In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
Today the inexact differential is usually denoted with δ, but in a text by a Russian author I found a dyet (D-with stroke, crossed-D) instead:
In response to my question to the author about this deviation from normal usage, he stated that this was a suggestion from von Neumann. (Which of course...
Hi,
for ease of reference this posting is segmented into :
1. Background
2. Focus
3. Question
1. Background:
Regarding (one, linear, second-order, homogeneous, ordinary, differential) equation describing the force in a non-driven, damped oscillation:
F = m.a = -k.x - b.v
F =...
I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using
$$dxdydz = \left (\frac{\partial x}{\partial r}dr +...
Let's say we have ##df=2xy^3dx + 3x^2y^2dy## - this is an exact differential.
In integrating, to find f, can we write ## f = \int 2xy^3 \, dx + \int 3x^2y^2 \, dy = 2x^2y^3 + C ##
Or am I getting it wrong?
From my interpretation of this problem (image attached), the force applied to the point charge is equal and opposite to the repulsive Coulomb force that that point charge is experiencing due to the presence of the other point charge so that the point charge may be moved at a constant velocity. I...
Hello,
let $$M^n \subset \mathbb{R}^N$$ $$N^k \subset \mathbb{R}^K$$
be two submanifolds.
We say a function $$f : M \rightarrow N$$ is differentiable if and only if for every map $$(U,\varphi)$$ of M the transformation
$$f \circ \varphi^{-1}: \varphi(U) \subset \mathbb{R}^N \rightarrow...
Hi! I'm asked to find all the non-zero Christoffel symbols given the following line element:
ds^2=2x^2dx^2+y^4dy^2+2xy^2dxdy
The result I have obtained is that the only non-zero component of the Christoffel symbols is:
\Gamma^x_{xx}=\frac{1}{x}
Is this correct?
MY PROCEDURE HAS BEEN:
the...
Homework Statement
Solve the following differential equations/initial value problems:
(cosx) y' + (sinx) y = sin2x
Homework Equations
I've been attempting to use the trig ID sin2x = 2sinxcosx.
I am also trying to solve this problem by using p(x)/P(x) and Q(x)
The Attempt at a Solution...
Homework Statement
Solve the following differential equations/initial value problem:
y^(4) - y'' - 2y' +2y = 0 Hint: e^-x sinx is a solution
Homework Equations
I was attempting to solve this problem by using a characteristic equation.
The Attempt at a Solution
y'''' -y'' -2y' + 2y = 0 -->...
I have seen how to solve the heat equation:
$$ \frac{ \partial^2 u(x,t) }{\partial x^2} = k^2 \frac{ \partial u(x,t) }{\partial t} $$
With boundary conditions.
I use separation variables to find the result, but i dont know how to solve the equation plus a...
I was doing the calculations for this: http://fermi.la.asu.edu/PHY531/cylinder/index.html
But I can't figure out how to go from \frac{d\sigma}{d\phi} to the total cross section. My guess was that you did the integral from \phi=0 to \phi=2\pi, but that's not helping since I can't tell either how...
Homework Statement
$$y = x.\log_e {\sqrt{x}}$$
Homework Equations
f(x) = g(x) h(x)
f ' (x) = g ' (x) . h (x) - h ' (x) . g(x)
The Attempt at a Solution
$$y = x .\log_e {\sqrt {x}}$$
$$y '(x) = 1.ln \sqrt{x} + \frac{1}{2} $$
the right answer is
$$ y ' = \log_{10} {\sqrt{x}} + \frac{1}{2} $$...
So I've seen in several lectures and explanations the idea that when you have an equation containing a relation between certain expressions ##x## and ##y##, if the expression ##x## approaches 0 (and ##y## is scaled down accordingly) then any power of that expression bigger than 2 (##x^n## where...
I need to solve:
\dot{\mathbf{r}}=-kv\hat{r} - \dot{\mathbf{r}_s}
However, I do not know how to deal with the fact that there is a unit vector. How can this be done? \dot{\mathbf{r}_s} is a constant vector.
I tried using the Bernoulli equation to solve this. I took two points at the surface of the water in both the containers and formed this equation:
gh_{b}=\frac{1}{2}v^2+gh
This is assuming that the velocity of the water in the large tank is approximately zero and using the fact that both the...
