This condition can be broken down into two separate conditions which have to be satisfied simultaneously,

1. The condition of homogeneity:

$\displaystyle F(\alpha u)=\alpha F(u)$

(1.6.4)

2. The condition of linearity

$\displaystyle F(u+v)=F(u)+F(v)$

(1.6.5)

As both of these conditions have to be satisfied simultaneously, an operator or function that does not satisfy any one of the two conditions above can be proved as nonlinear.

Initially, checking the condition of homogeneity (Equation 1.6.4)

So the given term $\displaystyle c_{3}(Y^{1},t){\ddot {Y}}^{1}$ is not homogenous with respect to $\displaystyle Y^{1}$. As it is one of the two conditions to be simultaneously satisfied for linearity, we can say that term $\displaystyle c_{3}(Y^{1},t){\ddot {Y}}^{1}$ is also not linear with respect to $\displaystyle Y^{1}$.