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 just started learning about DE's and am impressed by how much information they contain. Are they usually just a relationship between several simpler functions?
For instance, a basic kinematic problem like v = at + vo seems quite hard to come up with experimentally but v = d/t would be...
Homework Statement
the full question is asking me to solve
dy/dx + (1/x)y = 3cos2x
i think i know what i am doing up to a point,
but for me to continue with the question i have to integrate
exp(x^-1)3cos2x
and I am not sure how to do this, once i get this part i would know how to...
Hi.
Anyway, learning this sort of makes me feel like I've chosen the wrong school for myself, but I'd like to try and see if I can understand this nevertheless.
See, I'm supposed to learn various numerical methods for solving systems of linear equations, like the Jacobi or the Gauss-Seidel...
Homework Statement
(2x*y-5)dx+(x^2+y^2)dy=0 test for exactness and solve
Homework Equations
The Attempt at a Solution
I've been following an example to try and solve this but I'm not sure if its right or how to finish it of.
\delta M/\delta y=2x
\delta N/\delta x=2x...
Hello
I am kinda confused when it comes to finding a general solution of equation.
Here is the question.
Y'' + 6Y' +9Y = 0
Solution
Y'' + 6Y' +9Y = 0
I used "x" instead of Lamda...ANYWAY
X^2 + 6X + 9 = 0
FACTOR IT WHICH IS (X+3) (X+3) = 0
X IS EQUAL TO -3.
HERE IS WHERE I...
Homework Statement
Q1) (dy/dx)= 2x(y2+9); y(0)=0
Q2) (x4+y2) dx - xy dy =0; y(2)=1
Q3) (dy/dt)= 4y+t
Q4) y"+2y'+y=0; y(0)=4 and y'(0)=-6
Q5) y"+3y'+2y= 30e2t
3. Solutions found
A1) y= tan (x2/3)
A2) this is not an exact differential and so cannot be solved
A3) couldn't solve this...
Studying a maths degree, going onto final year next year, am planning to do a 3rd year course in quantum mechanics.
I just want to ask, how much probability theory and differential equations are there in quantum mechanics? Someone said that ultimately quantum mechanics is about probability...
Homework Statement
((d^2)x)/(d(t^2)) + 5(dx/dt) + 6x = f(t)
f(t)=3 for (0<=t<6)
f(t)=0 for t>=6
Homework Equations
This I am not sure of, this is my question. Would I use the Second Order Differential equation to do this problem:
L{((d^2)f)/(d(t^2))} = (s^2)F(s) - sf(0") -...
Homework Statement
The pollutants are being added to the lake at a constant rate I, and the pollutants are thoroughly mixed into the lake.
The annual precipitation into the lake matches evaporation, so the flow rate F is also constant.
Let y(t) be the amount of pollutant in the lake and c(t)...
The rate of capital growth in a bank account is described by the differential equation
dM/dt = aM
Where dM/dt is the rate of change of the capital M and a is the annual interest rate.
Show that the general solution for the time dependence of capital M(t) is given by:
M(t) = M0 * e^at...
hi,
first let me inform you that I'm not very good maths but love to learn it. I've basic understanding of maths. u take my maths abilities as 9th or 10th grader high schooler. i don't exactly know what linear algebra is but someone told me that it's about matrices. i have 4x4 and 6x6...
Homework Statement
Given a system of differential equations and asked to find the equilibrium points and classify them.
Homework Equations
Equation 1 ... dx/dt=y(13-x^2-y^2 )
Equation 2 ... dy/dt=12-x(13-x^2-y^2 )
The Attempt at a Solution
I know the solution...
Homework Statement
I have two equations:
\frac{\partial}{\partial z}E_{L}\left(z,\omega \right) = i \frac{2 \mu d_{eff}(\omega+\omega_{0})^{2}}{k(\omega+\omega_{0})}\int E_{L}(z,\omega-\omega_{T})E_{T}(z,\omega_{T})d\omega_{T}
\frac{\partial}{\partial z}E_{T}\left(z,\omega_{T} \right) = i...
Hi everyone, I recently started studying heat conduction using differential equations and this has been stumping me for a while.
I am having trouble understanding what type of heat conduction problem this is.
We are given a 100cm long copper rod with ends maintained at 0 C. The center of the...
It is very well known that the result of varying some functional gives a differential equation which solutions minimizes the given functional. What about the other way around? Can one find a functional that is minimized given a differential equation? Is there a procedure for this?
The reason...
I have a feeling this is the wrong place to post this. I can never figure out where to put things :confused:
Anyway, my professor uses the 4th edition of Edwards and Penney Differential Equations and Boundary Value Problems (https://www.amazon.com/gp/product/0131561073/?tag=pfamazon01-20)...
