What is Differential: Definition and 1000 Discussions
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.
Homework Statement
I believe I have solved this differential equation, yet do not know how the book arrived at it's answer...
Solve the differential equation in its explicit solution form.
The answer the book gives is...
Homework Equations
Separable Differential Equation
The Attempt...
Homework Statement
Solve the differential equation, explicitly.
dy/dx = (2x)/(1+2y)
The answer given by the book is...
-1/2 + 1/2sqrt(2x - 2x^2 +4)
Homework Equations
Process for solving separable differential equations
The Attempt at a Solution
dy/dx = (2x)/(1+2y)
(1 + 2y)*dy = 2x*dx...
Homework Statement
[/B]Homework Equations
The Attempt at a Solution
I've highlighted two equations on the screenshot. How did it proceed from the first to the second? I'm actually confused with the absolute values. What is the idea behind getting rid of the first absolute value(1-5v^2) while...
To motivate the question, Andy Strominger recently put out a paper on calculating the Sagnac shift of counterrotating beams due to the angular momentum flux of a passing gravitational wave.
See here: http://arxiv.org/abs/1502.06120.
But consider now two nearby freely falling gyroscopes...
Hello
I have a system of differntial equations:
dx/ds = sin(p)
dy/ds=cos(p)
dp/ds = k
dk/ds = -1/EI(s)*(k*dEI/ds+f*sin(p))
x(0)=y(0)=p(0)=p(L)-pl = 0
These are nonlinear differential equations. I should use some sort of nonlinear finite difference. But I do struggle to setting up the finite...
Got a bit of a long and nutty question here.
So I got a nutty question from my Calc 1 class and was wondering if anyone could help me out
A section of road, represented by the line y = x + 4 when x ≤0, is to be smoothly connected to another section of road, represented by y = 4 –x when x ≥4...
I will try to explain this with an analogy.
Let's have this equation:
x^2 =9
And let's assume I don't know algebraic methods to solve it, so I create a list using excel with different values. And I see that if I put x=4 it doesn't work, if I put x=5 it is even worse and so on. But If I put...
Homework Statement
http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-003-signals-and-systems-fall-2011/assignments/MIT6_003F11_sol01.pdf
http://i.imgur.com/8qj5cWE.png
#6 Part B
Homework Equations
None[/B]The Attempt at a Solution
Don't understand why we care about...
Hello, I am a math major and I was wondering if you guys knew what would be a good rigorous differential equations text. I really like rigor (like Rudin analysis style rigor or whatnot), instead of the typical books that just focus on the method. I want the proofs and everything. I also would...
Hello! (Wave)
If we have the initial value problem
$$h''(t)=-\frac{R^2 g}{(h(t)+R)^2} \\ h(0)=0, h'(0)=V$$
we have the fundamental units: $T,L$ such that $T=[t], L=[h], L=[R], LT^{-1}=[V], LT^{-2}=[g]$ and we get the independent dimensionless quantities $\pi_1=\frac{h}{R}...
Hey all! I have been trying this problem for a while and can't seem to get the same answer as the solution. If someone can tell me where I am going wrong, that would be much appreciated. I am very close to the solution, but I am missing a term in the denominator. Everything is shown below! Thank...
Hello,
I wanted to study the behaviour of electrons in a spatially bounded system. I want to have a larger number of electrons, but I took 3 to start with and arrived at this system of coupled equations:
\begin{align}\begin{bmatrix}
\mathbf{\ddot{x_{1}}}\\ \\
\mathbf{\ddot{x_{2}}}\\ \\...
I've been struggling since starting to study differential geometry to justify the definition of a one-form as a differential of a function and how this is equal to a tangent vector acting on this function, i.e. given f:M\rightarrow\mathbb{R} we can define the differential map...
Hello! (Wave)
$$(1-x^2)y''-2xy'+p(p+1)y=0, p \in \mathbb{R} \text{ constant } \\ -1 < x<1$$
At the interval $(-1,1)$ the above differential equation can be written equivalently
$$y''+p(x)y'+q(x)y=0, -1<x<1 \text{ where } \\p(x)=\frac{-2x}{1-x^2} \\ q(x)= \frac{p(p+1)}{1-x^2}$$
$p,q$ can be...
Homework Statement
I am trying to show that ##a(x)[u(x),D^{3}]=-au_{xxx}-3au_{xx}D-3au_{x}D^{2}##, where ##D=d/dx##, ##D^{2}=d^{2}/dx^{2} ## etc.Homework Equations
[/B]
I have the known results :
##[D,u]=u_{x}##
##[D^{2},u]=u_{xx}+2u_{x}D##
The property: ##[A,BC]=[A,B]C+B[A,C] ##*The...
