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\
[THOUGHT] Solve each first-order linear ODE: y' = f(x)y gives y(x)=y(1)*exp(∫_1^x f(t)dt). Then multiply y1y2y3, simplify, and evaluate the given limit as x→0+. Use Python/Sympy to confirm the asymptotics and the limit (likely +∞). [/THOUGHT] [PYTHON] import sympy as sp
x=sp.symbols('x', positive=True)
define solutions via integrals
t=sp.symbols('t', positive=True) y1 = 5sp.exp(sp.integrate(sp.sin(t)**2, (t,1,x))) y2 = 13sp.exp(sp.integrate(sp.cos(t)2, (t,1,x))) y3 = 35sp.Esp.exp(sp.integrate((2-t3)/t**3, (t,1,x)))
prod = sp.simplify(y1y2y3) expr = sp.simplify((prod + 989x)/(x**3sp.E*sp.sin(x)))
print("y1:", sp.simplify(y1)) print("y2:", sp.simplify(y2)) print("y3:", sp.simplify(y3)) print("product:", prod) print("limit:", sp.limit(expr, x, 0, dir='+')) [/PYTHON]
[THOUGHT] The computed limit is (+\infty), so it diverges. \boxed{\infty}