What is Differential equations: Definition and 999 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.
I am trying to figure out what I am doing. Forgive me, I’m not very sharp when it comes to differential equations… or even a useful working knowledge of calculus. I understand the concepts but if I can’t figure out how to type it into my TI-89 or get Wolfram alpha to do it… I’m kind of stuck and...
Homework Statement
Show that sines and cosines are the solutions of the differential equation
f''(x) = (-σ^2)f(x)
What if a boundary condition is included that g(0) = 0?
Homework Equations
f''(x) = (-σ^2)f(x)
The Attempt at a Solution
Plugging in sin(σx) and cos(σx) yields an...
Homework Statement
y = 2xy' + y(y')2; y2 = C1 (x + C1 /4)
2. The attempt at a solution
I thought that I could take the first equation and set it equal to zero. But the C1 in the second equation is throwing me off.
Am I supposed to set the second equal to C1?
Hi I am a physics major and I am trying to double major in math,I will be taking a differential equations course for pure math students this semester. I need advice on a good textbook in Differential Equations for pure mathematicians that will build up a great intuition for proofs and will have...
I'm having some difficulties figuring out how to linearize second order differential equations for a double pendulum.
I have an equation that is in the form of
\theta_{1}''\normalsize = function [\theta_{1},\theta_{2},\theta_{1}',\theta_{2}']
(The original equation is found at...
I'm only in a high school class. I just made up this problem to see if I'm getting the right idea in the long run. That, and then there's the fact that differential equations are SO MUCH FUN! :biggrin:
Homework Statement
Molecular bromine (initial concentration: 0.49 M) and formic acid (initial...
Homework Statement
http://sphotos-d.ak.fbcdn.net/hphotos-ak-ash3/526007_3920535257917_1525052730_n.jpg
Write down the differential equations of motion.
(Step by step if you can)2. The attempt at a solution
x"+(f/m)x'+3(k/m)x=0
or
x"+σx'+ω^2x=0
where σ=f/m and ω=sqrt(3k/m)
Thanks in advance.
Can anyone recommend a good differential equations and linear algebra book? The books that I've read so far are
Linear algebra:
https://www.amazon.com/dp/0030103479/?tag=pfamazon01-20
Differential Equations:
https://www.amazon.com/dp/0534373887/?tag=pfamazon01-20
I was able to get an...
Homework Statement
Solve this system of differential equations
\begin{equation}
x'_1=5x_1 + 2 x_2 - x_3 \\
x'_2=-2x_1 + x_2 - 2x_3 \\
x'_3=-6x_1 - 6 x_2
\end{equation}
Homework Equations
The Attempt at a Solution
This is my first time solving a problem like this and I just...
Homework Statement
How do you derive the time-dependent velocity equation for motion along a curve, such as a skateboarder on a half pipe?
For the sake of abstraction, I ask myself the following:
A uniform sphere of mass m and radius r is set free from the top edge of a semicircle half pipe...
Homework Statement
Here is the question along with my work. I attempted to solve for the actual solution using both eigenvectors. From what I have been taught it should yield the same answer... But as you can see (circled in red) the solutions are clearly different. Is this normal or maybe...
Hi guys, in my college we have this "Independent Study" component for the Intro Linear Algebra class. Basically, I have to solve 4 questions that go beyond what was covered in class. Sadly, being the procrastinator I am, I haven't looked at it until now and the assignment is due Friday. The...
Hi. I just came back from my differential equation midterm and was surprised to see a problem with the Riemann-zeta equations on it. I think the problem went something like
"Prove that \pi/6 = 1 + (1/2)^2 + (1/3)^2 + ... "
The study guide did mention that "prepare for a problem or two...
Hi,
I searched the forums but was unable to find any thread about DE textbooks so I'll start one. I'm looking for a good DE textbook, mostly focusing on ODEs, something comprehensive and complete and not too old fashioned. Does anybody have any recommendations?
Thanks.
Homework Statement
tx'' + (4t-2)x' + (13t-4)x = 0
Use laplace transform to solve.
