[한빛아카데미] 최신 공학수학 with Sage 실습실

예제 2-1

Sage 코딩

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var('x')             # 변수지정
y = function('y')(x) # y는 x의 함수
de1 = desolve(diff(y, x) == -x/y, y)
print de1
de2 = desolve(diff(y, x) == x*y^(1/2), y)
print de2

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예제 2-4

Sage 코딩

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x = var('x')
y = function('y')(x)
de = lambda y : diff(y, x) + y - 1
desolve (de(y), [y,x])

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예제 2-5

Sage 코딩

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x = var('x')
y = function('y')(x)
de = desolve (diff(y, x)-x*y^(1/2), y)
de

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Sage 코딩

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x = var('x')
y = function('y')(x)
de = desolve (diff(y, x) - y/x , y)
de

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예제 3-1

Sage 코딩

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x, r = var('x, r')
y = function('y')(x)
f = x^2 + y^2 - r^2
df = diff(f, x)
de=desolve(df, y)
print (de)
​
var('x, y')
@interact
def _(c = slider(-10, -1, 0.2, 1)):
    show(implicit_plot(-1/2*y^2 == 1/2*x^2 + c, (x, -5, 5), (y, -5, 5)))

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예제 3-2

Sage 코딩

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x, a = var('x, a')
y = function('y')(x)
f = (x^2)/(4*a^2) + (y^2)/(a^2) - 1
df = diff(f, x)
print (df)
​
de = desolve(2*y*diff(y,x) + 1/2*x, y)
print (de)
​
var('x, y')
@interact
def _(c = slider(-8, -1, 0.2, 1)):
    show(implicit_plot(-2*y^2 == 1/2*x^2 + c, (x, -5, 5), (y, -5, 5)))

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예제 3-3

Sage 코딩

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x, a = var('x, a')
y = function('y')(x)
f = (x^2)/(4*a^2) - (y^2)/(a^2) - 1
df = diff(f, x)
print (df)
​
de = desolve(-2*y*diff(y,x) + 1/2*x, y)
print (de)
​
var('x, y')
@interact
def _(c = slider(-8, -1, 0.2, 1)):
    show(implicit_plot(2*y^2 == 1/2*x^2 + c, (x, -5, 5), (y, -5, 5)))

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예제 3-4

Sage 코딩

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x,y = var("x,y")
f(x,y) = x*y
P1 = plot_slope_field(f(x,y), (x,0, 5),(y,0, 5))
x,y = PolynomialRing(RR,2,"xy").gens()
f_xy = x*y
pts = eulers_method(f_xy, 1, 2, 0.1, 4, algorithm="none")
P2 = line(pts, xmin=0,xmax=5,ymin=0,ymax=5, thickness=2)
show(P1+P2, aspect_ratio=1)

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예제 4-2

Sage 코딩

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x = var('x')
y = function('y')(x)
de = lambda y: diff(y,x) + y
soln = desolve(de(y),[y,x],[0,2])
P = plot(soln,0,5)
print (soln) 
soln; show(P, xmin=0)

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예제 4-3

Sage 코딩

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x = var('x')
y = function('y')(x)
soln = desolve(diff(y, x, 2) + 2*y, [y, x], [0, 0, 1])
P = plot(soln,0,5)
print (soln)
soln; show(P, ymin = -0.7)

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예제 4-4

Sage 코딩

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x = var('x')
y = function('y')(x)
de = desolve (diff(y, x, 2) - y, [y, x], [0, 1, 0])
print (de.simplify())

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Sage 코딩

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x = var('x')
y = function('y')(x) 
de = desolve( diff(y, x, 2) + y, y),
show (de)

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Sage 코딩

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x = var('x')
y = function('y')(x) 
de = desolve( diff(y, x, 2) + diff(y, x), y), 
show (de)

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Sage 코딩

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var('x')
y = function('y')(x)
de = desolve(diff(y, x, 2) +10*diff(y,x)+ 25*y==0, [y, x])
show (de)

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Sage 코딩

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var('x')
y = function('y')(x)
de = desolve(diff(y, x, 2) +9*y==0, [y, x])
show (de)

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Sage 코딩

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x = var('x')
y = function('y')(x) 
de = desolve(diff(y, x) + x/y, y) 
show (de)

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Sage 코딩

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x = var('x')
y = function('y')(x) 
de = desolve( diff(y, x) + y == sin(x), y)
show (de)

