Question

A system of diﬀerential equations having the form

t(~x)' = A ~x,

where A is a matrix with constant entries, is known as a
**Cauchy-Euler system.**

(a) Suppose λ is an eigenvalue of A and ~ v is an eigenvector corresponding to λ. Show that the function

**x**(t) = t^λ (**v**)

is a solution to the Cauchy-Euler system t(**x**)'
= A(**x**).

(b) Solve the following Cauchy-Euler system:

t(**x**)' =

3 | -2 |

2 | -2 |

(**x**)

(t > 0)

Answer #1

[Cauchy-Euler equations] For the following equations with the
unknown function y = y(x), find the general solution by changing
the independent variable x to et and re-writing the equation with
the new unknown function v(t) = y(et).
x2y′′ +xy′ +y=0
x2y′′ +xy′ +4y=0
x2y′′ +xy′ −4y=0
x2y′′ −4xy′ −6y=0
x2y′′ +5xy′ +4y=0.

Two functions, u(x,y) and v(x,y), are said to verify the
Cauchy-Riemann
differentiation equations if they satisfy the following
equations ∂u\dx=∂v/dy and ∂u/dy=−(∂v/dx)
a. Verify that the Cauchy-Riemann differentiation equations can
be written in the polar coordinate form as
∂u/dr=1/dr ∂v/dθ and ∂v/dr =−1/r ∂u/∂θ
b. Show that the following functions satisfy the Cauchy-Riemann
differen- tiation equations
u=ln sqrt(x^(2)+y^(2)) and v= arctan y/x.

Write the system of equations as an augmented matrix. Then solve
the system by putting the matrix in reduced row echelon form.
x+2y−z=-10
2x−3y+2z=2
x+y+3z=0

Consider the following system of equations.
3x
−
2y
=
b1
4x
+
3y
=
b2
(a) Write the system of equations as a matrix equation.
x
y
=
b1
b2
(b) Solve the system of equations by using the inverse of the
coefficient matrix.(i) where
b1 = −6, b2 = 11
(x, y) =
(ii) where
b1 = 4, b2 = −2
(x, y) =

Solve the system of equations using diagonalization x'(t)=(5
-1)x(t)
3 1

A particular system is found to obey the following two equations
of state: T= 3As2/v P=As3/v2 where A is a constant. a. Find the
fundamental equation of this system by direct integration of its
molar total differential form: du(s,v)=Tds - Pdv . (This is an
integration over multiple variables, but the variables are not
independent of each other. Your solution will have only one term,
not two.) b. Find the chemical potential \mu as a function of s and
v...

3) For the given system of equations:
x+y-z=-6
x+2y+3z=-10
2x-y-13z=3
Rewrite the system as an augmented matrix. [4 pt]
Find the reduced row echelon form of the matrix using your
calculator, and write it in the spacebelow. [4 pt]
State the solution(s) of the system of equations. [3 pt]

Solve the following system of linear equations: 3x2−9x3 = −3
x1−2x2+x3 = 2 x2−3x3 = 0 If the system has no solution, demonstrate
this by giving a row-echelon form of the augmented matrix for the
system. If the system has infinitely many solutions, your answer
may use expressions involving the parameters r, s, and t. You can
resize a matrix (when appropriate) by clicking and dragging the
bottom-right corner of the matrix.

1. solve the system of equations:
x+3y=9
3x+9y=27
please solve to get the x and y solution
2. solve the system of equations, please state the x and y
solution
9x-4y=23
15x+4y=41
3. solve the system of equations please state the x and y
solution
x - y - z = 1
5x + 6y + z = 1
30x + 25y = 0

1. If x1(t) and x2(t) are solutions to the differential
equation
x" + bx' + cx = 0
is x = x1 + x2 + c for a constant c always a solution? Is the
function y= t(x1) a solution?
Show the works
2. Write sown a homogeneous second-order linear differential
equation where the system displays a decaying oscillation.

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