The general solution

For each

$$G_n (x) = \sin(\frac{ \pi n x }{ L } ),\;\;n=1,\;2,\;\dots$$

we can find $F_n(t)$ by solving the differential equation

$$\frac{d} {dt } F(t) = -(\frac{ \pi n }{ L })^2 F(t)$$

which follows from (1.1) with $\lambda = -(\frac{ \pi n }{ L })^2.$ The general solution for this equation is

$$F_n(t) = C_n e^{-(\frac{ \pi n }{ L })^2 t },$$

where the constant $C_n$ is arbitrary. Hence, we have infinitely many special solutions

$$C_n e^{-(\frac{ \pi n }{ L })^2 t } \cdot \sin(\frac{ \pi n x }{ L } ) ,\;\;n=1,\;2,\;\dots$$

for the heat equation with the boundary conditions

$$u(t,0)=0,\;\; u(t,L) = 0$$

The general solution corresponding to these boundary conditions is given by

$$u(t,x) = \sum_{n=1}^\infty C_n e^{-(\frac{ \pi n }{ L })^2 t } \cdot \sin(\frac{ \pi n x }{ L } )$$ (1.3)