If $ x^4 \, dy + (4x^3 y + 2 \sin x) \, dx = 0 $ and $ f\left( \frac{\pi}{2} \right) = 0 $, then find the value of $ \pi^4 f\left( \frac{\pi}{3} \right) $ (where $ y = f(x) $):

Question

If $y = f(x)$ is the solution of the differential equation $(1 + \sin x) \frac{dy}{dx} + \cos x = 0$, such that $f(0) = 0$, then $f\left(\frac{\pi}{2}\right)$ is:

Answer 1

We are given the differential equation:

x⁴ dy + (4x³y + 2 sin x) dx = 0

First, rearrange the equation to express dy/dx:

x⁴ dy = −(4x³y + 2 sin x) dx

dy/dx = −(4x³y + 2 sin x) / x⁴

Simplifying:

dy/dx + (4/x) y = −(2 sin x) / x⁴

This is a first-order linear differential equation of the form:

dy/dx + P(x)y = Q(x)

where P(x) = 4/x.

Step 1: Find the integrating factor

IF = e^{∫(4/x) dx} = e^{4 ln x} = x⁴

Step 2: Multiply the equation by the integrating factor

x⁴ dy/dx + 4x³y = −2 sin x

The left side becomes the derivative of x⁴y:

d(x⁴y)/dx = −2 sin x

Step 3: Integrate both sides

x⁴y = ∫(−2 sin x) dx

x⁴y = 2 cos x + C

Step 4: Apply the initial condition

Given f(π/2) = 0, substitute x = π/2 and y = 0:

(π/2)⁴ · 0 = 2 cos(π/2) + C

0 = 0 + C

C = 0

Step 5: Write the final solution

x⁴y = 2 cos x

y = f(x) = 2 cos x / x⁴

Step 6: Evaluate π⁴ f(π/3)

f(π/3) = 2 cos(π/3) / (π/3)⁴

Since cos(π/3) = 1/2:

f(π/3) = 1 / (π/3)⁴

π⁴ f(π/3) = π⁴ / (π/3)⁴ = 3⁴

π⁴ f(π/3) = 81

Final Answer:

81

Answer 2