Let solve a PDE, if it happens that , i.e. any linear combination is a solution, then we say the PDE islinear.

For example, consider the wave equation,

.

Confirm that if are solutions then is a solution as well.

Another way to think of linear is that , the solution, is just like what linear means in an ordinary differencial equation. So is linear. But is not, because by analogy we never have being considered linear ODE.

Quasi-linear means (this is how I understand it) that thehighest order termsare linear.

For example,

Is not linear but it is quasi-linear. Because the highest order terms are linear.