Answer
For $f\left( {x,t} \right) = {{\rm{e}}^{Ax + Bt}}$ to satisfy Eq. (3) we must have $B = {A^2}$ for all values of $A$ and $B$.
Work Step by Step
1. Take the partial derivative of $f\left( {x,t} \right)$ with respect to $t$:
$\frac{{\partial f}}{{\partial t}} = B{{\rm{e}}^{Ax + Bt}}$
2. Take the partial derivatives of $f\left( {x,t} \right)$ with respect to $x$:
$\frac{{\partial f}}{{\partial x}} = A{{\rm{e}}^{Ax + Bt}}$
$\frac{{{\partial ^2}f}}{{\partial {x^2}}} = {A^2}{{\rm{e}}^{Ax + Bt}}$
For $f$ to satisfy the heat equation, Eq (3) we write
$\frac{{\partial f}}{{\partial t}} = \frac{{{\partial ^2}f}}{{\partial {x^2}}}$
$B{{\rm{e}}^{Ax + Bt}} = {A^2}{{\rm{e}}^{Ax + Bt}}$
Divide both sides by ${{\rm{e}}^{Ax + Bt}}$ we obtain $B = {A^2}$.
Thus, for $f\left( {x,t} \right) = {{\rm{e}}^{Ax + Bt}}$ to satisfy Eq. (3) we must have $B = {A^2}$ for all values of $A$ and $B$.
