{\displaystyle x\in \mathbb {R} ^{n}}
is the optimization variable.
‖
x
‖
2
{\displaystyle \lVert x\rVert _{2}}
is the Euclidean norm and
T
{\displaystyle ^{T}}
indicates transpose. The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function
(
A
x
+
b
,
c
T
x
+
d
)
{\displaystyle (Ax+b,c^{T}x+d)}
to lie in the second-order cone in
R
n
i
+
1
{\displaystyle \mathbb {R} ^{n_{i}+1}}
.SOCPs can be solved by interior point methods and in general, can be solved more efficiently than semidefinite programming (SDP) problems. Some engineering applications of SOCP include filter design, antenna array weight design, truss design, and grasping force optimization in robotics.
Consider the second order linear ODE with parameters ##a, b##:
$$
xy'' + (b-x)y' - ay = 0
$$
By considering the series solution ##y=\sum c_mx^m##, I have obtained two solutions of the following form:
$$
\begin{aligned}
y_1 &= M(x, a, b) \\
y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\
\end{aligned}
$$...
Hi everybody.
I need to learn how to solve this kind of equation by decomposing y in a serie of functions. All the examples I have seen are of homogeneous functions. I would be extremely thankfull if someone pointed me to some text in which this is done-explained.
Thanks for reading.
Homework Statement
Solve (x-1)y''-xy'+y=0 , given x>1 and y1=ex
Homework EquationsThe Attempt at a Solution
I've tried solving it by multiplying everythin for (1/x-1), so my equation looks like:
y''-(x/x-1)y'+y=0 (because x-1 is unequal to 0) (1)
so now the equation has the...
So, I had studied oscillatory motion for a while and I found it unpleasant to have to remember the various different solutions for the equations of motion. I began to learn about second order linear differential equations and now I know how to solve this kind of stuff. But there is a problem...
Homework Statement
Solve the following differential equation and compare computer solutions
\begin{equation*}
4y''+12y'+9=0
\end{equation*}
Homework Equations
None
The Attempt at a Solution
[/B]
First of all looking at this equation, even though it is in the section where we learned about...
Homework Statement
I've been stuck on this problem for three days now, and I have no clue how to solve it.
Construct a linear differential equation of order 2, for which { y_1(x) = sin(x), y_2(x) = xsin(x)} is a set of fundamental solutions on I = (0,\pi) .
Homework Equations
Wronskian for...
Homework Statement
y′′=−20⋅4x^3
Homework Equations
Undetermined coefficients method
The Attempt at a Solution
so at first, solving the associated homogeneous equation I find the fundamental set of solutions to be: y1=1 and y2=x.
I know that these are correct. Now for the part that confuses...
I'm having a lot of trouble with this problem. I'm also having a lot of trouble inputting it into LaTeX. I hope you can follow even though the markup isn't good.
I'm trying to find a formula for the general solution of $ax^2y''+bxy'+cy=0$ where $y=x^r\ln(x)$ when $(b-a)^2-4ac=0$;
using...
Hi, all. While solving a second order linear differential equation why do we have to use linear independent but two solutions. For example, when solving y''- y = 0 , y(0) = 5 and y'(0) = 3 , we use ex and e-x and then we write y = c1*ex+ c2*e-x-
Question about Solutions of second order linear PDEs
I don't have very much formal knowledge of this topic, this is something I have been thinking about, so excuse me if my notation is off. I have a question about second order linear PDEs, do all have a separable solution? It seems that we can...
Hey, I'm not sure how to even approach this problem. It's not a simple ODE.
Basically, I want to find the solution for Θ in terms of ε. The equation is
\frac{1}{ε}*\frac{d}{dε}*(ε*\frac{dΘ}{dε})-β^{2}Θ=0
I tried to move the B^2 to the other side and I wasn't able to solve it that way. I...
Homework Statement
Let y(x)=\sumckxk (k=0 to ∞) be a power series solution of
(x2-1)y''+x3y'+y=2x, y(0)=1, y'(0)=0
Note that x=0 is an ordinary point.
Homework Equations
y(x)=\sumckxk (k=0 to ∞)
y'(x)=\sum(kckxk-1) (k=1 to ∞)
y''(x)=\sum(k(k-1))ckxk-2 (k=2 to ∞)
The Attempt at a Solution...
Homework Statement
Find the general solution to the differential equation y'' -16y = 0, where y is a function of x. Give initial conditions that would give a unique solution to the eqution.
For the differential equation y'' - k2y = R(x), with k ≠ 0 a real constant, show that it has a...
Homework Statement
Find the general solution for the following differential equation:
y'' + x(y')^2 = 0.
Homework Equations
Integration, differentiation...
