Question

give a simple odes system with a time-dependent coefficients

Answer #1

time-dependent ordinary differential equations:-

Let x(t) be the position of a particle moving on the x-axis at time t.

1st order ODE

dx/dt = f(x) : velocity is a function of position

x(0) : initial position

The problem is to find the position x(t) for t > 0.

ex

1.

dx/dt = x, x(0) = 1 ) x(t) = et

2.

dx/dt = x2 , x(0)= 1 x(t) = 1/1-t

3.

dx/dt = sin x, x(0)=1 x(t)=?

The simplest numerical method is Euler’s method.

choose Δt: time step

define wn : numerical solution at time tn = ndefine wn:nΔt

wn+1-wn/Δt= f(Wn)

Wn+1=wn + Δtf(Wn)

Given W0, W1, W2.......

questions : accuracy , stability , efficiency

2nd order ODE

d2x/dt2 = f(x) : acceleration is a function of position
(Newton’s equation)

x(0) , x0

(0) : initial position , velocity

(wn+1 - 2wn+wn-1)/(Δt)2 = f(wn)

given w0 and w1, compute w2, w3,...

Formulate a system of linear 1st-order ODEs from the higher
order ODEs below.
y^(4) + 2y''' + 3y'' 4y' + 5y = f(t)

For the following nonhomogeneous system of ODEs
y' = [TABLE 1 MATRIX] y + [TABLE 2 MATRIX] t,
Table 1 & 6,1 is Table 2
1
4
1
1
6
1
(c) Find the general solution
y(k), for the
corresponding homogeneous ODE.
(d) Use the method of undetermined coefficients
to find the particular solution,
y(p).

give an example of linear system of differential equations with
constant coefficients. thanks

Homog. linear ODEs with constant coefficients: Find the solution
of the IVP 25y''-20y'+4y=0, y(0)=10 and y'(0)=9

Find the (real-valued) general solution to the system of ODEs
given by:
X’=[{-1,-1},{2,-3}]X (X is a vector)
-1,-1, is the 1st row of the matrix
2,-3, is the 2nd row of the matrix.
Then, determine whether the equilibrium solution x1(t) = 0,
x2(t) = 0 is stable, a saddle, or unstable.

(a) For a polynomial x2 + bx + c, give the
coefficients in terms of its roots: α1 and α2.
(b) For a monic, cubic polynomial, give the coefficients in
terms of its roots.
(c) Generalize these result to monic polynomials of higher
degree

Solve the system. If there's no unique solution, label the
system as either dependent or incomsistent.
2x+y+3z=12
x-y+4z=5
-4x+4y-4z=-20
a. Dependent system
b. inconsistent system
c.(1,4,2)
d. (4,1,2)

The work done on a system is time dependent, where W(t) = (2.00
J/s)t + (0.500 J/s3)t3. (a) What is the instantaneous power at t =
0.665s and (b) t = 1.16s? Now let us assume that we know the
instantaneous power to be P(t) = (3.50 J/s2)t + (1.50 J/s3)t2.
Assuming that there is no work done on the system at t = 0, what is
work done at (c) t = 0.865s and (d) t = 1.25s?

In a simple linear regression model, if the independent and
dependent variables are negatively linearly related, then the
standard error of the estimate will also be negative.

1. Give three linear systems and three non-linear system, and
specify whether one is time variant or time invariant.

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