## Quiz: Linearize ODEs

1. What is a method for linearizing a nonlinear function? Select all that apply.

A. Newton's Method
Incorrect. Newton's method is used to find the root of an equation where it is equal to zero f(x)=0.
B. Gauss Elimination
Incorrect. Gauss elimination is an algorithm in linear algebra for solving a system of linear equations.
C. Taylor Series Expansion
Correct. Linearization of the right-hand side of a differential equation is the first two terms of a Taylor series expansion with f(x)=f(x_0)+f'(x_0)(x-x_0).
D. Power Series Expansion
Correct. A power series is an example of a Taylor series. Linearization is using the first two terms of a Taylor series expansion with f(x)=f(x_0)+f'(x_0)(x-x_0).

2. Determine the steady state value of x in the following equation, given a steady state input of u = 2.

$$\frac{dx}{dt}=\sqrt{x}-u^3$$

A. 8
Incorrect. Set the derivative equal to zero to find the steady state solution. Square both sides of the equation to solve for x.
B. 2
Incorrect. Set the derivative equal to zero to find the steady state solution.
C. 68
Incorrect. Set the derivative equal to zero to find the steady state solution.
D. 64
Correct. 0=\sqrt{x}-2^3 is rearranged to find the solution x=8^2

3. Find the value \alpha for the function f(y,u,d) with u_{ss}=4 and y_{ss}=3.

$$\frac{dy}{dt}=3y^3-(u^2-\sin(u))^{1/3}+ln(d)$$

The constant \alpha is the partial derivative of f(y,u,d) with respect to y and evaluated at steady state conditions.

$$\frac{dy'}{dt} = \alpha y' + \beta u' + \gamma d'$$

$$\alpha = \frac{\partial f}{\partial y}\bigg|_{\bar y,\bar u,\bar d}$$

A. 81
Correct. The partial derivative with respect to y is 9y^2. Evaluating at y_{ss}=3 gives the answer \alpha=81.
B. 273
Incorrect. Try again.
C. 27
Incorrect. Try again.
D. 144
Incorrect. You have swapped the values of u_{ss}=4 and y_{ss}=3.

4. Find the value \beta for the function f(y,u,d) with u_{ss}=4 and y_{ss}=3.

$$\frac{dy}{dt}=3y^3-(u^2-\sin(u))^{1/3}+ln(d)$$

The constant \beta is the partial derivative of f(y,u,d) with respect to u and evaluated at steady state conditions.

$$\frac{dy'}{dt} = \alpha y' + \beta u' + \gamma d'$$

$$\beta = \frac{\partial f}{\partial u}\bigg|_{\bar y,\bar u,\bar d}$$

Use the following Python source code to calculate the partial derivative, if needed.

# analytic solution with Python
import sympy as sp
# define symbols
y,u,d = sp.symbols(['y','u','d'])
# define equation
dydt = 3*y**3-(u**2-sp.sin(u))**(1/3)+sp.log(d)
# partial derivative with respect to u
beta = sp.diff(dydt,u)
# evaluate at steady state condition
print(beta.subs(u,4).evalf())
A. -0.4432
Incorrect. Run the code to check your answer. Type pip install sympy to install SymPy.
B. -0.4405
Correct.
C. 0.4402
Incorrect. Run the code to check your answer. Type pip install sympy to install SymPy.
D. -0.5442
Incorrect. You have swapped the values of u_{ss}=4 and y_{ss}=3.