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The problem you're dealing with involves the heat transfer by radiation governed by the Stefan-Boltzmann law, simplified to a differential equation:

\[
\frac{du}{dt} = -\alpha u^4
\]

where \( u(t) \) is the temperature of the body as a function of time, and \( \alpha \) is a constant given as \( 6 \times 10^{-12} \, \text{K}^{-3}/\text{s} \). The initial condition is \( u(0) = 2500 \, \text{K} \), and we are tasked with solving this equation for \( u(t) \).

### Steps to solve the equation:

1. We have a separable differential equation:

\[
\frac{du}{dt} = -\alpha u^4
\]

2. Rearrange it to:

\[
\frac{du}{u^4} = -\alpha dt
\]

3. Integrate both sides:

\[
\int u^{-4} du = -\alpha \int dt
\]

The integral of \( u^{-4} \) is:

\[
\frac{u^{-3}}{-3} = -\alpha t + C
\]

4. Multiply both sides by -3 to simplify:

\[
u^{-3} = 3\alpha t + C'
\]

5. Solve for \( u(t) \):

\[
u(t) = \left( \frac{1}{3\alpha t + C'} \right)^{1/3}
\]

6. Use the initial condition \( u(0) = 2500 \, \text{K} \) to find \( C' \):

\[
2500^{-3} = C'
\]

Now I'll calculate the final expression for \( u(t) \).The constant \( C' \) is \( 6.4 \times 10^{-11} \).

Thus, the equation for the temperature of the body as a function of time becomes:

\[
u(t) = \left( \frac{1}{3\alpha t + 6.4 \times 10^{-11}} \right)^{1/3}
\]

Substitute \( \alpha = 6 \times 10^{-12} \, \text{K}^{-3}/\text{s} \) to get the complete expression:

\[
u(t) = \left( \frac{1}{18 \times 10^{-12} t + 6.4 \times 10^{-11}} \right)^{1/3}
\]

This is the solution for the temperature \( u(t) \) of the body over time.

Would you like further clarifications or additional steps? Here are some related questions:

1. How do we verify the solution satisfies the initial condition?
2. What does the behavior of \( u(t) \) look like for large \( t \)?
3. How does the solution change if the surrounding temperature is not negligible?
4. Can this method be extended to solve equation (1) directly?
5. What happens if the body has a different initial temperature?

**Tip:** Separable differential equations, like the one above, are solved by integrating both sides after rearranging them into appropriate functions of each variable.

Suppose that a body with initial temperature 2500 K is surrounded by a medium with temperature 270 K and that α = 6 × 10^−12 K^−3/s. Determine the temperature of the body at any time by solving the differential equation du/dt = -αu^4.

Learn how to solve the differential equation du/dt = -αu^4, a heat transfer problem based on the Stefan-Boltzmann law, with an initial temperature of 2500 K and surrounding temperature of 270 K. This solution involves separable differential equations and the Stefan-Boltzmann constant α = 6 × 10^−12 K^−3/s. Suitable for university-level students studying physics or engineering.

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