You probably is confusing boundary conditions and initial conditions. For both wave equation and heat equation we need just 1 boundary condition at each end, however for other types of equations we may need more

Now, we need two initial conditions (as $t=0$) for wave equation, and one for heat equation because the former has an order 2 with respect to $t$, and the latter has order $1$. This was explained in details during the lecture.

Please do not post big chunks of screenshots but only exact equations you are talking about, and better type them. Then you questions would not require guesswork to answer