What is Second order: Definition and 602 Discussions
In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence
∀
P
∀
x
(
P
x
∨
¬
P
x
)
{\displaystyle \forall P\,\forall x(Px\lor \neg Px)}
says that for every formula P, and every individual x, either Px is true or not(Px) is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables as explained in the section Syntax and fragments. Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified.
Hey,
I'm studying Predicate Logic at the moment and I can't seem to wrap my head around the way that english sentences would convert into second order logic. What kind of sentence can be faithfully represented in PL2 but not in PL1? Sorry if this isn't the appropriate section; I'm actually in...
Homework Statement
Find the Particular Solution to the differential Equation satisfying the initial conditions.
y''+7y'+10y= -30
y(0)= 3
y'(0) = -27Homework Equations
Characteristic Equation for Homogenous Solution
y''+7y'+10y=0
roots are -2 and -5
General Solution is
C1e^-2t + C2e^-5t...
I am a Matlab rookie. I need to solve numerically the following second order differential equations
d^2x/dt^2 + w0_(el) * x = e/m_e * E - K3/m_e * x *y;
d^2y/dt^2 + w0_(v) * y = - K_3/2M * x^2;
I have started to deal with only the harmonic part of the problem. So I tried to solve...
Hi all,
I've been doing a lot of thinking and I was wondering precisely how the 2nd order correction to the wave function from perturbation theory is derived:
I mean, I can see where bits and pieces come from and I've tried to work through it as an exercise. Does anyone have a...
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.
Hello forum,
I am trying to solve a differential equation for the last four hours and I can't figure out how...
here it is
\frac{d^2x(t)}{dt^2} + \frac{dx(t)}{dt} + c x(t) = d e^{-a t^2}
actually my problem is how to handle the Gaussian term...
if anyone can help please...
Hello all,
I'm having a little trouble getting the Latex to work, so I'm hoping this won't be too hard for everyone to understand.
Homework Statement
I am given the second order differential equation
x2*y''(x)+(2*b+1)*x*y'(x)+c*y(x) = 0
Use the transform x=ez to find the general solution...
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
Let V denote a vector space of twice differentiable functions on R. Define a linear map L on V by the formula:
L(u)=au''+bu'+cu
Suppose that u_{1},u_{2} is a basis for the solution space of L(u)=0. Find a basis for the solution space of the fourth order equation L(L(u))=0...
Homework Statement
Here is my issue. I am an EE student that has been part time for almost five years. I took diff eq and signals a long time ago. I am currently taking a feedback and controls class, and my lack of memory in these topics is killing my study time. I am looking for relevant...
Could you please help me or give me any hint to solve this ODE..
\frac{d^2y}{d x^2} + ( 2\rm{sech}^2 x - a^2 ) y = 0
where a is a constant.
I want only even function solution. (y(x) = y(-x))
BTW, this is a homework problem. I encountered this equation while considering surface waves...
I'm trying to solve equation in the attached pdf, which describes anistropic diffusion in 3D with an additional term to account for hydrogen bonding and unbonding of the diffusing substance to the medium. I've considered Laplace transforms, then solving in the Laplace domain, then inverting...
Homework Statement
Evaluate
\int^{2\pi}_{0}\frac{1}{r}\frac{d^{2}f}{d\varphi^{2}}d\varphi
Homework Equations
n/a
The Attempt at a Solution
I get the integral as
\frac{1}{r}\frac{df}{d\varphi}
Not sure how to evaluate this.
Hey guys, I am new to PF. I need to be able to model a stiff differential equation in MatLab. I haven't used MatLab before so I am not really sure how to set the function and boundary conditions for the equation:
y'' + (2/x)*Y' = (.7/x^2)*( (y^(-1/2)) - (.067)((1-y)^(-1/2) )
y(0)=0...
