Solution to a PDE (heat equation) with one initial condition

[frenchkiki]

Homework Statement

By trial and error, find a solution of the diffusion equation du/dt = d^2u / dx^2 with
the initial condition u(x, 0) = x^2.

Homework Equations

The Attempt at a Solution

Given the initial condition, I tried finding a solution at the steady state (du/dt=0) which gives me d^2u / dx^2 = 0, from I found u(x,0)=c_1 + c_2 x = x^2 and now I am stuck. Thanks in advance!

 

[Dustinsfl]

$u_t = u_{xx}$ with $u(x,0) = x^2$.

The basic solution is of the form $u(x,t) = \varphi(x)\psi(t)$.
You would use the method of separation of variables.
$$
\begin{cases}
\varphi = A\cos\lambda x + B\frac{\sin\lambda x}{\lambda}\\
\psi = C\exp(-\lambda^2t)
\end{cases}
$$
The above is basic and nothing ground breaking.
Right now your solution is of the form
$$
a_0 + \sum_{n = 1}^{\infty}\left[\left(A_n\cos\lambda_n x + B_n\frac{\sin\lambda_n x}{\lambda}\right)\exp(-\lambda^2t)\right]
$$
When you have non-homogeneous boundary conditions, you would solve the steady state solution. What are your BC?
I would then use the orthogonality and Fourier series to finish the problem solving for the Fourier coefficients.