A semilinear partial differential equation is a special kind of quasi-linear equation. It is still linear in the first partial derivatives. But its derivative coefficients do not depend on the unknown function $u$. This is the main difference between a semilinear equation and a general quasi-linear equation. For an unknown function $u=u(x,y)$, the standard semilinear first-order equation is $$\begin{equation*} a(x,y)u_x+b(x,y)u_y=c(x,y,u). \quad \text{(1)} \end{equation*}$$ Here $a$ and $b$ are functions of $x$ and $y$ only. The right side may still depend on $x$, $y$, and $u$.

A first-order partial differential equation is called semilinear if it is linear in the first-order partial derivatives and the coefficients of those derivatives are independent of the unknown function $u$.