Quiz: Linearize ODEs
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1. What is a method for linearizing a nonlinear function? Select all that apply.
- Incorrect. Newton's method is used to find the root of an equation where it is equal to zero f(x)=0.
- Incorrect. Gauss elimination is an algorithm in linear algebra for solving a system of linear equations.
- 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)`.
- 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.
$$\frac{dx}{dt}=\sqrt{x}-u^3$$
- Incorrect. Set the derivative equal to zero to find the steady state solution. Square both sides of the equation to solve for x.
- Incorrect. Set the derivative equal to zero to find the steady state solution.
- Incorrect. Set the derivative equal to zero to find the steady state solution.
- 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.
$$\frac{dy'}{dt} = \alpha y' + \beta u' + \gamma d'$$
$$\alpha = \frac{\partial f}{\partial y}\bigg|_{\bar y,\bar u,\bar d}$$
- Correct. The partial derivative with respect to `y` is `9y^2`. Evaluating at `y_{ss}=3` gives the answer `\alpha=81`.
- Incorrect. Try again.
- Incorrect. Try again.
- 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.
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())
- Incorrect. Run the code to check your answer. Type pip install sympy to install SymPy.
- Correct.
- Incorrect. Run the code to check your answer. Type pip install sympy to install SymPy.
- Incorrect. You have swapped the values of `u_{ss}=4` and `y_{ss}=3`.
