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.
Hi,
unfortunately, I have a problem to solve the following task
The equation looks like this:
$$\left(\begin{array}{c} \frac{d}{dt} x(t) \\ \frac{d}{dt} y(t) \end{array}\right)=\left(\begin{array}{c} -a y(t) \\ x(t) \end{array}\right)$$
Since the following is true ##\frac{d}{dt}...
Let ## \mathbf{x''} = A\mathbf{x} ## be a homogenous second order system of linear differential equations where
##
A = \begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
## and ##
\mathbf{x} = \begin{bmatrix}
x(t)\\ y(t))
\end{bmatrix}
##
Now to solve this equation we transform it into a 4x4...
I shall not begin with expressing my annoyance at the perfect equality between the number of people studying ODE and the numbers of ways of solving the Second Order Non-homogeneous Linear Ordinary Differential Equation (I'm a little doubtful about the correct syntactical position of 'linear')...
Hi all,
I have another second order ODE that I need help with simplifying/solving:
##p''(x) - D\frac{e^{\gamma x}}{A-Ae^{\gamma x}}p'(x) - Fp(x) = 0##
where ##\gamma,A,F## can all be assumed to be nonzero real numbers and ##D## is a purely nonzero imaginary number.
Any help would be appreciated!
I believe it is the case that any linear second order ode with at most 3 regular singular points can be transformed into a hypergeometric function. I am trying to solve the following equation for a(x):
where E, m, v, k_{y} are all constants and I believe turning it into hypergeometric form will...
Hello! Consider this partial differential equation
$$ zu_{xx}+x^2u_{yy}+zu_{zz}+2(y-z)u_{xz}+y^3u_x-sin(xyz)u=0 $$
Now I've got the solution and I have a few questions regarding how we get there. Now we've always done it like this.We built the matrix and then find the eigenvalues.
And here is...
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:
$$
\begin{aligned}
y_1 &= M(x, a, b) \\
y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\
\end{aligned}
$$...
Hello everyone,
I am struggling with calculating the coefficients for second order transient analysis.
For example, when analyzing a underdamped circuit, we know that the equation for voltage or current is xt=e-αt(K1cos(sqrt(ω2-α2)t ) + K2sin(sqrt(ω2-α2)t)).
Then in order to determine for...
i am new to MATLAB and and as shown below I have a second order differential equation M*u''+K*u=F(t) where M is the mass matrix and K is the stifness matrix and u is the displacement.
and i have to write a code for MATLAB using ODE45 to get a solution for u. there was not so much information on...
Any idea how to solve this equation:
## \ddot \sigma - p e^\sigma - q e^{2\sigma} =0 ##
Or
## \frac{d^2 \sigma}{dt^2} - p e^\sigma - q e^{2\sigma} =0 ##
Where p and q are constants.Thanks.
source
Change the second-order IVP into a system of equations
$y''+y'-2y=0 \quad y(0)= 2\quad y'(0)=0$
let $u=y'$
ok I stuck on this substitution stuff
I know the solution to the equation (1) below can be written in terms of exponential functions or sin and cos as in (2). But I can't remember exactly how to get there using separation of variables. If I separate the quotient on the left and bring a Psi across, aka separation of variables (as I...
We choose an approximative solution given by
$$
u_N(x) = \frac{a_0}{2} + \sum_{n=1}^N a_n \cos nx + b_n \sin nx
$$
Comparing this approximative solution with the differential equation yields that
$$
\frac{a_0}{2} = a
$$
and the boundary conditions yields the equation system
$$
a + \sum_{n=1}^N...
Question:
So I got around on doing this example, and I'm pretty sure I messed up somewhere, would appreciate if someone could point out what I did wrongly.
1) For any second ODE, I should let:
##y_{1}= y ## and ##y_{2}= y' ##
Hence,
##y_{1}'= y' = y_{2} ## and ##y_{2}'= y'' = xy(x)+x^2-y(x) =...
With the new variable, I got:
$$p^2 (p'_y)^{2}=k^2(1+p^2)$$ where ##p'_y## is ##\frac{dp}{dy}##.
