Illustrations of solutions of 2nd order linear DEs with constant coefficients¶
In [ ]:
from resources306 import *
x = sp.symbols('x')
e = sp.exp(1)
Cases of distinct real roots¶
In [3]:
y1 = e**(-17*x)
y2 = e**(-2*x)
c1,c2 = -3,2
y = c1*y1 + c2*y2
expressionplot( c1*y1 , x, 0,1,lw=2 )
expressionplot( c2*y2, x, 0,1,lw=2 )
expressionplot( c1*y1 + c2*y2, x, 0,1,lw=5 )
plt.title('$' + sp.latex(y) + '$',fontsize=20)
plt.grid()
In [4]:
y1 = e**(-17*x)
y2 = e**( 2*x)
c1,c2 = -3,2
y = c1*y1 + c2*y2
expressionplot( c1*y1 , x, 0,1,lw=2 )
expressionplot( c2*y2, x, 0,1,lw=2 )
expressionplot( c1*y1 + c2*y2, x, 0,1,lw=5 )
plt.title('$' + sp.latex(y) + '$',fontsize=20)
plt.grid()
Case of repeated root¶
In [5]:
y1 = e**(-2*x)
y2 = x*e**(-2*x)
c1,c2 = -1,2
y = c1*y1 + c2*y2
expressionplot( c1*y1 , x, 0,2,lw=2 )
expressionplot( c2*y2, x, 0,2,lw=2 )
expressionplot( c1*y1 + c2*y2, x, 0,2,lw=5 )
plt.title('$' + sp.latex(y) + '$',fontsize=20)
plt.grid()
Case of complex roots¶
In [11]:
alpha = -1/2
beta = 5
y1 = e**(alpha*x)*sp.cos(beta*x)
y2 = e**(alpha*x)*sp.sin(beta*x)
c1,c2 = 1,1
y = c1*y1 + c2*y2
expressionplot( c1*y1 , x, 0,5,lw=2 )
expressionplot( c2*y2, x, 0,5,lw=2 )
expressionplot( c1*y1 + c2*y2, x, 0,5,lw=5 )
plt.title('$' + sp.latex(y) + '$',fontsize=20)
plt.grid()
In [9]:
alpha = 1/5
beta = 20
y1 = e**(alpha*x)*sp.cos(beta*x)
y2 = e**(alpha*x)*sp.sin(beta*x)
c1,c2 = 1,1
y = c1*y1 + c2*y2
expressionplot( c1*y1 , x, 0,5,lw=2 )
expressionplot( c2*y2, x, 0,5,lw=2 )
expressionplot( c1*y1 + c2*y2, x, 0,5,lw=5 )
plt.title('$' + sp.latex(y) + '$',fontsize=20)
plt.grid()
In [ ]:
alpha = -1/5
beta = 1/20
y1 = e**(alpha*x)*sp.cos(beta*x)
y2 = e**(alpha*x)*sp.sin(beta*x)
c1,c2 = 1,1
y = c1*y1 + c2*y2
expressionplot( c1*y1 , x, 0,15,lw=2 )
expressionplot( c2*y2, x, 0,15,lw=2 )
expressionplot( c1*y1 + c2*y2, x, 0,15,lw=5 )
plt.title('$' + sp.latex(y) + '$',fontsize=20)
plt.grid()
In [ ]: