Section 1
Universality of Mathematical Models
Using a replacement $\rho(x, t)=\varphi(t) \cdot u(x, t)$ reduce (3) to the equation of thermal conductivity.
Prove that the square equation (6) always has real roots, and that a necessary and sufficient condition of negativity of both roots is the condition $c>0$.
Deduce equations (16) and (17) using the same assumptions as in the deduction of (15).
Differentiating the integral on the right hand side of (20) by $t, x$, and using equation (18), prove that the function $u(t, x)$ is a solution of equation (18).