I know a PDE is linear when the dependent variable $u$ and its derivatives appear only to the first power. So, $u_t + u_x +5u = 1$ would be linear.
However, I do not quite understand the other two.
My professor described
"semilinear" PDE's as PDE's whose highest order terms are linear, and
"quasilinear" PDE's as PDE's whose highest order terms appear only as individual terms multiplied by lower order terms.
No examples were provided; only equivalent statements involving sums and multiindices were shown, which I do not think I could decipher by tomorrow.
Can someone provide some examples of "semilinear" and "quasilinear" PDE's?