Linearization of Non-Linear Ordinary Differential Equations

Question 2: Linearization using Taylor series 1st order

Using Taylor series 1st

order expansion,

But, at steady state,  so that:

Therefore, the linearized differential equation becomes:

Question 3: Linearized ODE using Integrating factor

where

Multiplying equation 1 by the IF yields:

Integrating both sides with respect to t results to:

Question 4:

Plot of Question 1 and Question 3 Solutions

The graphs of solutions to question 1 and question 3 are shown in figure 1.

Figure 1: Plot of Q1 and Q3

From Figure 1, it is evident that the values of the linearised ODE about  are approximately close to the ones of the non-linearized function. However, moving further from the neighbourhood (when ) decreases the accuracy of the linearised ODE.