Hi guy,
I have this ODE that I'm having problems with
y"+4y'+4y= e^(-2x)logx
Now, Using method of UC to get rid of the RHS I've tried using Ae^(-2x) x^2 logx
However, I'm not quite sure whether that is correct or not as I have never had a question containing logs before
Hello
I am going to post here a problem that I've been working on for quite some time, and I haven't managed to obtain a good answer.
To approximate free-field conditions in the numerical solution of wave equation PDEs, the following coordinate transformation is often applied to implement...
Hi,
I have a differential equation of the form
d2 x
---------------- = g/z * x(t)
dt2
Here g and z are constants. So, this is a 2nd order ODE which has a closed form solution.
In fact, i know the solution for x in terms of cosh and sinh functions.
In the above differential...
Hi. I believe I understand separation of variables for a first order DE. But can anyone tell me how to use it on a second order DE? In particular I have been looking at this example
http://en.wikipedia.org/wiki/Integrating_factor#General_use"
where it is claimed that one can use separation of...
Homework Statement
Derive the second order differential equation relating x(t) and y(t).
Using the Laplace transform, find the total response as a function of the zero input response and the zero state response in the following form.Homework Equations
Y(s)=Yzs(s) + Yzi(s)
The Attempt at a...
Homework Statement
d2y/dx2-dy/dx+y*exp(2x) = x*exp(2x)-1
substitute t=exp(x) and set z(t)=y(x) and rewrite hence find all solutions
The Attempt at a Solution
Rewriting gives:
d2z/dt2-dz/dt+z*t^2=(ln(t) * t^2) - 1
however I don't see how this in any way helps us...
Hi, I'm trying to solve this exercise:
0=<x=<a
D*C''(x)= A*C(x)/ [K+C(x)]
D, A and K are constants
and we know the following IV:
C(0)= Known (number)
C''(a)=0
Do you have any idea how I may proceed with this?
Ly ≡ (x +1)^2y′′− 4(x +1)y′+6y =0
given y[1]=(x+1)^2 is a solution, use the ansatz y2(x)= u(x)(x+1)2 to reduce
the order of the differential equation and find a second independent solution y2
how to reduce !? and i can't find u ...can't solve (x+1)^2u''+6u=0
please help!
thx!
Hi, I just had my ordinary differential equation final exam today (summer class), the exam was fine. We had a bonus question though, it was a nonlinear second order differential equation, we didn't learn anything about those except for one existence and uniqueness theorem. I am curious if I got...
Homework Statement
Substitute p = \frac{dx}{dt} to solve x\prime\prime + \omega^2x = 0 Homework Equations
\frac{dp}{dx} = v + x\frac{dv}{dx}
v = \frac{p}{x}
The Attempt at a Solution
p = \frac{dx}{dt}, \frac{dp}{dt} = \frac{d^2x}{dt^2}
\frac{dp}{dt} = \frac{dp}{dx}\frac{dx}{dt} =...
Homework Statement
It will be great if someone could show me some options:
I need to prove the following:
A particle of mass m is attracted toward a fixed point O (the origin) with a force proportional
to its instantaneous distance from O raised to a positive integer power, i.e...
Hi,
I am sure I have mistaken my DE setup or my initial conditions assumption.
Homework Statement
http://ivila.net/E8.png
R1 will be the middle Resistor
R2 the top one.
Homework Equations
i1' + R1/L i1 = R1/L i2
i2(R1+R2)*1/(R1) + 1/(R1*C)*integral(i2)=i1
The Attempt at a...
second order nonlinear ODE -- not autonomous
This equation has arisen from a steady state problem of diffusion with nonlinear reaction:
(dependent var=c; independent variable=x)
c'' = ko + k1 c + k2 c^2 + k3 c x
ko, etc are constants.
I can obtain a solution if I drop the last term...
Homework Statement
Solve the following equation
f'' - 3f' + 2f = 3
For
f(0) = 0,\ and\ f'(0) = 1
Homework Equations
None.
The Attempt at a Solution
Ok, so, I know how to solve inhomogeneous ODEs when it's on this form
af'' + bf' + cf = g
Where g is a given function.