Homework Statement
\dot{x}=2x+5y+1, \dot{y}=-x+3y-4
Homework Equations
The Attempt at a Solution
Well, if system was: \dot{x}=2x+5y, \dot{y}=-x+3y we let a=2, b=5, c=-1, d=3.
Then p = a + d and q =ad - bc and we investigate p^2-4q
Don't know what to do when \dot{x}=2x+5y+1...
I am trying to solve a problem (not homework, too old for that! lol!) which involves the time dependent schrodinger equation for magnetic moment in time-dependent magnetic fields. I end up with the following that needs to be solved:
x' = -i*(b*t-a*t^2)*x - i*c*y
y' = -i*c*x -...
dy/dt = 2y-5, y(0) = y_0
The answer I got for this was y =5/2 + (y_0-5/2)e^t
but the solution is y =5/2 + (y_0-5/2)e^-2t
Where did this -2t come from?
My steps:
dy = (2y-5)dt Integrating both sides gives
1/2*ln|2y - 5| = t + C take exp of both sides gives
1/2(2y-5) = Cet...
Homework Statement
Verify that each given function is a solution of the differential equation.
1. ty' - y = t^2 ; y = 3t + t^2
2. y'' + y = sect , 0<t<pi/2 ; y = (cost)ln( cost ) + tsint
The Attempt at a Solution
int (tdy) = int(t^2 + y)dt
which isn't y=3t + t^2...
Homework Statement
Consider a population p of field mice that grows at a rate proportional to the current population, so that dp/dt=rp.
Find the rate constant r if the population doubles in 30 days.
Find r if the population doubles in N days.
Homework Equations
...The Attempt at a Solution...
Homework Statement
I can't solve the following problems:
38) Solve y' =y + et from yo =0 by Method 2, where the
deposit eT at time T is multiplied by et-T. The total output
at time t is y(t) = (Integral)(eT * et-TdT) (The integral goes from 0 to t). Substitute back to
check y' =y + et...
Homework Statement
2(2x^2+y^2)dx-xydy=0
Homework Equations
let x = yv
dx = ydv+vdy
The Attempt at a Solution
(4v^2y^2+2y^2)(ydv+vdy)-vy^2dy=0
4v^2y^3dv+4v^3y^2dy+2y^3dv+2vy^2dy-vy^2dy=0
4v^2y^3dv+2y^3dv+4v^3y^2dy+vy^2dy=0
2y^3(2v^2+1)dv+y^2(4v^3+v)dy=0
dividing all sides by...
Homework Statement
I have a series RLC circuit, no values given, connected to a voltage source Vs(t). I am asked to write the differential equations for:
a. that relates the inductor current iL(t) to the source voltage Vs(t).
b. that relates the capactor voltage Vc(t) to the source...
Homework Statement
Greetings –
I am wanting to solve the following system of differential equations:
K*\frac{d^2x}{dt^2} = -K2*\frac{dz}{dt}
K*\frac{dz^2}{dt^2} = K2*\frac{dx}{dt}
Homework Equations
The Attempt at a Solution
Now – I am not just...
According to my school, I got to take calc 3 and 1 semester of linear algebra before i can start differential equations. Is this so? Anything else you would advise to add? Or is this too much and I can deal with less? I ask because many have advised calc 3 before any physics even though my...
Homework Statement
The equation
2y'' - y' + y2(1 - y) = 0;
where y' = dy/dx
and y'' = d^2y/dx^2
represents a special case of an equation used as a model
for nerve conduction, and describes the shape of a wave of electrical activity
transmitted along a nerve fi bre.
Homework...
Thank you everyone for being a source of help in previous problems I've posted here. I'm starting an intermediate course in Differential Equations and I'm enjoying it so far, but this one problem on my homework seems to be giving me a problem and I think that I haven't fully grasped the...
Homework Statement
Let M(x,y) = yf(xy) and N(x,y) = xg(xy), where f(v) and g(v) are functions of a single real variable v, defined and continuously differentiable for all real values of v. Under what conditions on f and g is the form Mdx + Ndy exact for all values of x, y in the plane? In that...
My professor has us using C. Henry Edwards and David E. Penney, Differential Equation and Boundary Value Problems, Fourth Edition. However, I don't really like it as our textbook. Anyone else have any recommendations?
Homework Statement
I'm revising at the moment and a bit stumped on question 4 http://www.maths.ox.ac.uk/system/files/attachments/AC104.pdf
Homework Equations
The Attempt at a Solution
I think for the first part of the question, the regular singular points are 0 and -2...
