Substituting $u(x, t)=X(x) T(t)$ into the partial differential equation yields $a^{2} X^{\prime \prime} T=X T^{\prime \prime}+2 k X T^{\prime} .$ Separating
variables and using the separation constant $-\lambda$ we obtain
$$\frac{X^{\prime \prime}}{X}=\frac{T^{\prime \prime}+2 k T^{\prime}}{a^{2} T}=-\lambda.$$
Then
$$X^{\prime \prime}+\lambda X=0 \quad \text { and } \quad T^{\prime \prime}+2 k T^{\prime}+a^{2} \lambda T=0.$$
We consider three cases:
I. If $\lambda=0$ then $X^{\prime \prime}=0$ and $X(x)=c_{1} x+c_{2} .$ Also, $T^{\prime \prime}+2 k T^{\prime}=0$ and $T(t)=c_{3}+c_{4} e^{-2 k t},$ so
$$u=X T=\left(c_{1} x+c_{2}\right)\left(c_{3}+c_{4} e^{-2 k t}\right).$$
II. If $\lambda=-\alpha^{2}<0,$ then $X^{\prime \prime}-\alpha^{2} X=0,$ and $X(x)=c_{5} \cosh \alpha x+c_{6} \sinh \alpha x .$ The auxiliary equation
of $T^{\prime \prime}+2 k T^{\prime}-\alpha^{2} a^{2} T=0$ is $m^{2}+2 k m-\alpha^{2} a^{2}=0 .$ Solving for $m$ we obtain $m=-k \pm \sqrt{k^{2}+\alpha^{2} a^{2}}$
so $T(t)=c_{7} e^{(-k+\sqrt{k^{2}+\alpha^{2} a^{2}}) t}+c_{8} e^{(-k-\sqrt{k^{2}+\alpha^{2} a^{2}}) t} .$ Then
$$u=X T=\left(c_{5} \cosh \alpha x+c_{6} \sinh \alpha x\right)\left(c_{7} e^{(-k+\sqrt{k^{2}+\alpha^{2} a^{2}}) t}+c_{8} e^{(-k-\sqrt{k^{2}+\alpha^{2} a^{2}}) t}\right).$$
III. If $\lambda=\alpha^{2 }> 0,$ then $X^{\prime \prime}+\alpha^{2} X=0,$ and $X(x)=c_{9} \cos \alpha x+c_{10} \sin \alpha x .$ The auxiliary equation
of $T^{\prime \prime}+2 k T^{\prime}+\alpha^{2} a^{2} T=0$ is $m^{2}+2 k m+\alpha^{2} a^{2}=0 .$ Solving for $m$ we obtain $m=-k \pm \sqrt{k^{2}-\alpha^{2} a^{2}} .$ We
consider three possibilities for the discriminant $k^{2}-\alpha^{2} a^{2}$:
(i) If $k^{2}-\alpha^{2} a^{2}=0$ then $T(t)=c_{11} e^{-k t}+c_{12} t e^{-k t}$ and
$$u=X T=\left(c_{9} \cos \alpha x+c_{10} \sin \alpha x\right)\left(c_{11} e^{-k t}+c_{12} t e^{-k t}\right).$$
From $k^{2}-\alpha^{2} a^{2}=0$ we have $\alpha=k / a$ so the solution can be written
$$u=X T=\left(c_{9} \cos k x / a+c_{10} \sin k x / a\right)\left(c_{11} e^{-k t}+c_{12} t e^{-k t}\right).$$
(ii) If $k^{2}-\alpha^{2} a^{2}<0$ then $T(t)=e^{-k t}\left(c_{13} \cos \sqrt{\alpha^{2} a^{2}-k^{2}} t+c_{14} \sin \sqrt{\alpha^{2} a^{2}-k^{2}} t\right)$ and
$$
u=X T=\left(c_{9} \cos \alpha x+c_{10} \sin \alpha x\right) e^{-k t}\left(c_{13} \cos \sqrt{\alpha^{2} a^{2}-k^{2}} t+c_{14} \sin \sqrt{\alpha^{2} a^{2}-k^{2}} t\right)
$$
(iii) If $k^{2}-\alpha^{2} a^{2}>0$ then $T(t)=c_{15} e^{(-k+\sqrt{k^{2}-\alpha^{2} a^{2}}) t}+c_{16} e^{(-k-\sqrt{k^{2}-\alpha^{2} a^{2}}) t}$ and
$$
u=X T=\left(c_{9} \cos \alpha x+c_{10} \sin \alpha x\right)\left(c_{15} e^{(-k+\sqrt{k^{2}-\alpha^{2} a^{2}}) t}+c_{16} e^{(-k-\sqrt{k^{2}-\alpha^{2} a^{2}}) t}\right)
$$