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Sage 코딩

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x = var('x')
y = function('y')(x)
de1 = desolve( diff(y, x, 2) + y, [y,x])
de2 = desolve( diff(y, x, 2) + y, [y,x], [0,0,0])
show (expand(de1))
show (expand(de2))

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Sage 코딩

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x = var('x')
y = function('y')(x)
de1 = desolve( diff(y, x) + x/y, [y,x])
de2 = desolve( diff(y, x) + x/y, [y,x], ics=[0,2])
show (expand(de1))
show (expand(de2))

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Sage 코딩

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x = var('x')
y = function('y')(x)
de1 = desolve(diff(y, x)+y-sin(x), [y,x])
de2 = desolve(diff(y, x)+y-sin(x), [y,x], ics=[0,1/2])
show (expand(de1))
show (expand(de2))

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Sage 코딩

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var('x')
y = function('y')(x)
de1 = desolve(diff(y, x, 1) + y== e^x, [y, x])
de2 = desolve(diff(y, x, 1) + y== e^x, [y, x], ics= [0,0])
print de1
print de2

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Sage 코딩

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x, c = var('x, c')
y = function('y')(x)
f = y^2 + x + c
df = diff(f, x)
print (df)

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Sage 코딩

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x, b = var('x, b')
y = function('y')(x)
f = y - x - b*x^3
df = diff(f, x)
print (df)

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Sage 코딩

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x, c = var('x, c')
y = function('y')(x)
f = (x - c)^2 + y^2 -c
df = diff(f, x)
print (df)

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Sage 코딩

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x, a = var('x, a')
y = function('y')(x)
f = (1/a^2)*x^2 + (1/4*a^2)*y^2 - 1
df = diff(f, x)
print (df)

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Sage 코딩

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x, a = var('x, a')
y = function('y')(x)
f = (1/a^2)*x^2 - (1/4*a^2)*y^2 + 1
df = diff(f, x)
print (df)

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Sage 코딩

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x,y = var("x,y")
f(x,y) = 1 - x*y
P1 = plot_slope_field(f(x,y), (x,0, 5),(y,0, 5))
x,y = PolynomialRing(RR,2,"xy").gens()
f_xy = 1 - x*y
pts = eulers_method(f_xy,2,2,0.1,4, algorithm="none")
P2 = line(pts,xmin=0,xmax=5,ymin=0,ymax=5,thickness=2)
show(P1+P2, aspect_ratio=1)

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Sage 코딩

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x,y = var("x,y")
f(x,y) = exp(-x*y^2)
P1 = plot_slope_field(f(x,y),(x,-3, 3),(y, -3, 3))
x,y = PolynomialRing(RR,2,"xy").gens()
f_xy = exp(-x*y^2)
pts = eulers_method(f_xy,0,1,0.1,4,algorithm="none")
P2 = line(pts,xmin=-3,xmax=3,ymin=-3,ymax=3,thickness=2)
show(P1+P2, aspect_ratio=1)

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Sage 코딩

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x = var('x')
y = function('y')(x)
de = desolve (diff(y, x, 2) + y, [y, x], [0, -1, -2] )
print (de.full_simplify())

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Sage 코딩

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x = var('x')
y = function('y')(x)
de = desolve (diff(y, x, 2) + 4*y, [y, x], [pi/4, 0, 1] )
print (de.full_simplify())

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Sage 코딩

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x = var('x')
y = function('y')(x)
de = desolve (diff(y, x, 2) - y, [y, x], [0, 1, 2] )
print (de.simplify())

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Sage 코딩

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x = var('x')
y = function('y')(x)
de = desolve (diff(y,x) - y + y^2, [y, x], [0,1/3] )
print (de.full_simplify())

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Sage 코딩

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x = var('x')
y = function('y')(x)
de = desolve (diff(y,x) + 2*x*y^2, [y, x], [0,1] )
show (de.full_simplify())

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Sage 코딩

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var('x')
y = function('y')(x)
de = desolve(diff(y, x, 1) + 1 + y^2 == 0, [y, x], ics= [2, pi/4])
de.full_simplify()

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Chapter 02 미분방정식의 기초

예제 2-1

예제 2-4

예제 2-5

예제 3-1

예제 3-2

예제 3-3

예제 3-4

예제 4-2

예제 4-3

예제 4-4

연습문제