Why are PDEs harder than ordinary differential equations?

PDEs are harder because the unknown function depends on more than one variable. An ODE might ask for $$y(t).$$ A PDE might ask for $$u(x,t),$$ which changes across space and time. That adds several complications: - boundary conditions matter, - the shape of the domain matters, - different directions can behave differently, - there are fewer universal recipes, - solutions can form waves, diffusion patterns, shocks, or singularities. For an ODE, an initial value often determines a curve. For a PDE, you may need initial data plus boundary data, and the geometry of the boundary can change the entire problem. So yes, the notation is heavier, but the real difficulty is that PDEs describe fields, not just curves.