To solve the Cauchy problem for the heat equation using the integral formula, let's proceed systematically.

Problem:

The heat equation is given as: $u_t - u_{xx} = 0, \quad -\infty < x < \infty, \, t > 0$ with the initial condition: $u(x,0) = x^2, \quad -\infty < x < \infty.$

Integral formula for the heat equation solution:

The solution to the Cauchy problem is given by: $u(x,t) = \frac{1}{\sqrt{4\pi t}} \int_{-\infty}^\infty e^{-\frac{(x-y)^2}{4t}} u(y,0) \, dy.$

Substituting the initial condition $u(y,0) = y^2$, the formula becomes: $u(x,t) = \frac{1}{\sqrt{4\pi t}} \int_{-\infty}^\infty e^{-\frac{(x-y)^2}{4t}} y^2 \, dy.$

Solution:

To compute this integral, follow these steps:

1. Expand $(x-y)^2$ in the exponent: $(x-y)^2 = x^2 - 2xy + y^2.$ Substituting this into the exponential gives: $e^{-\frac{(x-y)^2}{4t}} = e^{-\frac{x^2}{4t}} e^{-\frac{-2xy}{4t}} e^{-\frac{y^2}{4t}} = e^{-\frac{x^2}{4t}} e^{\frac{xy}{2t}} e^{-\frac{y^2}{4t}}.$

2. Rewrite the integral: $u(x,t) = \frac{1}{\sqrt{4\pi t}} e^{-\frac{x^2}{4t}} \int_{-\infty}^\infty y^2 e^{\frac{xy}{2t}} e^{-\frac{y^2}{4t}} \, dy.$

3. Evaluate the integral using standard Gaussian integral results. For simplicity, define: $A = \int_{-\infty}^\infty y^2 e^{\frac{xy}{2t}} e^{-\frac{y^2}{4t}} \, dy.$ This can be solved using advanced techniques such as completing the square and properties of Gaussian integrals.

After solving, the result is: $u(x,t) = x^2 + 2t.$

Final Answer:

The solution to the given Cauchy problem is: $u(x,t) = x^2 + 2t.$

Would you like further details on any step? Here are some follow-up questions for deeper understanding:

1. How does the integral formula for the heat equation work in general?
2. Why does the Gaussian integral play a central role in the solution?
3. Can this solution method handle other types of initial conditions?
4. How does the solution $u(x,t) = x^2 + 2t$ reflect the heat diffusion process?
5. What are the key assumptions in the derivation of the integral formula?

Tip: When solving heat equations, symmetry and properties of Gaussian functions simplify many