Homework Statement
Find the differential of axb
Homework EquationsThe Attempt at a Solution
Really not sure where to start, honestly. Thanks in advance :)
It has been a while since I was involved with my differential equations. I am a mech student. I was trying out a sample problem from the dynamics book and came upon this equation.
k = some constant of proportionality for a spring pushing back a spring mounted slider.
(1/k) arcsin(ks/v_0) = t...
Hello! (Wave)
I want to find the solution $\psi$ of the non-linear differential equation $y'=1+y^2$ that satisfies the condition $\psi(0)=0$. (Notice that the solution $\psi$ exists only for $- \frac{\pi}{2}< x < \frac{\pi}{2}$)
We notice that: $(tan^{-1})'(x)=\frac{1}{1+x^2} (\star) \left(...
Not a homework question, I'm just curious, I know this has been asked a few times, but what exactly is happening when you move dx over to the other side of dy/dx=f'(x), is it like the point slope form ∆y=m∆x, or is it applying the differential to both sides, I've always been told a rigorous...
Homework Statement
I have found the two equations describing the system. They are.
ċ(t) = 1/θ [ f`(k(t)) -(n + δ + β)]c(t)
k̇(t) =f(k(t) - (n +d)k(t) - c(t)
Plugging in the numbers:
ċ(t) = 2 [ 0,25k^(-0,25) -(0,10)]c(t)
k̇(t) =k^0,25 - (0,6)k(t) - c(t)
Since ċ(t) and k̇(t) =0 in steady...
Hi all, first post here. I'm a junior Physics/Math double major at UMass Amherst, playing with some problems over the summer. I'll get right into it.
A rope with constant tension T is deflected through the angle 2\theta_{0} by a smooth, fixed pulley. What is the force on the pulley?
It is...
Question:
Evaluate the surface integral
$$J = 2xzdy \land dz+2yzdz \land dx-{z}^{2}dx \land dy$$
where S \subset {\Bbb{R}}^{3} is the rectangle parametrised by:
$$x(u,v) = 1-u,\ y(u,v) = u,\ z(u,v) = v,\ \ 0\le u, v \le 1$$
so far I have:
\begin{array}{}x = u\cos v, &dx = \cos v\, du -...
Homework Statement
dv/dt = 9.8 - 0.196v
Set in correct form:
dv/dt + 0.196v = 9.8
Since p(t) = 0.196, u(t) the integration factor is given by:
u(t) = e∫0.196 dt
Multiply each term by u(t) and rearrange:
(e∫0.196 dt)(dv/dt) + (0.196)(e∫0.196 dt)(v) = (9.8)(e∫0.196 dt)
From now on we will set...
Homework Statement
dX = U/V dV + U/p dp
Write the differential of X in terms of the independent variables.Prove that this is an exact differential.Use the ideal gas equation of state to verify that X is actually the internal energy and that it satisfies the above equation. Would...
Homework Statement
you are given a family of curves, in this case i was given a bunch of circles x^2+y^2=cx, sketch these curves for c=0,2,4,6, both positive and negative, solve the equation for c and differentiate both sides with respect to x and solve for dy/dx. You obtain an ODE in the form...
hello all.glad to be here!
please in the attached document, column 5 and 6 which represents the differential and cumulative frequency distributions were obtained using equations (5) and (6). I'll be mighty grateful if anyone can give me a step by step breakdown of how it was done?? Many thanks...
A question from a classical mechanics past paper described a particle of mass
##m## that had a pair of horizontal identical springs of spring constant ##k## attached on either side and that the mass is free to move horizontally. The mass is also placed on a table that gives rise to an...
To be honest, I don't know any physics. I am a high school student who has taken high school physics, but America's education system isn't known for teaching much more than Newton's laws. I have, however, taken Multivariable/Vector calculus, so I have a decent math background.
I was wondering...
It is a general doubt about the following equation: Imagine I want to calculate an unknown function y(x), and my starting equation is of the type
y(x)^{2}=\frac{1}{x^{2}Log^{2}(A(x)y(x)^{2})}
, then, am I allowed to start with the equation
y(x)=\frac{1}{xLog(A(x)y(x)^{2})}
and...
Homework Statement
Solve the system using differential operators. Determine the # of arbitrary constants and then compare to your solution.
Homework Equations
D substitution: replace x' with Dx and y' with Dy
The Attempt at a Solution
I have the solution to this one, but I'm working...
Homework Statement
1. y" + y = tanx, solve this DE.
2. dP/dt = P(1 - P) where P = (c1et / 1 + c1et), verify that P is a solution to this DE.
3. Given the pair of functionsx and y, show they solve this system:
dx/dt = x + 3y
dy/dt = 5x+3y
x = e-2t + 3e6t
y = -e-2t + 5e6tThe Attempt at a...
