Solving $u_t=4u_{xx}+\sin{(\pi x)}+\sin{(2\pi x)}$ using an eigenfunction expansion
Solution 1:
The differential equations for $c_1(t)$ and $c_2(t)$ should be \begin{align*} \dot{c_1} & = -4\pi^2c_1 + 1 \\ \dot{c_2} & = -16\pi^2c_2 + 1 \end{align*}
The differential equations for $c_1(t)$ and $c_2(t)$ should be \begin{align*} \dot{c_1} & = -4\pi^2c_1 + 1 \\ \dot{c_2} & = -16\pi^2c_2 + 1 \end{align*}