Differential equation Definition and 146 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.
The characteristic equation ## m^3 -6m^2 + 12m -8 = 0## has just one single, I mean all three are equal, root ##m=2##. So, one of the particular solution is ##y_1 = e^{2x}##. How can we find the other two? The technique ##y_2 = u(x) e^{2x}## doesn't seem to work, and even if it were to work how...
I am currently looking at section IIA of the following paper: https://arxiv.org/pdf/gr-qc/0511111.pdf. Eq. (2.5) proposes an ansatz to solve the spheroidal wave equation (2.1). This equation is
$$ \dfrac{d}{dx} \left((1-x^2) \dfrac{d}{dx}S_{lm} \right) + \left(c^2x^2 + A_{lm} -...
I have a few questions about the negative Bendixon criterion. In order to present my doubts, I organize this post as follows. First, I present the theorem and its interpretation. Second, I present a worked example and my doubts.
The Bendixson criterion is a theorem that permits one to establish...
(x cos(y) + x2 +y ) dx = - (x + y2 - (x2)/2 sin y ) dy
I integrated both sides
1/2x2cos(y) + 1/3 x3+xy = -xy - 1/3y3+x2cos(y)
Then
I get x3 + 6xy + y3 = 0
Am I doing the calculations correctly?
Do I need to solve it in another way?
So I am a bit uncertain what approach is best for solving this problem and how exactly I should approach it, but my strategy right now is:
1. Solve the time-independent Schrödinger Equation with the given Hamiltonian and find energy eigenvalues of system:
-Here I struggle a bit with actually...
the differential equation that describes a damped Harmonic oscillator is:
$$\ddot x + 2\gamma \dot x + {\omega}^2x = 0$$ where ##\gamma## and ##\omega## are constants.
we can solve this homogeneous linear differential equation by guessing ##x(t) = Ae^{\alpha t}##
from which we get the condition...
I want to solve the heat equation below:
I don't understand where the expression for ##2/R\cdot\int_0^R q\cdot sin(k_nr)\cdot r \, dr## came from. The r dependent function is calculated as ##sin(k_nr)/r## not ##sin(k_nr)\cdot r##. I don't even know if ##sin(k_nr)/r## are orthogonal for...
for example, I want to know velocity of a person when time is equal to t, that person start running from 0m/s (t=0s) to max velocity of 1m/s (t=1s). I am thinking that this is like rain droplet that affected by gravity and drag force, where force is directly proportional to its velocity, to make...
Hi! I am looking into a mechanical problem which reduces to the set of PDE's below. I would be very happy if you could help me with it.
I have the following set of second order PDE's that I want to solve. I want to solve for the generic solutions of the functions u(x,y) and v(x,y). A, B and C...
Hello forum, i want to make a samulation of a body. The body will be moved horisontal on y,x axis. I want on my simulation the body to change direction many times(for example i want to go for 10sec right and then left end right...). My question is does i need more than one differential equation...
I am trying to solve a PDE (which I believe can be approximated as an ODE). I have tried to solve it using 4th Order Runge-Kutta in MATLAB, but have struggled with convergence, even at an extremely high number of steps (N=100,000,000). The PDE is:
\frac{\partial^2 E(z)}{\partial z^2} +...
I have a problem. The task is to develop an differential equation of the airflow of a balloon. I know that it is dependent on the volume and pressure. But I can't get a good differential equasion. Can someone help me?
[Thread moved from the technical forums, so no Homework Help Template is shown]
I am unable to complete the first part of the question. After I plug in the function for psi into the differential equation I am stuck:
$$\frac {d \psi (r)}{dr} = -\frac 1 a_0 \psi (r), \frac d{dr} \biggl(r^2 \frac {d\psi (r)}{dr} \biggr) = -\frac 1 {a_0}\frac d {dr} \bigl[r^2 \psi(r) \bigr] =...
How does one solve the partial differential equation Uxx+Uyy+Uzz=C when C is non-zero. Here U is a function of x,y and z where (x,y,z) lies in the ball centered at 0 of radius 1 and U=0 on the boundary. Uxx, Uyy and Uzz denote second partial derivatives with respect to x, y and z.
Any hints on...
