The solution of the partial differential equation $\dfrac{\partial u}{\partial t} = \alpha \dfrac{\partial ^2 u}{\partial x^2}$ is of the form
1. $C \: \cos (kt) \lfloor C_1 e^{(\sqrt{k/\alpha})x} +C_2 e^{-(\sqrt{k/\alpha})x} \rfloor \\$
2. $C e^{kt} \lfloor C_1 e^{(\sqrt{k/\alpha})x} +C_2 e^{-(\sqrt{k/\alpha})x} \rfloor \\$
3. $C e^{kt} \lfloor C_1 \cos \big( \sqrt{k/ \alpha} \big) x + C_2 \sin ( – \sqrt{k / \alpha}) x \rfloor \\$
4. $C \sin(kt) \lfloor C_1 \cos \big( \sqrt{k/ \alpha} \big) x + C_2 \sin ( – \sqrt{k/ \alpha} ) x \rfloor$