User:Egm6321.f10.team5.steinberg

Given
The vehicle component equation of motion for a Maglev train is represented by a second-order nonlinear partial differential equation (Eq. 2.5e).

The coefficient for the second-order term is defined as follows (assuming $$ I_w = 0 $$):

$$ C_3(Y^1,t) := M \left [ 1 - \bar{R} u,_{SS}^2 (Y^1,t) \right ]^2 $$

Find
Show that $$ C_3 \! $$ is non-linear with respect to $$ Y_1 \! $$.

Solution
If the coefficient $$ C_3 \! $$ is linear, it must satisfy both properties of homogeneity and additivity.

Homogeneity $$ \Rightarrow C_3(\alpha Y^1,t) = \alpha C_3(Y^1,t) $$, where $$ \alpha \! $$ is an arbitrary constant.

$$ C_3(\alpha Y^1,t) = M \left [ 1 - \bar{R} u,_{SS}^2 (\alpha Y^1,t) \right ]^2 = M \left [ 1 - \alpha \bar{R} u,_{SS}^2 (Y^1,t) \right ]^2 \not= \alpha M \left [ 1 - \bar{R} u,_{SS}^2 (Y^1,t) \right ]^2 $$

$$ C_3(\alpha Y^1,t) \not= \alpha C_3(Y^1,t) $$

Since $$ C_3 \! $$ is not homogeneous, it is therefore nonlinear with respect to $$ Y_1 \! $$.