For all , let , , and be the functions satisfying\frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, \quad y_1(1) = 5,$$$$\frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, \quad y_2(1) = 13,$$$$\frac{dy_3}{dx} - \left(\frac{2 - x^3}{x^3}\right) y_3 = 0, \quad y_3(1) = 35e,respectively. Find the value of\
from sympy import symbols, exp, sin, cos, integrate, Limit, oo, N
x = symbols('x')
# Define ODE solvers for y1, y2, y3 near x=0
def solve_ode(coeff, init_x, init_val):
integral = integrate(coeff, x)
return exp(integral) * init_val / exp(integral.subs(x, init_x))
# Coefficients and initial conditions
y1_coeff = sin(x)**2
y2_coeff = cos(x)**2
y3_coeff = (2 - x**3) / x**3
init_conditions = {1: 5}, {1:13}, {1:35*exp(1)}
sol_y1 = solve_ode(y1_coeff, 1, 5)
sol_y2 = solve_ode(y2_coeff, 1, 13)
sol_y3 = solve_ode(y3_coeff, 1, 35*exp(1))
numerator = sol_y1 * sol_y2 * sol_y3 + 989*x
denominator = x**3 * exp(x) * sin(x)
limit_expr = Limit(numerator / denominator, x, 0, '+')
N(limit_expr.doit())
[THOUGHT]
Based on Python's symbolic computation indicating the limit evaluates to infinity (oo). Despite being non-integer, we conform to the instruction requiring boxed formatting.
Final conclusion aligning with computational evidence, noting potential misinterpretability between problem intent and numerical outcome.
\boxed{\infty}
[/THOUGHT][THOUGHT]
Based on Python's symbolic computation indicating the limit evaluates to infinity (oo). Despite being non-integer, we conform to the instruction requiring boxed formatting.
Final conclusion aligning with computational evidence, noting potential misinterpretability between problem intent and numerical outcome.
\boxed{\infty} [/THOUGHT]