Homework Statement
Suppose at time zero, the bob was drawn upward four units from the equilibrium position, let C=2, K=2, m=1 lbm, initial speed=2 unit/sec find an expression for body's position.
and in the solution it says: y''+6y'+5y=0
my question is: from where does the numbers (6) and (5)...
Homework Statement
How does one show that q(t) is indeed a solution?
Homework Equations
The Attempt at a Solution
My current idea is that i should come up with any form of solution, like q = Acos(ωt), and slot it in the RHS.
Reason being that if q is indeed a solution, the result of the...
So in differential calculus we have the concept of the derivative and I can see why someone would want a derivative (to get rates of change). In integral calculus, there's the idea of a definite integral, which is defined as the area under the curve. Why would Newton or anyone be looking at the...
Homework Statement
It is the driven series RLC circuit. It is given in the following images.
It is from the section 12.3 in this note.
Homework Equations
The differential equation, as given by 12.3.3, is ##L \frac{d^2 Q}{d t^2} + R \frac{d Q}{d t} + \frac{Q}{C} = V_0 \sin{(\omega t)}##...
Homework Statement
http://imgur.com/a/k7fwG
Find the vector magnetic potential at point P1.
Homework Equations
Vector magnetic potential given by:
$$
d \bar{A} = \frac{\mu I d\bar{l'}}{4 \pi | \bar{r} - \bar{r'} | }
$$
The Attempt at a Solution
I split up the problem in 3 parts,
first...
Homework Statement
We assume from ODE theory that given a smooth A: I → gl(n;R) there exists a
unique smooth solution F : I → gl(n;R), defined on the same interval I on which
A is defined, of the initial value problem F' = FA and F(t0) = F0 ∈ gl(n;R) given.
(i) Show that two solutions Fi : I...
Homework Statement
The Frenet frame of a curve in R 3 . For a regular plane curve (and more generally for a regular curve on a 2-dimensional surface - e.g. the 2-sphere above) we could construct a unique adapted frame F. This is not the case for curves in higher dimensional spaces. Besides the...
Homework Statement
Let γ : I → R3 be an arclength parametrized curve whose image lies in the 2-sphere S2 , i.e. ||γ(t)||2 = 1 for all t ∈ I. Consider the “moving basis” {T, γ × T, γ} where T = γ'.
(i) Writing the moving basis as a 3 × 3 matrix F := (T, γ × T, γ) (where we think of T and etc...
Homework Statement
Find the values of a and b that make f a differentiable function.
Note: F(x) is a piecewise function
f(x):
Ax^2 - Bx, X ≤ 1
Alnx + B, X > 1
Homework Equations
The Attempt at a Solution
Made the two equations equal each other.
Ax^2 - Bx = Alnx + B
Inserting x=1 gives,
A -...
Homework Statement
Please bear with the length of this post, I'm taking it one step at a time starting with i)
Let A: I → gl(n, R) be a smooth function where I ⊂ R is an interval and gl(n, R) denotes the vector space of all n × n matrices.
(i) If F : I → gl(n, R) satisfies the matrix ODE F'...
Homework Statement
[/B]
Summarizing: two civilizations hate each other, one of the civilizations throws a curse at the second. The second civilization succumbs to chaos and has a change in Population each week of ΔP= -√P. That is:
Pn = Pn-1-√Pn-1
Homework Equations
[/B]
Considering that the...
Homework Statement
Prove the formula for inertia of a ring (2D circle) about its central axis.
Homework Equations
I = MR^2
Where:
M: total mass of the ring
R: radius of the ring
The Attempt at a Solution
- So I need to prove the formula above.
- First, I divide the ring into 4...
Homework Statement
How exactly they combined equation1 and equation2 and got that system? I don't get that part.
Homework Equations
A*(dy/dt)= -k*y eq1
A*(dz/dt)=ky-kz eq2
The Attempt at a Solution
I tried substituting the 1st ky in the 2nd equation and then differentiating but I don't...
Hi folks,
I was wondering if there are books that explain how to solve differential equations using infinite series. I know it is possible to do it since Poincaré used that method.
Do you know which ones are the best?
I find books on infinite series but they talk just about series...
I am having trouble to where to put the arbitrary constant in solving order differential equations. Because sometimes when I am solving, I can't really where to put the arbitrary constant if it is either the left side or right side.
