def generator():
t = var("t")
s = var("s")
x = var("x")
y,yp,ypp = mi_vars("y","y'","y''")
d = dirac_delta
u = unit_step
def gen_delta():
"""
Generates a nice IVP involving delta(t-a)
"""
a = randrange(1,4)
b = randrange(1,4)
m = choice([-1,1])*randrange(1,5)
n=m
while n==m:
n = choice([-1,1])*randrange(1,5)
k=m*b^2
y_0=choice([-1,1])*randrange(1,6)
if randrange(0,2)==0:
y_0=0
else:
n=0
yp_0 = n*b
ode = shuffled_equation(ypp,b^2*y,-k*u(t-a))*randrange(2,4)
ly = (y_0*s+yp_0)/(s^2+b^2)+k*exp(-a*s)/(s*(s^2+b^2))
ly_simp = y_0*s/(s^2+b^2)+yp_0/(s^2+b^2)+m*exp(-a*s)/s-m*s*exp(-a*s)/(s^2+b^2)
partial_fractions = (1/(x^3+x*b^2)==(m/k)/x-(m*x/k)/(x^2+b^2))
sol = (y==y_0*cos(b*t)+n*sin(b*t)+m*u(t-a)-m*cos(b*(t-a))*u(t-a))
return {
"ode": ode,
"y0": y_0,
"yp0": yp_0,
"ly": ly,
"ly_simp": ly_simp,
"sol": sol,
"partial_fractions": partial_fractions
}
def gen_u():
"""
Generates a nice IVP involving u(t-a)
"""
a = randrange(1,4)
m = choice([-1,1])*randrange(1,5)
n=m
while n==m:
n = choice([-1,1])*randrange(1,5)
alpha = choice([-1,1])*randrange(1,5)
beta = choice([-1,1])*randrange(1,5)
k=alpha*(m-n)
yp_0 = beta*(m-n)
y_0=0
b=-m-n
c=m*n
ode = shuffled_equation(ypp,b*yp+c*y,-k*d(t-a))*randrange(2,4)
ly = (y_0*s+yp_0)/(s^2+b*s+c)+k*exp(-a*s)/(s^2+b*s+c)
ly_simp = beta/(s-m)-beta/(s-n)+exp(-a*s)*alpha/(s-m)-exp(-a*s)*alpha/(s-n)
partial_fractions = (1/(x^2+b*x+c)==(alpha/k)/(x-m)-(alpha/k)/(x-n))
sol = (y==beta*exp(m*t)-beta*exp(n*t)+alpha*exp(m*(t-a))*u(t-a)\
-alpha*exp(n*(t-a))*u(t-a))
return {
"ode": ode,
"y0": y_0,
"yp0": yp_0,
"ly": ly,
"ly_simp": ly_simp,
"sol": sol,
"partial_fractions": partial_fractions
}
return choice([gen_delta(),gen_u()])
