 \, z \sin\left(x {{\xi}}\right) + \sin\left(x {{\xi}}\right)
s = a.expand()

s.factor()
(z+1)3sin⁡(xξ)\displaystyle {\left(z + 1\right)}^{3} \sin\left(x {{\xi}}\right)
s
z3sin⁡(xξ)+3 z2sin⁡(xξ)+3 zsin⁡(xξ)+sin⁡(xξ)\displaystyle z^{3} \sin\left(x {{\xi}}\right) + 3 \, z^{2} \sin\left(x {{\xi}}\right) + 3 \, z \sin\left(x {{\xi}}\right) + \sin\left(x {{\xi}}\right)
s.simplify_full()
z3sin⁡(xξ)+3 z2sin⁡(xξ)+3 zsin⁡(xξ)+sin⁡(xξ)\displaystyle z^{3} \sin\left(x {{\xi}}\right) + 3 \, z^{2} \sin\left(x {{\xi}}\right) + 3 \, z \sin\left(x {{\xi}}\right) + \sin\left(x {{\xi}}\right)
s = (x+1)^2 - x^2 - 2*x - 1

s
(x+1)2−x2−2 x−1\displaystyle {\left(x + 1\right)}^{2} - x^{2} - 2 \, x - 1
s.simplify_full()
0\displaystyle 0


File: /projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/functions/orthogonal_polys.py
Signature : chebyshev_T(*args, **kwds)
Docstring :
Chebyshev polynomials of the first kind.

REFERENCE:

* [ASHandbook] 22.5.31 page 778 and 6.1.22 page 256.

EXAMPLES:

   sage: chebyshev_T(5,x)
   16*x^5 - 20*x^3 + 5*x
   sage: var('k')
   k
   sage: test = chebyshev_T(k,x)
   sage: test
   chebyshev_T(k, x)

g1 = plot(chebyshev_T(8,x), [x, -1, 1], axes_labels=[r'$\xi$', '$y$'],
          aspect_ratio=1, color='red', thickness=2, linestyle=':')

g1



g2 = plot(sin(x^3), (x,-1,1)) ; g2

g = g1 + g2

g

g.save("plot_g.png")

g = plot3d(sin(x*y^2), (x,-4,4), (y,-3,3), plot_points=100)

g.save("plot3d.png")
3D rendering not yet implemented
solve([x+y==6, x-y==4], x, y)
[[x=5\displaystyle x = 5, y=1\displaystyle y = 1]]
f(x) = x^2 + x - 1

solve(f(x)==0, x)[0]
x=−12 5−12\displaystyle x = -\frac{1}{2} \, \sqrt{5} - \frac{1}{2}
y = function('y')(x)

y
y(x)\displaystyle y\left(x\right)
eq = diff(y, x) + 3*x*y(x)^2

eq
3 xy(x)2+D[0](y)(x)\displaystyle 3 \, x y\left(x\right)^{2} + D[0]\left(y\right)\left(x\right)
desolve(eq == 0, y)
13 y(x)=12 x2+C\displaystyle \frac{1}{3 \, y\left(x\right)} = \frac{1}{2} \, x^{2} + C
desolve?

File: /projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/calculus/desolvers.py
Signature : desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False)
Docstring :
Solves a 1st or 2nd order linear ODE via maxima. Including IVP and
BVP.

*Use* "desolve? <tab>" *if the output in truncated in notebook.*

INPUT:

* "de" - an expression or equation representing the ODE

* "dvar" - the dependent variable (hereafter called "y")

* "ics" - (optional) the initial or boundary conditions

  * for a first-order equation, specify the initial "x" and "y"

  * for a second-order equation, specify the initial "x", "y",
    and "dy/dx", i.e. write [x_0, y(x_0), y'(x_0)]

  * for a second-order boundary solution, specify initial and
    final "x" and "y" boundary conditions, i.e. write [x_0, y(x_0),
    x_1, y(x_1)].

  * gives an error if the solution is not SymbolicEquation (as
    happens for example for a Clairaut equation)

* "ivar" - (optional) the independent variable (hereafter called
  x), which must be specified if there is more than one independent
  variable in the equation.

* "show_method" - (optional) if true, then Sage returns pair
  "[solution, method]", where method is the string describing the
  method which has been used to get a solution (Maxima uses the
  following order for first order equations: linear, separable,
  exact (including exact with integrating factor), homogeneous,
  bernoulli, generalized homogeneous) - use carefully in class, see
  below for the example of the equation which is separable but this
  property is not recognized by Maxima and the equation is solved
  as exact.

* "contrib_ode" - (optional) if true, desolve allows to solve
  Clairaut, Lagrange, Riccati and some other equations. This may
  take a long time and is thus turned off by default.  Initial
  conditions can be used only if the result is one SymbolicEquation
  (does not contain a singular solution, for example)

OUTPUT:

In most cases return a SymbolicEquation which defines the solution
implicitly.  If the result is in the form y(x)=... (happens for
linear eqs.), return the right-hand side only.  The possible
constant solutions of separable ODE's are omitted.

EXAMPLES:

   sage: x = var('x')
   sage: y = function('y')(x)
   sage: desolve(diff(y,x) + y - 1, y)
   (_C + e^x)*e^(-x)

   sage: f = desolve(diff(y,x) + y - 1, y, ics=[10,2]); f
   (e^10 + e^x)*e^(-x)

   sage: plot(f)
   Graphics object consisting of 1 graphics primitive

We can also solve second-order differential equations.:

   sage: x = var('x')
   sage: y = function('y')(x)
   sage: de = diff(y,x,2) - y == x
   sage: desolve(de, y)
   _K2*e^(-x) + _K1*e^x - x

   sage: f = desolve(de, y, [10,2,1]); f
   -x + 7*e^(x - 10) + 5*e^(-x + 10)

   sage: f(x=10)
   2

   sage: diff(f,x)(x=10)
   1

   sage: de = diff(y,x,2) + y == 0
   sage: desolve(de, y)
   _K2*cos(x) + _K1*sin(x)

   sage: desolve(de, y, [0,1,pi/2,4])
   cos(x) + 4*sin(x)

   sage: desolve(y*diff(y,x)+sin(x)==0,y)
   -1/2*y(x)^2 == _C - cos(x)

Clairaut equation: general and singular solutions:

   sage: desolve(diff(y,x)^2+x*diff(y,x)-y==0,y,contrib_ode=True,show_method=True)
   [[y(x) == _C^2 + _C*x, y(x) == -1/4*x^2], 'clairault']

For equations involving more variables we specify an independent
variable:

   sage: a,b,c,n=var('a b c n')
   sage: desolve(x^2*diff(y,x)==a+b*x^n+c*x^2*y^2,y,ivar=x,contrib_ode=True)
   [[y(x) == 0, (b*x^(n - 2) + a/x^2)*c^2*u == 0]]

   sage: desolve(x^2*diff(y,x)==a+b*x^n+c*x^2*y^2,y,ivar=x,contrib_ode=True,show_method=True)
   [[[y(x) == 0, (b*x^(n - 2) + a/x^2)*c^2*u == 0]], 'riccati']

Higher order equations, not involving independent variable:

   sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y).expand()
   1/6*y(x)^3 + _K1*y(x) == _K2 + x

   sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,1,3]).expand()
   1/6*y(x)^3 - 5/3*y(x) == x - 3/2

   sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,1,3],show_method=True)
   [1/6*y(x)^3 - 5/3*y(x) == x - 3/2, 'freeofx']

Separable equations - Sage returns solution in implicit form:

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

   sage: desolve(diff(y,x)*sin(y) == cos(x),y,show_method=True)
   [-cos(y(x)) == _C + sin(x), 'separable']

   sage: desolve(diff(y,x)*sin(y) == cos(x),y,[pi/2,1])
   -cos(y(x)) == -cos(1) + sin(x) - 1