I modified the equation so the variable p and dp can be separated from dy. Here what I got:
$$\frac{p}{\sqrt{p^2+1}} dp=k dy$$
I substitute ##p^2+1=u## so I got
$$\sqrt{u}=ky+c_1$$
Back substitution...
I use metric, which describes spacetime upto second order terms in rotation. It is solution of Einstein equations expanded upto second order. My query is, how to manipulate with such metric during calculations? Concetrly I make inverse metric, produce effective potential (ie, multiplying...
Hi,
Could you please help me to solve a second-order differential equation given below
∂M/r∂r+∂2M/∂r2 = A
[Moderator's note: Moved from a technical forum and thus no template.]
Goldrei's Propositional and Predicate Calculus states (in my words; any mistake is mine) that first-order logic is complete, i.e. any logic deduction from a set of axioms (written in first-order logic) is equivalent to proving the theorem for all models satisfying the axioms.
Completeness is...
I was reading about this 2nd order op-amp circuit which is essentially a cascaded integrator and got confused with the explanation of the book regarding the rate of change of the outputs. The book said that when the initial energy stored in the circuit is zero then
this rate of change is zero...
I was trying to understand the way this problem was solved and I got confused with the latter part of the solution. I encircled the part that confused me. They seem to contradict each other. If dv(0+)/dt = 0 why is it dv(0+)/dt = -1 in the other one? Please explain. TIA...
I have been struggling with a problem for a long time. I need to solve the second order partial differential equation
$$\frac{1}{G_{zx}}\frac{\partial ^2\phi (x,y)}{\partial^2 y}+\frac{1}{G_{zy}}\frac{\partial ^2\phi (x,y)}{\partial^2 x}=-2 \theta$$
where ##G_{zy}##, ##G_{zx}##, ##\theta##...
Homework Statement
Homework Equations
V(t) = V(∞)+( V(0+) - V(∞) )e^-t/τ
3. The Attempt at a Solution
Hello again! I've already solved the problem depicted in the picture above and below are the following unknowns that I managed to solved:
These results checked out with the answers...
At the end of a long proof I came across something in tensor calculus that seems too good to be true. And if something seems too good to be true ...
The something is that a second order partial derivative vanishes if one of the parts in the denominator is in the same reference frame as the...
I'm trying to solve the following nonlinear second order ODE where ##a## and ##b## are constants: $$\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}-\frac{y}{ay+b}=0$$ It looks somewhat like the modified Bessel equation, except the third term on the left makes it nonlinear. I've been trying to...
I have currently been reading a book called 'Mathematical Methods In Physical Sciences'. Whilest I was looking at the differential section I came across a differential which I have never thought about before, which is of the form...
I see comments such as "explains ... to the first order" or "to the second order" quite a bit in physics discussions. Can someone explain in lay terms, what first order and second order refer to?
Homework Statement
The question I am working on is the one in the file attached.
Homework Equations
y = u1y1 + u2y2 :
u1'y1 + u2'y2 = 0
u1'y1' + u2'y2' = g(t)
The Attempt at a Solution
I think I have got part (i) completed, with y1 = e3it and y2 = e-3it. This gives a general solution to the...
Homework Statement
Hi there, I have an assignment which involves using reduction of order to solve for a second solution to an ode (the one attached). However this is a method I am new to, and though I have tried several times, I'm somehow getting something wrong because the LHS and RHS are not...
$\tiny{3.1.1}$
find the general solution of the second order differential equation.
$$y''+2y'-3y=0$$
assume that $y = e^{rt}$ then,
$$r^2+2r-3=0\implies (r+3)(r-1)=0$$
new stuff... so far..
1. The problem statement, all variables, and given/known data
Given is a second order partial differential equation $$u_{xx} + 2u_{xy} + u_{yy}=0$$ which should be solved with change of variables, namely ##t = x## and ##z = x-y##.
Homework Equations
Chain rule $$\frac{dz}{dx} = \frac{dz}{dy}...
I tried to convert the second order ordinary differential equation to a system of first order differential equations and to write it in a matrix form. I took it from the book by LM Hocking on (Optimal control). What did I do wrong in this attachment because mine differs from the book?. I've...