But how do I...
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
---->...
show that y1(x) = e^(2+i)x and y2(x) = e^(2-i)x, i=sqrt(-1) are two linearly independent functions
hence obtain a second order linear differential equation with constant coefficients each that y1(x) and y2(x) are its two fundamental solutions.
my attempt :
for the first part, I use the...
Hello All,
I am stuck on the following question. Can you please help to find the solutions
Using the complementary function and particular integral method, find the solution of the diffential equation which satisfies y(0) = 1 and y'(0) = 0.
y'' + 3y' + 2y = 20cos2x
and then can you...
ok, for 2nd order differential equations, if we have repeat roots, like for example y" +4y = cos2x
, we would have repeat roots +2i,-2i.
so how do we try for the particular solution?
i tried Acos2x + Bsin2x, it all canceled out.
i tried (Ax+B)cos2x + (Cx+D)sin2x it all canceled out too
then...
I need to solve the following system of differential equations:
\frac{\partial^2 y}{\partial t^2} + A\frac{\partial y}{\partial t} - B \frac{\partial^2 y}{\partial z^2} = Cq
\frac{\partial^2 q}{\partial t^2} + D\frac{\partial q}{\partial t} + q = E\frac{\partial^2 y}{\partial t^2}...
Homework Statement
if given the general solution to a differential equation
y(t)=a cos t + b sin t + ((cos wt)/1-w^2),
Find out how long it takes the solution to approach the steady state solution.
original second order differential equation:
y'' + y = cos wt such that 0 < w < 10...
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...
Hi everybody,
I'm reading about sigma delta adc, the base theory. I've read about the first order modulator and now the second order, but I have a problem (maybe stupid) with a phrase of my book (Data Converters by F.Maloberti), when he describe the 'passage' from first order to second order...
Homework Statement
Please see attachment for block diagram of the system.
Starting with the block diagram given in Figure 2 derive an expression for the overall transfer
function of the system, where Gc(s) is in the form given by Eq. (1). Show that the system is a
Second Order system...
Hello.
I took a class on ODEs and learned about solving second order homologous equations by writing down the characteristic equation.
http://www.sosmath.com/diffeq/second/constantcof/constantcof.html
I am now learning about PDEs on my own and I came across parabolic, hyperbolic, and...
Homework Statement
Show that:
\theta (\xi) = 1-\frac{1}{6} \xi^2 + \frac{n}{120} \xi^4 +...
is a general solution to the Lane-Emden equation. (assuming that the above equation converges)
Homework Equations
Lane-Emden equation:
\frac{1}{\xi^2}\frac{d}{d \xi} \left ( \xi^2...
Homework Statement
y''[x] = y'[x] + x
Homework Equations
We were taught two special types of second order diff. equations:
Type 1: Supposed to be when x is missing
v = y'[x]
v'[x] = y''[x]
Type 2: Supposed to be when y is missing
v = y'[x]
v v'[x] = y''[x]
The Attempt at...
Hello, I was wondering if anyone could shed some light upon solving this:
s(x)'' = (a b s(x)) / ||s(x)||^3
Where s is a n dimensional vector, || || is operation finding the magnitude and a and b are constants.
Is this solveable or will I have to use alternative numerical methods?
Out of boredom* I derived some simplified equations to describe the dynamics of a pneumatic gun. These equations are extremely simplified, but they should provide some insight into the system that my numerical models can't. This system had one non-linear equation that could be rewritten as the...
I want to write a program in C that's going to solve 3 second order differential equations, and I have little to no experience in solving differential equations in C (however, I know C well enough). I'm having a bit of trouble.
First of all, I don't really know what method I'm going to use to...
Homework Statement
What is the general solution to y'' - 2(y')^2 = 0 ?
Homework Equations
The Attempt at a Solution
Let u = y '
u ' - 2u^2 = 0
u ' = 2u^2
u = (2 / 3)u^3 + C
This cannot be solved using separation of variables, what is done?