# What is Second order linear: Definition and 49 Discussions

A second-order cone program (SOCP) is a convex optimization problem of the form

minimize

f

T

x

{\displaystyle \ f^{T}x\ }

subject to

A

i

x
+

b

i

2

c

i

T

x
+

d

i

,

i
=
1
,

,
m

{\displaystyle \lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i},\quad i=1,\dots ,m}

F
x
=
g

{\displaystyle Fx=g\ }
where the problem parameters are

f

R

n

,

A

i

R

n

i

×
n

,

b

i

R

n

i

,

c

i

R

n

,

d

i

R

,

F

R

p
×
n

{\displaystyle f\in \mathbb {R} ^{n},\ A_{i}\in \mathbb {R} ^{{n_{i}}\times n},\ b_{i}\in \mathbb {R} ^{n_{i}},\ c_{i}\in \mathbb {R} ^{n},\ d_{i}\in \mathbb {R} ,\ F\in \mathbb {R} ^{p\times n}}
, and

g

R

p

{\displaystyle g\in \mathbb {R} ^{p}}
.

x

R

n

{\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.

View More On Wikipedia.org
1. ### I Solve second order linear differential equation

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}...
2. ### I Solution:Second Order Linear Non-Homogenous ODEs in Physics

Hello, could someone please give me some examples of where order linear non homogenous ordinary differential equations are used in physics[emoji4]
3. ### I Question about second order linear differential equations

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.
4. ### Help Solving the following Differential equation

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...
5. ### Solving Second Order Linear Diff Eqns: Proving Theorems & More

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...
6. ### Second Order Linear D.E W/ Constant Coefficients and Zero RHS

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...
7. ### Construct a second order ODE given the solutions?

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...
8. ### Y′′=−20⋅4x^3, Second order linear ordinary DE

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...
9. ### Wronskian Equation for y1 and y2 with Initial Conditions

Homework Statement W(t) = W(y1, y2) find the Wronskian. Equation for both y1 and y2: 81y'' + 90y' - 11y = 0 y1(0) = 1 y1'(0) = 0 Calculated y1: (1/12)e^(-11/9 t) + (11/12)e^(1/9 t) y2(0) = 0 y2'(0) = 1 Calculated y2: (-3/4)e^(-11/9 t) + (3/4)e^(1/9 t)Homework Equations W(y1, y2) = |y1 y2...
10. ### MHB How Can I Solve This Second Order Linear ODE Problem?

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...
11. ### A question on second order linear equations

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-
12. ### Solutions of second order linear PDEs

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...
13. ### Second Order Linear Differential Equation

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...
14. ### Second Order Linear ODE - Power Series Solution to IVP

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...
15. ### MHB Jimmy Mai's ODEs Questions at Yahoo Answers

Here are the questions: I have posted a link there to this topic so the OP can see my work.
16. ### Solving this second order linear differential equation

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...
17. ### Second Order Linear Differential Equation (Non-constant coefficients)

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...
18. ### Sum of Second Order Linear PDEs

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...
19. ### System of second order linear coupled pde with constant coefficient

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?
20. ### MHB Solve Second Order Differential Eq. With Variable Coefficient

$(1-x^2)y'' - xy' + 4y =2 x \sqrt{1-x^2}$ 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
21. ### Solving second order linear homogeneous differential equation

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...
22. ### Solving second order linear homogeneous differential equation

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...
23. ### Obtaining particular solution of second order linear DE from first

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...
24. ### Second Order Linear Homogeneous DE

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...
25. ### Second order linear differential equation with constant coefficients

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...
26. ### Where does the general solution for second order linear ODEs come from?

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...
27. ### Finding the green's function for a second order linear DE and solve it

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...
28. ### Solving second order linear homogeneous differential equation HELP?

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...
29. ### Second order linear differential equation

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
30. ### Second order Linear ODE - NEED HELP to finish

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)=...
31. ### Second order Linear ODE

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...
32. ### Nonhomogeneous second order linear differential equations

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)...
33. ### Second Order Linear Nonhomogeneous Differential Equations

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...
34. ### Second order linear differential equations nonhomogeneous equations

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.
35. ### Second Order Linear Differential Equation Question

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...
36. ### Series Solutions of Second Order Linear Equations

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 ---->...
37. ### System of second order linear homogenous differential coupled equations

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.
38. ### Second order linear differential equations

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...
39. ### Second order linear system and power series: Differential Equations

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...
40. ### Therefore, y(t) = k1y1(t) + k2y2(t) is a solution to the given DE.

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...
41. ### Second order linear differential operator

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...
42. ### Solving Second Order Linear ODE with Parabolic Cylinder Functions

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.
43. ### Solving a Second Order Linear Differential Equation

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...
44. ### Homogeneous second order linear ODE

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...
45. ### Second Order Linear Homogeneous differential equations

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...
46. ### Second Order Linear Diff Eq

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?
47. ### Second order linear equations

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)
48. ### How to solve a second order linear homeogeneous ODE with Frobenius?

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...
49. ### Finding Second Order Linear Equation with x & x*ln(x) Solutions

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.