Linear equation - Sage returns the expression on the right hand
side only:

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

   sage: desolve(diff(y,x)+(y) == cos(x),y,show_method=True)
   [1/2*((cos(x) + sin(x))*e^x + 2*_C)*e^(-x), 'linear']

   sage: desolve(diff(y,x)+(y) == cos(x),y,[0,1])
   1/2*(cos(x)*e^x + e^x*sin(x) + 1)*e^(-x)

This ODE with separated variables is solved as exact. Explanation -
factor does not split e^{x-y} in Maxima into e^{x}e^{y}:

   sage: desolve(diff(y,x)==exp(x-y),y,show_method=True)
   [-e^x + e^y(x) == _C, 'exact']

You can solve Bessel equations, also using initial conditions, but
you cannot put (sometimes desired) the initial condition at x=0,
since this point is a singular point of the equation. Anyway, if
the solution should be bounded at x=0, then _K2=0.:

   sage: desolve(x^2*diff(y,x,x)+x*diff(y,x)+(x^2-4)*y==0,y)
   _K1*bessel_J(2, x) + _K2*bessel_Y(2, x)

Example of difficult ODE producing an error:

   sage: desolve(sqrt(y)*diff(y,x)+e^(y)+cos(x)-sin(x+y)==0,y) # not tested
   Traceback (click to the left for traceback)
   ...
   NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True."

Another difficult ODE with error - moreover, it takes a long time

   sage: desolve(sqrt(y)*diff(y,x)+e^(y)+cos(x)-sin(x+y)==0,y,contrib_ode=True) # not tested

Some more types of ODE's:

   sage: desolve(x*diff(y,x)^2-(1+x*y)*diff(y,x)+y==0,y,contrib_ode=True,show_method=True)
   [[y(x) == _C*e^x, y(x) == _C + log(x)], 'factor']

   sage: desolve(diff(y,x)==(x+y)^2,y,contrib_ode=True,show_method=True)
   [[[x == _C - arctan(sqrt(t)), y(x) == -x - sqrt(t)], [x == _C + arctan(sqrt(t)), y(x) == -x + sqrt(t)]], 'lagrange']

These two examples produce an error (as expected, Maxima 5.18
cannot solve equations from initial conditions). Maxima 5.18
returns false answer in this case!:

   sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,2]).expand() # not tested
   Traceback (click to the left for traceback)
   ...
   NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True."

   sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,2],show_method=True) # not tested
   Traceback (click to the left for traceback)
   ...
   NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True."

Second order linear ODE:

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y)
   (_K2*x + _K1)*e^(-x) + 1/2*sin(x)

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,show_method=True)
   [(_K2*x + _K1)*e^(-x) + 1/2*sin(x), 'variationofparameters']

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,1])
   1/2*(7*x + 6)*e^(-x) + 1/2*sin(x)

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,1],show_method=True)
   [1/2*(7*x + 6)*e^(-x) + 1/2*sin(x), 'variationofparameters']

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,pi/2,2])
   3*(x*(e^(1/2*pi) - 2)/pi + 1)*e^(-x) + 1/2*sin(x)

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,pi/2,2],show_method=True)
   [3*(x*(e^(1/2*pi) - 2)/pi + 1)*e^(-x) + 1/2*sin(x), 'variationofparameters']

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y)
   (_K2*x + _K1)*e^(-x)

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,show_method=True)
   [(_K2*x + _K1)*e^(-x), 'constcoeff']

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,1])
   (4*x + 3)*e^(-x)

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,1],show_method=True)
   [(4*x + 3)*e^(-x), 'constcoeff']

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,pi/2,2])
   (2*x*(2*e^(1/2*pi) - 3)/pi + 3)*e^(-x)

   sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,pi/2,2],show_method=True)
   [(2*x*(2*e^(1/2*pi) - 3)/pi + 3)*e^(-x), 'constcoeff']

AUTHORS:

* David Joyner (1-2006)

* Robert Bradshaw (10-2008)

* Robert Marik (10-2009)