The Attempt at a Solution
Usually for these sort of DE you could use the substitution v(x) = y'(x) and this would simplify...
Suppose we have two multivariate functions, u_{1}(x,t) and u_{2}(x,t). These functions are solutions to second-order linear equations, which can be written as follows:
Au_{xx}+Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu=G
Each of the coefficients are of the form A(x,y). Now, the linearity of these...
Someone know how to uncouple this system of pde?
Δu_{1}(x) + a u_{1}(x) + b u_{2}(x) =f(x)
Δu_{2}(x) + c u_{1}(x) + d u_{2}(x) =g(x)
a,b,c,d are constant.
I would like to find a solution in one, two, three dimension, possibily in terms of Green function...someone could help me, please?
$(1-x^2)y'' - xy' + 4y =2 x \sqrt{1-x^2} $
Hint use the substitution $x =\sin t$
I used it and end with
$\cos t y'' + \sin t y' - \frac{\sin t}{\cos t} y' + 4y = 2\sin t |\cos t| $
how to solve this i just want the name of the method
Homework Statement
Find the set of functions from (-1,1)→ℝ which are solutions of:
(x^{2}-1)\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-4y = 0
Homework Equations
The Attempt at a Solution
There is a hint which says to use the change of variable:
x=cos(θ)
doing this I get...
Homework Statement
Find the set of functions from (-1,1)→ℝ which are solutions of:
(x^{2}-1)y''+xy'-4y = 0
Homework Equations
The Attempt at a Solution
OK, I'm not really sure how to go about solving this equation, I have only previously attempted problems where the functions in...
In cullen-zill chapter 6 equation 23 it says that
y_{2}(x)=y_{1}(x)\int\frac{e^{-\int P(x)dx}}{y_{1}^{2}(x)}dx
is a solution of
y''+P(x)y'+Q(x)y=0
whenever y_{1}(x) is a known solution
Where does this come from? I would like to be able to prove this or find a proof somewhere.
My...
Homework Statement
3\frac{d^{2}y}{dx^{2}} + 2\frac{dy}{dx} + y = 0
Homework Equations
The Attempt at a Solution
3y'' + 2y' +y = 0
I know the solution is going to be in the form of y=Ce^{mx}+De^{nx}+...
(Unless there is a multiplicity, in which case I understand that too)
So I'll just skip...
Homework Statement
For the differential equation y'' - 4y' + y = 0,
(a) Show that if we let x = y' (i.e. x(t) = y'(t)), then this leads to the system:
x' = 4x -y
y' = x
(b) Conversely, show that the system in (a) leads to y'' - 4y' + y = 0 (and x'' -4x' + x = 0 also).
Homework...
If
ay+b\int^y_0ydy+cy'=0
then
ay'+by+cy''=0
now, let
y=e^{sx}
thus,
s^2+a/cs+b/c=0
and then one solves for s. It is then plugged into what sources are deeming a "general solution"
y=C_1e^{s_1x}+C_2e^{s_2x}
however, none of these texbooks explain or derive where this comes from, and I have not...
Homework Statement
So I'm trying to get a grip about those Green functions and how to aply them to solve differential equations. I've searched the forums and read the section on green's functions in my course book both once and twice, and I think I start to understand at least som of it...
Solving second order linear homogeneous differential equation! HELP!?
Solve the second order linear homogeneous differential equation with constant coefficients by reqriting as a system of two first order linear differential equations. Show that the coefficient matrix is not similar to the...
Homework Statement
Solve the following second order linear differential equation
d2x/dt2 + x = 2 cos(t)
subject to the initial condition x(0) = 0 and dx/dt (0) = 0. What type of motion do you find?
Homework Equations
The Attempt at a Solution
I don't know where to start
Homework Statement
22e^{2t}=y''+8y'-9y
Homework Equations
The Attempt at a Solution
The directly previous question to this was the same but homogeneous, i.e. the 22e^(2t) was replaced with a 0.
So I know the general solution to the homogeneous ode is C_1e^t+C_2e^{-9t}
I know that r(x)=...
Homework Statement
22e^{2t}=y''+8y'-95
Homework Equations
The Attempt at a Solution
I've been reading a textbook on this and think that I should use "method of undetermined coefficients" I know r(x)=ke^{\gamma*x} so y_p(x)=Ce^{\gamma*x}
The trouble is after reading the entire chapter I...
Homework Statement
i'm supposed to find the general solution of the equation: y'' + 3y' + 2y = e^x + e^-x
Homework Equations
I have no problem with solving this equation however, i am confused with the step they are taking in the solutions (circled)...
Homework Statement
y''+7y'=392sin(7t)+686cos(7t) with y(0)=4 and y'(0)=9
Homework Equations
No real relevant equations
The Attempt at a Solution
I assumed since the g(t) has function of both sine and cosine the solution would be both the real and non real parts of the solution to...
