For x = eye(4) what shall be the output of the following statement: diff(x,1), diff(x,2), diff(x,1,2), diff(x,2,2)? For the differential equation d²y1/dt² - μ(1 - y1²)(dy1/dt) + y1... For x = eye(4) what shall be the output of the following statement: diff(x,1), diff(x,2), diff(x,1,2), diff(x,2,2)? For the differential equation d²y1/dt² - μ(1 - y1²)(dy1/dt) + y1 = 0 with the conditions at t = 0, y1(2) and dy1/dt = 0, write the suitable statements to solve the above equation. How will you get the solution at t's given by: t = 0:10? For the equation: ∂u1/∂t = 0.024 ∂²u1/∂x² - F(u1 - u2) with the condition: F(y) = (y(5.73) - exp(1.69)) and initial conditions u1(x,0) = 1.0, u2(x,0) = 0, ∂u1/∂x(0,t) = 1.5, u2(0,t) = 0, u1(1,t) = 0. Write the functions to define the differential equations, initial conditions, and boundary conditions. Read more

Steps to Solve

1. Understanding the Outputs for diff Functions

To solve the first part about the output of diff(x, n) for a symbolic environment, we note that this function computes the nth derivative of the variable $x$. For $x = \text{eye}(4)$, which represents a 4x4 identity matrix:

• $ \text{diff}(x, 0) = x $
• $ \text{diff}(x, 1) = 0 $ (derivative of a constant)
• $ \text{diff}(x, 2) = 0 $ (second derivative)
• $ \text{diff}(x, 1, 2) = 0 $ (first derivative of a derivative)
• $ \text{diff}(x, 2, 2) = 0 $ (second derivative of a derivative)

2. Setting Up the Differential Equation

For the second part, we need to set up the differential equation:

$$ \frac{d^2y_1}{dt^2} - \mu(1 - y_1^2)\frac{dy_1}{dt} + y_1 = 0 $$

with initial conditions given for $t = 0$:

• $y_1(0) = 2$
• $\frac{dy_1}{dt}(0) = 0$

3. Writing Suitable Statements

These initial conditions can be defined in a symbolic computation environment. The function could look something like this:

from sympy import symbols, Function, dsolve, Derivative

t = symbols('t')
y1 = Function('y1')(t)
mu = symbols('mu')

# Define the equation
eq = Derivative(y1, t, 2) - mu * (1 - y1**2) * Derivative(y1, t) + y1

# Solve the equation with initial conditions
init_conditions = {y1.subs(t, 0): 2, Derivative(y1, t).subs(t, 0): 0}
solution = dsolve(eq, y1, ics=init_conditions)

4. Analyzing the Next Differential Equation

The third part involves another differential equation:

$$ \frac{du_1}{dt} = 0.024 \frac{\partial^2 u_1}{\partial x^2} - F(u_1 - u_2) $$

and the corresponding boundary and initial conditions for $u_1$ and $u_2$.

5. Defining Initial and Boundary Conditions

The initial and boundary conditions are provided:

• $u_1(x, 0) = 1.0$, $u_2(x, 0) = 0$
• $\frac{\partial u_1}{\partial x}(0, t) = 1.5$, $u_2(0, t) = 0$
• $u_1(1, t) = 0$

These conditions can also be written similarly using symbolic tools.