I have the following system of PDEs:
\hat{\rho}\hat{c}_{th}\frac{\partial\hat{T}}{\partial\hat{x}}-\alpha_{1}\frac{\partial}{\partial\hat{x}}\left(\hat{k}(\hat{x})\frac{\partial\hat{T}}{\partial\hat{x}}\right)=\alpha_{1}\hat{\sigma}(\hat{x})\hat{E}...
Homework Statement
Use the power series method to solve the initial value problem:
##(x^2 +1)y'' - 6xy' + 12y = 0, y(0) = 1, y'(0) = 1##
Homework EquationsThe Attempt at a Solution
The trouble here is that after the process above I end up with ##c_{k+2} = -...
Homework Statement
We have a second order reaction: A + B → C + D with -rA = k[A][ B]
[A]0 = [ B]0 = 300 mol / m3; τ = 11 minutes.
k = 4.0 × 10-4 m3 / (mol × minutes)
What is the conversion in a CSTR?Homework Equations
I think:
τ = ([A]0 - [A]1) / -rA,1
τ = (XA × [A]0) / -rA,1
But since I...
Homework Statement
Consider a power series solution about x0 = 0 for the differential equation y'' + xy' + 2y = 0.
a) Find the recurrence relations satisfied by the coefficients an of the power series solution.
b) Find the terms a2, a3, a4, a5, a6, a7, a8 of this power series in terms of the...
Homework Statement
d2u/d2x + 1/2Lu = 0 where L is function of x
Homework Equations
I am try to find solutions y1 and y2 of this equation.
The Attempt at a Solution
y = [cos √(L/2) x] + [sin √(L/2) x]
y' = - [√(L/2) sin √(L/2) x] + [ √(L/2) cos √(L/2) x]
y'' = -[(L/2) cos √(L/2) x] -...
Homework Statement
If d^2/dx^2 + ln(x)y = 0[/B]Homework Equations
included in attempt
The Attempt at a Solution
I was confused as to whether I include the power series for ln(x) in the solution. It makes comparing coefficients very nasty though.
Whenever I expand for m=0 for the a0 I end...
Hi, I want to solve the p.d.e.:
##\frac{\partial u(x,t)}{\partial t} - \frac{\partial^2 u(x,t)}{\partial x^2}=f(x,t)##,
with periodic boundary conditions ##u(x,t)=u(L,t)##.
using a second order Runge-Kutta method in time. However, I am not having the proper results when I apply this method to...
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
A cantilever of length ##L## is rigidly fixed at one end and is horizontal in the unstrainted position. If a load is added at the free end of the beam, the downward deflection, ##y##, at a distance, ##x##, along the beam satisfies the differential equation...
Hello,
I have an equation of the form:
##\partial_t f(x,t)+a\partial_x^2 f(x,t)+g(x)\partial_xf(x,t)=0 ##
(In my particular case ##g(x)=kx## with ##k>0## and ##a=2k=2g'(x)##)
I'd like to know if there is some general technique that i can use to solve my problem (for example: in the first...
I have a physics project at university, we designed an experiment to measure the effectiveness of Poiseuilles law in a 'quasi non-steady state'. Poiseuilles law, simply being the measurement of the flow rate of a fluid in a pipe, holding only under steady state though. So by quasi steady state I...
Homework Statement
When a rocket launches, it burns fuel at a constant rate of (kg/s) as it accelerates, maintaining a constant thrust of T. The weight of the rocket, including fuel is 1200 kg (including 900 kg of fuel). So, the mass of the rocket changes as it accelerates:
m(t) = 1200 - m_ft...
Homework Statement
Given a set of fundamental solutions {ex*sinx*cosx, ex*cos(2x)}
Homework Equations
y''+p(x)y'+q(x)=0
det W(y1,y2) =Ce-∫p(x)dx
The Attempt at a Solution
I took the determinant of the matrix to get
e2x[cos(2x)cosxsinx-2sin(2x)sinxcosx-cos(2x)sinxcosx-...
My notes state that the method is constructed based on the idea:
yk+1=yk+∫f(x,y)dx where the integral is taken from xk to xk+1
We can estimate the integral by considering
∫f(x)dx (from xk to xk+1) =c0fk+c1fk-1
To simplify the equation, we move xk to the origin such that
∫f(x)dx (from 0 to h)...