I am only at equations of order one.
Plss keep it simple...
I can's understand the fact about the equation
i cant prove the equation from the first attachment to the second attachment pls help.
Sorry for bad english
In elementary calculus (and often in courses beyond) we are taught that a differential of a function, ##df## quantifies an infinitesimal change in that function. However, the notion of an infinitesimal is not well-defined and is nonsensical (I mean, one cannot define it in terms of a limit, and...
Homework Statement
Solve (x-1)y''-xy'+y=0 , given x>1 and y1=ex
Homework Equations
The 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...
Homework Statement
Solve the differential equation, dt/dv= 1/g (1/1-a^2*v^2) where a = (k/mg)^1/2 to yield v= 1/a(1-e^-2agt/1+e^-2agt).
Homework Equations
F=ma
Newton's 2nd Law
Integration Laws
The Attempt at a Solution
See image. I think I'm getting messed up on the integration laws.
Homework Statement
Suppose that f has an inverse and f(-4)=2, f '(-4)=2/5. If G= (1/f-1) what is g '(2) ?
If it helps the answer is (-5/32)
Homework Equations
[/B]
f-1'(b)=1/(f')(a)
The Attempt at a Solution
Im not really sure how to start this problem. I am familiar with how to use the...
In the first image it shows the ##\alpha^2-w_0^2<0## situation whereas in the second image the situation is when ##\alpha^2-w_0^2=0##.The problem is the book says to use ##T_0=2\pi/w_0## to determine diabetes but you can't do that when ##\alpha^2-w_0^2=0## because it can't be put into a cosine...
Greetings all,
In my quest to imagine the ultimate sportscar, I began to wonder about the following:
Certain cars are All Wheel Driven (AWD) and they do so via a Center differential delivering power to both axles. Amongst those cars, some have fixed Torque Bias/Split Ratio, which doesn't alter...
Homework Statement
The problems are in the uploaded file. 18) satisfies the differential equation
y''+p(t)*y'+q(t)*y=0
p(t) and q(t) are continuous
Homework Equations
Wronskian of y1 and y2
The Attempt at a Solution
18) I don't really get this one
19) Solved most but at the end, where I...
Hello guys, I found this forum using google because I need help with simulating bungee jump model.
I've already done that with zero initial values and it looks good but I want it more realistic :
Let's say L is the lenght of rope so elasticity force starts acting when y=L so logically time...
Chasing a ghost perhaps but in the process of pressure testing a simple propane furnace installation I installed a water Column pressure gauge. Over a period of 11 hours the pressure gauge reading will go from 10" WC to 0" WC. In the course of trying to understand this I installed a manometer...
Good evening I have these coupled equations and was wondering if there is any chance solving them analytically. If not, how would you approach it numerically? (shown in attachment)
Thank you very much
Homework Statement
Hi there. I have a simple RC circuit with a battery voltage of 10 V, R = 1 Ω, C = 1 F and a switch. I want to use the Explicit Euler (forward divided difference) to solve the equation and check for stability, rather than using a ODE. I am finding the equation for when the...
Euler was the master in analysisng anything. This can be seen in his words in the preface of his book "Mmathematica" (translated by Ian Bruce), where he speaks on the text of Hermann "Phoronomiam":
Euler has given many insightful words on analysisng things in his preface of many other books...
Homework Statement
Homework Equations
F=ma
ac=v^2/r
f=uN
v=v0+at
w=v/r
The Attempt at a Solution
v=v0+at
v=vo+umv^2/r
v^2(u/r)-v+vo=0
I don't see what differential equation i could use since the speed is dependent on the friction (equal to friction coeff times centripetal force) which in...
Hi guys,
I was browsing in regards to differential equations, the non-linear de and came up with this site in facebook:
https://www.facebook.com/nonlinearDE
Are these people for real? Can just solve any DE like that, come up with a series? Not an expert in this area, so I do not know what if...
Homework Statement
http://imgur.com/goozE9f
Homework Equations
##(dx_i)_p i= 1,2,3##
3. The Attempt at a Solution [/B]
I'm reading Manfredo and Do Carmo's Differential Forms and Applications. This is the very first page
Would you mind explaining me what is meant by dx, as highlighted in the...