I could not get LaTex to format properly, so I typed out the question and my work using Microsoft Word's equation editor. Please see the attached document, apologies for any inconvenience! These problems involve the techniques for the method of undetermined coefficients and variation of parameters.
Homework Statement
Solve the IVP, \frac{1}{4}y'' + 16y = 0
y(0)=\frac{1}{4}
y'(0)=0
Answer is given... y(t) = \frac{1}{4}cos 8t
Homework Equations
The Attempt at a Solution
This has the characteristic equation \frac{1}{4} \lambda^2 +16\lambda=0
Solving for lambda, I got...
Homework Statement
(1+x2)y'' - 4xy' + 6y = 0
Homework Equations
I'm going to assume y can be written as [n=0 to ∞] ∑anxn
The Attempt at a Solution
y = [n=0 to ∞] ∑anxn
----> y' = [n=0 to ∞] ∑(n+1)an+1xn
----> y'' = [n=0 to ∞] ∑(n+2)(n+1)an+2xn
---->...
my question is: what is the general solution of this system of coupled diff. equations:
f ''i = Cijfj
C is a matrix, fj(z) are functions dependent of z.
Homework Statement
a) ẍ + 5ẋ + 4x = 0
x(0) = 0, ẋ(0)=1
What type of damping?
b) ẍ + x = cos(t)
x(0) = 0, ẋ(0)= 1
What type of motion?
The Attempt at a Solution
a) Let x = R Eᴿᵀ
R = -4
R = -1
x(t) = CE**-4T + DE**-t
C + D = 0
-4C - D = 1
C = 1/5
D = 1/5
And it is...
Homework Statement
Find a third degree polynomial approximation for the general solution to the differential equation:
\frac{d^{2}y}{dt^{2}} +3\frac{dy}{dt}+2y= ln(t+1)
Homework Equations
Power series expansion for ln(t+1)
The Attempt at a Solution
The system to the...
I understand how to solve a normal second order linear equation, but this question in the homework is a bit more theoretical and I'm a bit confused.
"Suppose y1(t) and y2(t) are solutions of y'' + py' + qy = 0
Verify that y(t) = k1y1(t) + k2y2(t) is also a solution for any choice of...
Homework Statement
Suppose that L is a second order linear differential operator over the interval J, that f is a function defined on J, and that the function v has the property that
Lv = f on J
(a) Show that if y = u + v and that Lu = 0 on J, then Ly = f on J
(b) Show that if Ly = f on...
Can anyone help me to solve the following second order linear 'ODE' for V(x,s,t):
\frac{\partial^2 V(x,s,t)}{\partial s^2} = g(s) V(x,s,t)
where
g(s)=\frac{a^2}{B^4}+\frac{s^2 w^2}{B^2}.
Here, a, b and w are (real) constants.
Hi,
I'm having problems solving this equation;
y'' -2y' +3y =0 y(0)=-1 , y'(0)=(root 2) -1
I found the auxiliary equation r^2 -2r +3 = 0
and since b^2 -4ac is less than zero this the case where r1 and r2 are complex numbers.
This is as far as I get without getting stuck...
Hey; not much of a homework question, but something i was wanting to find out.
Im still a first year undergraduate and just started on differential equations. We have just finished going over homogeneous 2nd order ODE's of the form:
ay'' + by' + cy = 0
My texbook briefly outlines the...
hey I am having a little trouble with this topic. Here are the questions I was set.
a) Find the general solution of d^2y/dt^2 - 2(dy/dt) + y = 0. Verify your answer.
b) Solve the initial value problem y'' + 4y' + 5y = 0; y(0)= -3, y'(0) = 0
c) Find a DE that has the given functions as...
Quadratic Polynomial can't satisfy second order linear differential equation with constant coefficients... WHY?
and
if possible, how would I show this? Does it have to do with independence?
I need some help on finding the general solution.
I can find the complimentary solution, I'm having trouble finding the particular solution. Can anyone give me any tips.
y"+9y=t^2e^(3t)+6
y"-2y'-3y=-3te^-t
y"+2y'=3+4sin(2t)
A simple question i think although i can't find in any books
What do u when u are solving a second order linear hoemoeneous differential equation with frobenius and there is no shift.
(X^2)(y^{''}) (-6y)=0 it should be normal minus -6y
I only know what to do if there...
Hi ,
I am stuck with the following problem:
Find a second order linear homogeneous equation having the pair as a fundamental set of solutions:
y1(x)=x , y2(x)=x*ln(x).
My problem here is that I don't have the exponential form for the proposed solutions.
Thank you for your help
B.