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Text: (2) Linearization

The governing differential equations of a system are:

dx1(t)/dt = u(t) – bvx1(t) – x2(t)

dx2(t)/dt = bvx1(t) – x2(t) – @Vx2(t)

where u(t) is the input to the system. The input is represented as u(t) = u0, where u0 is the constant operating point value of u.

Find the relations between the state variables at the operating point(s) of the system. State whether this operating point is unique and justify your answer.

Linearize the non-linear differential equations about the equilibrium point(s) x1, x2. Represent the equations in state space form. You must use the following state vector: x = [x1, x2].

The output for this problem is X = +5X2. Determine the state space matrix vector form of the perturbation output fX.

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11:09

Solving the differential equations that arise from modeling may require using integration by parts. [See formula (1).]

A person deposits an inheritance of $\$ 100,000$ in a savings account that earns $4 \%$ interest compounded continuously. This person intends to make withdrawals that will increase gradually in size with time. Suppose that the annual rate of withdrawals is $2000+500 t$ dollars per year, $t$ years from the time the account was opened.(a) Assume that the withdrawals are made at a continuous rate. Set up a differential equation that is satisfied by the amount $f(t)$ in the account at time $t.$(b) Determine $f(t).$(c) With the help of your calculator, plot $f(t)$ and approximate the time it will take before the account is depleted.

10:03

Find all equilibria for the following system of differential equations and determine the stability of each equilibrium:dx1/dt = 12*1*(1-X1) – 6*X1*X2dx2/dt = X2*(2 – X2) – X1

Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice:A. The stable equilibria are X1, X2 and the unstable equilibria are X1, X2 (Type integers or fractions. Type ordered pairs. Use a comma to separate answers as needed.)B. The unstable equilibria are X1, X2. There are no stable equilibria. (Type an integer or a fraction. Type an ordered pair. Use a comma to separate answers as needed.)C. The stable equilibria are X1, X2. There are no unstable equilibria. (Type an integer or a fraction. Type an ordered pair. Use a comma to separate answers as needed.)D. There are no equilibria.

04:07

Equilibrium solutions $A$ differential equation of the form $y^{\prime}(t)=f(y)$ is said to be autonomous (the function $f$ depends only on y. The constant function $y=y_{0}$ is an equilibrium solution of the equation provided $f\left(y_{0}\right)=0$ (because then $y^{\prime}(t)=0$ and the solution remains constant for all $t$ ). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.a. Find the equilibrium solutions.b. Sketch the direction field, for $t \geq 0$.c. Sketch the solution curve that corresponds to the initial condition $y(0)=1$.$$y^{\prime}(t)=6-2 y$$

04:22

Equilibrium solutions $A$ differential equation of the form $y^{\prime}(t)=f(y)$ is said to be autonomous (the function $f$ depends only on y. The constant function $y=y_{0}$ is an equilibrium solution of the equation provided $f\left(y_{0}\right)=0$ (because then $y^{\prime}(t)=0$ and the solution remains constant for all $t$ ). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.a. Find the equilibrium solutions.b. Sketch the direction field, for $t \geq 0$.c. Sketch the solution curve that corresponds to the initial condition $y(0)=1$.$$y^{\prime}(t)=y(y-3)(y+2)$$

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