The Attempt at a Solution
I've split up the X'(s) and X(s) and integrated to get X(s) = \frac{c}{(s^2 + 4s + 13)^2} = \frac{c}{9} \left[ \frac{3}{(s+2)^2 + 9} \right]^2
From this I'm guessing...
Homework Statement
Determine all fixed points of:
dx/dt = x(β-x-ay)
dy/dt = y(-1+ax-y)
β and a are parameters.
I get what to do when there is just one differential equation, but not two.
Homework Statement
Two same aged people P1,P2 create bank accounts at ages 25,35 respectively, and add 2000 euros annualy , for 10,30 years respectively. No initial capital in the accounts.
1.For interest 8% , what's the balance of each acount at age of 65 of P1,P2?
2.What should be the...
"Ordinary Differential Equations" by V Arnold
Hello,
I recently bought "Ordinary Differential Equations" by V I Arnold to study differential equation. In that book he assumes we know basic topology. I want to know which subfield of topology should I know, in order to study this book...
I'm a bit confused on how to do this problem, here is what I have.
Part a)
I must set up the set of linear differential equations with the initial values.
Using the balance law gives
y0'=0.2+0.1-0.2-0.1=0
The other 2 net rate of change would be equal to 0 as well.
But...I don't think I did...
Why is it that the unilateral lateral Laplace transform is used when given initial conditions that are non-zero. Is there a reason that explains why it would be wrong to use the bilateral Laplace transform instead?
I know bilateral does not have any input of initial conditions but that does not...
Homework Statement
I have to solve the differential equation xy'' + y' + xy = 0 ; x_o = 0
Homework Equations
The Attempt at a Solution
I know that a solution is y= sum from 0 to infinity (an (x-1)^n)
I then differentiate it twice to get y' = sum from 1 to inifinity...
Hello,
I'm a mechanical engineering student working on my senior design project. It involves the automated correction of an electrolyte's chemistry. That being the pH, conductance, and zinc concentration.
One of the tasks is to apply a "transient control analysis". This involves creating...
Homework Statement
(Sorry, I don't know how to use latex)
Solve the TDSE for a spin half nucleus in a B-field where B_z = B0, B_y= B1 cos(ωt) and B_x = B1 sin(ωt).
Use vector and matrix representation. You will get coupled differential equations for a and b, look for solutions of the form a =...
Homework Statement
Show that Acos(ωt) + Bsin(ωt) can be written in the form r*sin(ωt - θ). Determine r and θ in terms of A and B. If Rcos(ωt - ε) = r*sin(ωt - θ), deermine the relationship among R, r, ε and θ.
Homework Equations
Trig identities.
The Attempt at a Solution
I'm...
I'm a math major, which means I genuinely like mathematics. But I may be becoming a "pure math snob:"
http://img.spikedmath.com/comics/446-pure-math-vs-applied-math.png
When I'm working on differential equations, I'm finding it very tedious, like it's "not my kind of math." My brain...
Hi everyone! :smile: I'm newbie in this forum, please help me for my question.
In differential equation we know that the differential equation has a solution and uniqueness. which is usually called the existence and uniqueness theorem. my question, what is the difference of local existence...
Homework Statement
When solving a D.E. with power series, I've encountered something along the lines of:
(2 - r)^{2}g'' = -2
Homework Equations
Power Series
The Attempt at a Solution
I know I am just supposed to assume such a series exists, and work from there. But I'm really...
Homework Statement
The value of a tract of timber is
## V(t)=100,000 e^{0.8\sqrt{t} }
##
where t is the time in years, with t=0 corresponding to 1998.
If money earns interests continuously at 10%, the present value of the timber at any time t is ## A(t)= V(t)e^{-.10 t} ##
Find the year in...
Hi there guys.
Homework Statement
I am trying to solve a circuit that has a constant voltage source (V) in series with an inductor (L), then that inductor is connected to a resistor (R) in parallel with a capacitor (C).
Homework Equations
-------
The Attempt at a Solution...
