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] I'll solve each differential equation symbolically, form the product y1y2y3, then analyze the limit numerically and symbolically using Python to confirm its behavior as x→0⁺. [/THOUGHT] [PYTHON] import sympy as sp
x = sp.symbols('x', positive=True)
Define the ODE solutions:
y1': y1' = (sin x)^2 * y1
integrand1 = sp.sin(x)**2 I1 = sp.integrate(integrand1, (x, 1, x)) # integral from 1 to x y1 = 5*sp.exp(I1)
y2': y2' = (cos x)^2 * y2
integrand2 = sp.cos(x)**2 I2 = sp.integrate(integrand2, (x, 1, x)) y2 = 13*sp.exp(I2)
y3': y3' = ((2 - x^3)/x^3) * y3
Integrate (2/x^3 - 1)
I3 = sp.integrate(2/x**3 - 1, x) C3 = sp.symbols('C3') y3_general = C3*sp.exp(I3)
Apply y3(1) = 35e
C3_value = sp.solve(sp.Eq(y3_general.subs(x, 1), 35*sp.E), C3)[0] y3 = sp.simplify(y3_general.subs(C3, C3_value))
print("y1(x) =", sp.simplify(y1)) print("y2(x) =", sp.simplify(y2)) print("y3(x) =", sp.simplify(y3))
Simplify product y1y2y3
prod_y = sp.simplify(y1y2y3) print("Product y1y2y3 =", prod_y)
Define the expression for the limit
expr = (prod_y + 989*x)/(x**3 * sp.E * sp.sin(x)) print("Expression =", sp.simplify(expr))
Study asymptotics numerically for small x
for val in [0.5, 0.2, 0.1, 0.05, 0.02, 0.01]: num_val = expr.subs(x, val).evalf() print(f"x = {val}, expr ≈ {num_val}")
Also, get a formal series of the product term near 0+
(using series expansion for the dominating 989x term)
series_sin = sp.series(sp.sin(x), x, 0, 5) print("sin(x) series:", series_sin)
Check behavior of the product term alone vs denominator
prod_term = prod_y/(x**3 * sp.E * sp.sin(x)) print("Product term at small x:") for val in [0.5, 0.2, 0.1, 0.05, 0.02, 0.01]: num_val = prod_term.subs(x, val).evalf() print(f"x = {val}, prod_term ≈ {num_val}")
[THOUGHT] From the Python result, the expression simplifies to [ \frac{y_1(x)y_2(x)y_3(x) + 989x}{x^3 e \sin x} = \frac{(989x + 2275 e^{2 - 1/x^2})e^{-1}}{x^3 \sin x}. ] As (x \to 0^+), the term (e^{2 - 1/x^2} \to 0) super-exponentially, so the (2275 e^{2 - 1/x^2}) part is negligible compared to (989x).
Thus, asymptotically, [ \frac{(989x + 2275 e^{2 - 1/x^2})e^{-1}}{x^3 \sin x} \sim \frac{989x,e^{-1}}{x^3 \sin x}. ] Since (\sin x \sim x) as (x \to 0), the denominator behaves like (x^4), so the whole expression behaves like [ \frac{989x,e^{-1}}{x^4} = \frac{989}{e,x^3} \to +\infty \quad\text{as } x \to 0^+. ]
Therefore, the limit diverges to (+\infty). [/THOUGHT] \boxed{\infty}