Homework Statement
r" = 1/r^2
Homework Equations
A friend gave this to me, he was just wondering how we'd approach it.
The Attempt at a Solution
I don't think the equation is linear, so I don't know how to approach it. My friend suggested integrating both sides with respect to r...
greetings,
is it true that a exact differential equation is for two independent variable whereas linear differential equation is for one variable?
advanced thanks,
Can anyone recommend an elementary but rigorous text on differential equations that takes care to distinguish between a function, f: X \rightarrow Y, and its value f(x) \in Y, as Spivak does in his Calculus. I find I'm getting rather confused by the traditional y=y(x) type dual-meaning notation.
Hi guys, I am trying to linearize a coupled non-linear ode. I used partial derivative, and then jacobian matrix, i have seen paper using state-space model of jacobian matrix. I can't get a proper reference on this state-space model.
Attached is the non-linear ode, the partial derivative of the...
Homework Statement
http://img228.imageshack.us/i/cimg5162f.jpg/
This isn't the usual format, so I hope it's still ok. It's just easier than writing it out on here. I can't work out how to form a simultaneous equation from (1) and (2).
When solved it should form the du/dx and dv/dx...
By all techinical standards, I am not required to take Differential Equations. I am currently a Physics major with a 2nd major in Mathematics Education. I want to take DE before I start any of my upper level Physics courses next fall since 2 of my academic advisors (one for Physics, one for...
Homework Statement
i'm supposed to find the general solution of the equation: y'' + 3y' + 2y = e^x + e^-x
Homework Equations
I have no problem with solving this equation however, i am confused with the step they are taking in the solutions (circled)...
Homework Statement
In aerodynamics, one encounters the following initial value problem for Airy’s equations:
y''(x) + xy = 0, y(0) = 1, y'(0) = 0
Using the Runge-Kutta method with h=0.005 and determine values between x=0 and x=10 sufficient to sketch the relationship...
Homework Statement
The function F is defined by F (r, θ) = f (x(r, θ), y(r, θ)), where f is twice continuously
differentiable and
x(r, θ) = r cos θ, y(r, θ) = r sin θ.
Use the chain rule to find
d2F/dθ2Homework Equations
The Attempt at a Solution
I know that dF/dθ = (df/dx)(dx/dθ) +...
Which one would you recommend me to take over the summer? Or should I take both? Has anyone ever done that?
Next semester I will be taking multivariable calculus and i just wanted to take a math class over the summer because I really enjoy it.
Each class is 3 hours long and meets MTWTR.
i am told to investagate the nature of the critical points of the system:
x'=e^y
y'=(e^y)*cos(x)
i am not sure where to begin because x' is always non-zero.
Need Help with a couple diff eq problems:
2.) show y=cx^2 - x is a solution of xy' - 2y + x =0
I was trying to separate the variables so i could integrate but i cannot get it to work out.
I have tried so many things from adding 2y to both sides and and multiply by the inverse dx/dy. but...
Homework Statement
Consider the implicit (not actually sure wether that's the correct english word, my material is in Finnish and I'm Swedish-speaking :smile:) method
xj+1=xj+h/2(f(tj,xj)+f(tj+h,xj+1))
a)Write an appropriate Runge-Kutta scheme
b) What is the methods rank when we use the...
1. Suppose we have two tanks, each with an inflow pipe and an outflow pipe and also connected to each other by pipes. At the start, tank A contains 40 gallons of clean water and Tank B contains 25 gallons of clean water. At t = 0 brine containing 0.5 lbs of salt per gallon begins to flow into...
Homework Statement
solve the system of first-order linear differential equations:
(y1)' = (y1) - 4(y2)
(y2)' = 2(y2)
using the equation:
(λI -A)x = 0
Homework Equations
using eigenvectors and eigenvalues
in the book 'Elementary Linear Algebra' by Larson and Falvo - Section 7.4 #19...
Hey.. I ran across some problems and the notation used is a little different from what I've seen before.
considering U(x,y)=X(x)Y(y)
Sometimes I'll see Uxx for \frac{d^{2}u}{dt^{2}} which equals X''Y
Or Ux for \frac{du}{dt} which equals X'Y
But what about U'x
Is that a redundant way of...
Homework Statement
Given that y1(t) = t is a solution of t2y'' + ty' - y= 0 find all solutions.
The Attempt at a Solution
I already put it into mathbin asking somewhere else for help and it doesn't look like i can copy paste that syntax to here.
http://mathbin.net/56336
Could...
I want to preface this by saying that these questions are not to find an exact answer, just to build intuition. If you find them ill-posed or incorrect, it would be most helpful if you could show me a "better way" of looking at it.
So, I'm trying to gather a geometric viewpoint of...