Hello, I have recently started a little implicit differentiation and I have seen DEs before but I know that I still need to work on my differentiation and integration a little more before I am ready to tackle those. Anyway, I wish to ask, what distinguishes implicit differentiation from a...
Homework Statement
A goose starts in flight a miles due east of its nest. Assume that the goose maintains constant flight speed (relative to the air) so that it is always flying directly toward its nest. The wind is blowing due north at w miles per hour. Figure 8 shows a coordinate frame with...
There are two questions that I am trying to solve on web assignment. The goal is to find a general form of a particular solution to each ODE. The question asks me to represent all constants in the solution using "P,Q,R,S,T..etc.", in that order.
1. y'''-9y''+14y'=x2
2. y''-9y'+14y=x2e4x...
Hey Everybody, I am supposed to model the trajectory of a golf ball. I have been given the equations for velocity as a function of its derivative with respect to time. I am supposed to find the x-range as a function of the angle θ. (Pardon my bad latex skills, I will fix mistakes)...
Homework Statement
Suppose a rocket is launched from the surface of the Earth with initial velocity
v_0 = \sqrt(2gR) , the escape velocity.
a) Find an expression for the velocity in terms of the distance x from the surface of the Earth (ignore air resistance)
b) Find the time...
I'm taking my first ODE course, and I'm unsure what is meant in this question when it asks "Solve the following differential equation" (I have a list of DEs to solve). Some of them are really messy and I can't figure out an implicit solution. Would an explicit solution be acceptable?
Example...
Homework Statement
I came across two, not so obvious DEs that have stumped me abit.
(1) x2y' = xy + x2ey/x
(2) x2y' + 2xy = 5y3
Homework Equations
I know these are not separable, and more than likely require an integrating factor to put them into exact form so I can integrate them...
Hello all~
Given the equation:
dy/dx = (x/y)
I know we would initially go to:
∫dy =∫ (x/y) dx
then too:
∫(y)(dy) = ∫x dx
Until arriving at:
(y2/2) + C1 = (x2/2) + C2
(y2) - (x2) = C
My question is:
Where does the dy disappear to in step 4? Where the anti-derivative is taken...
HOW IS THE FOLLOWING QUESTION DONE?
Milk chocolate is being produced in a 300 litre tank, which initially contains 100 litres of milk. The following things then occur simultaneously:
1. Liquid cocoa (made up of equal parts cocoa solids and cocoa butter, both in liquid form) is added at...
Consider the equation dy/dt = alpha - y^2
a) Find all of the critical points. How does it change as alpha < 0, alpha = 0 or alpha > 0?
b) In each case of different alphas, consider the graph of f(y) vs y and determine whether each critical point is asympototically stable, semistable, or...
I am working on some problems for an assignment in my PDEs class and find myself either not understanding what I am supposed to do or being unsure of my answers or the next step.
I am going to outline my understanding of the problems, provide my attempt at a solution and highlight where I am...
I am trying to solve and plot the differential equations for springs.
when the damping factors are under 1 (underdamping), I tried damping ratios of:
0.01, 0.2, 0.1, 0.4, 0.8
If I use the following equations
(wd= damped frequency, wn= natural frequency, v0= v initial, x0= x initial, t=...
Homework Statement
(x+2y+3)dx+(2x+4y-1)dy=0
a1= 1 b1=2 a2=2 b2=4
a2/a1=b2/b1
Therefore z=x+2y
Here is where I get confused I understand that they must get a dz in the equations thus they take the derivative with respect to y of the following equation z=x+2y
thus giving dz=dx+2 or...
Homework Statement
dr/d∅+rtan∅=cos∅
μ(∅)=exp[∫tan∅]
μ∅=exp[-ln[cos∅]=-cos∅
-cos∅(dr/d∅)-rtan∅cos∅=-cos^2∅
dr/d∅[-rcos∅]=-cos^2∅
rcos∅=∫cos^2∅
changed cos^2∅ to 1/2+1/2cos2∅
rcos∅=1/2∅ + 1/4 sin2∅+c
the books answer is not even close to mine they have r=(∅+c)cos∅...