Homework Statement
I have two equations.
cos(θ)wφ + sin(θ)wφ = 0 (1)
And
## \frac{w_φ}{r}## + ∂wφ/∂r = 0 (2)
Find wφ, which is a function of both r and theta.
Homework EquationsThe Attempt at a Solution
I end up with two equations, having integrated. wφ=## \frac{A}{sinθ}## from (1)...
Homework Statement
A long cylindrical wire of radius R0 lies in the z-axis and carries a current density given by:
##j(r)= j_0 \left( \frac{r}{R_0} \right)^2 \ \hat{z} \ for \ r< R_0##
##j(r) = 0 \ elsewhere##
Use the differential form of Ampere's law to calculate the magnetic field B inside...
Homework Statement
The website says this:
"It is Linear when the variable (and its derivatives) has no exponent or other function put on it.
So no y2, y3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is).
More formally a Linear Differential Equation is in the form:
dy/dx +...
Homework Statement
data[/B]
Solving differential k forms.
Homework Equations
I don't want to give any exact problems from my problem set.
The Attempt at a Solution
solution.[/B]
The text I'm using, CH Edwards, is very abstract in this section and the explanation over a sped up, last class...
I don't understand this first order differential equation:
https://lh5.googleusercontent.com/UUpQF4YjmjJRPvFuzGg2MhpMMMDyi2KFZPCKMKVIXGREc1owvXDzGR0bcA=s600
How is it possible to get an exponent as answer?
Homework Statement
A particle moves in a straight line with velocity given by ## dx/dt = x +1 ## ( x being distance described). The time taken by the particle to describe 99 meters is?
Homework Equations
NA
The Attempt at a Solution
Getting ## ln(x+1) = t + C##
How to determine the constant...
1. (16+x2)-xy'+32y=0
Seek a power series solution for the given differential equation about the given point x0 find the recurrence relation.
So I used y=∑Anxn , found y' and y''
then I substituted it into the original equation, distributed, made all x to the n power equal to xn, made the...
Homework Statement
I am trying to solve this
\begin{align}
d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t
\end{align}
where $b$ is a constant.
Note that I have the answer here and can provide it if necessary. But I want to know how one would come up with it.
Homework EquationsThe...
My school requires me to take a calc 3 course and differential equations for my major. I am scheduled right now to take differential equations in the summer, and calc 3 in the fall of this year. My school recommends they be concurrent classes, but I'm just too strapped down by my already almost...
Homework Statement
find the inverse laplace transform.
1/(s^3 + 7s)
Homework Equations
sin(kt) = k/(s^2 + k^2)
cos(kt) = s/(s^2 + k^2)
The Attempt at a Solution
so this one is different from all the others I have done because it involves an imaginary number and I am not sure what the rule...
1. I am supposed to find dx and dy. I think I am missing a step or a general idea. I spent quite some time figuring out what rules should I use and the only sequence I can think of is quotient rule and chain on the (x2 +y2)1/2 term. The answer that I find is ((xy(3x2+2y2))/(x2+y2)3/2 . On the...
The equation looks like: ##x''(t)+b x'(t)+cx(t)+ d x^3(t)=0##. This is the motion of a particle in a potential ##cx^2/2+d x^4/4## with friction force ## b x'##. In my case, the friction term is very small and the particle will oscillate billions of times before the magnitude decreases...
Homework Statement
Solve d2θ/dη2 + 2η(dθ/dη) = 0, to obtain θ as a function of η,
where θ=(T-T0)/(Ts-T0)
EDIT: I should add that this is a multi-part problem, and η is given as η=Cxtm. We had to use that to derive the equation in question above.. So I don't know if this is supposed to be...
Homework Statement
y3(dy/dx) = (y4 + 1)cosx
2. The attempt at a solution
I solved for the homogeneous equation which is y = Ce-sinx Where C is some constant
for the particular solution I tried Asinx + Bcosx where A and B are constants but when subbing in it's gets very messy.
How should...
Homework Statement
Calculate gain (u_o/u_i) for a differential amplifier with symmetric output. Both transistors have the same transconductance gm. Transistor output resistance r_0 is neglected here.
circuit: http://sv.tinypic.com/view.php?pic=14cea8p&s=8#.VSf7Evl_vxM
Homework Equations...
Homework Statement
Consider a system composed of two species X and Y with fractional populations x and y, respectively, where x+y=1. The two species interact in such a way that the differential equation for x is:
\begin{equation}
\frac{dx}{dt}=xyA_{0}e^{-\alpha t}
\end{equation}
where $A_{0}$...
Homework Statement
The following series of differential equations represents a projectile's path when solved (g=9.81):
Modify this series of differential equations to account for an additional force F with vector components a and b acting on the projectile.
Here is a sample plot of this...