From solving the characteristic equations, I got that ##\lambda = 0.5 \pm 1.5i##. Since using either value yields the same answer, let ##\lambda = 0.5 - 1.5i##. Then from solving the system for the eigenvector, I get that the eigenvector is ##{i}\choose{1.5}##. Hence the complex solution is...
I'm having a hard time grasping the concept of reducing the two recursive relations at the end of the frobenius method.
For example, 2xy''+y'+y=0
after going through all the math i get
y1(x) = C1[1-x+1/6*x^2-1/90*x^3+...]
y2(x) = C2x^1/2[1-1/3*x+1/30*x^2-1/630*x^3+...]
I know those are right...
If we take a flat universe dominated by radiation, the scale factor is ##a(t)=t^{1/2}##
which can be derived from the first Friedmann Equation:$$(\dot a/a)^2 = \frac{8\pi G}{3c^2}\varepsilon(t)-\frac{kc^2}{R_0^2 a(t)^2}$$
But suppose I want to show this using the second Friedmann Equation
(Also...
When reading through Shankar's Principles of Quantum Mechanics, I came across this ODE
\psi''-y^2\psi=0
solved in the limit where y tends to infinity.
I have tried separating variables and attempted to use an integrating factor to solve this in the general case before taking the limit, but...
The ODE is:
\begin{equation}
(y'(x)^2 - z'(x)^2) + 2m^2( y(x)^2 - z(x)^2) = 0
\end{equation}
Where y(x) and z(x) are real unknown functions of x, m is a constant.
I believe there are complex solutions, as well as the trivial case z(x) = y(x) = 0 , but I cannot find any real solutions. Are...
Homework Statement
Given that ##x=\phi (t)##, ##y=\psi(t)## is a solution to the autonomous system ##\frac{dx}{dt}=F(x,y)##, ##\frac{dy}{dt}=G(x,y)## for ##\alpha < t < \beta##, show that
##x=\Phi(t)=\phi(t-s)##, ##y=\Psi(t)=\psi(t-s)##
is a solution for ##\alpha+s<t<\beta+s## for any real...
Homework Statement
Parameterize the part of the curve which allows an equilateral triangle, with the height 3R, to roll from one vertex to the next one, while its center travels at a constant height.
Homework Equations
I will include some pictures to show what I'm doing
The Attempt at a...
Homework Statement
I have to calculate the critical points of the following system.
$$x'=cx+10x^2$$
$$y'=x-2y$$
The Attempt at a Solution
So I solve the system
$$cx+10x^2=0$$
$$x-2y=0$$
So if $$x=2y$$ I have $$2yc+10*4y^2=2yc+40y^2=y(2c+40y)=0$$ and I get $$y=0$$ and $$y=-\frac{c}{20}$$ f I...
Homework Statement
Find Orthogonal Trajectories of ##\frac{x^2}{a}-\frac{y^2}{a-1}=1##
Hint
Substitute a new independent variable w
##x^2=w##
and an new dependent variable z
##y^2=z##
Homework Equations
The Attempt at a Solution
substituting ##x## and ##y## I get...
I've found the general solution to be y(x) = C1cos(x) + C2sin(x).
I've also found a recursion relation for the equation to be:
An+2 = -An / (n+2)(n+1)
I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this?
Any help would be...
Hey, this is how i tried solving the differential equation
The solution however does not match the general solution of the equation. Also differentiating it twice does not give me the previous equation. Please tell me if i did some mistake while solving.
I already know how to solve by finding...
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...
I've been having a sign problem while deriving the permittivity formula using Drude model,
and I found out that the problem came from the fact that complex field vectors are expressed with e-iwt, not eiwt, thus producing (-iwt) term when differentiated...
Hi all,
I'd appreciate help in calculating the voltages in the circuit shown. I thought it should be fairly straight forward, but it has me stumped. This is a sample-and-hold circuit for an ADC -- the switch is closed to charge the hold capacitor with the sample voltage and then opened to...
Hello all, I want to say thank you in advance for any and all advice on my question. My classical mechanics textbook (Marion Thornton) has been taking me through motion for a particle with retarding forces.
The example it keeps giving is:
m dv/dt = -kmv
which can be solved for:
v = v0e-kt...
I am aware that hypergeometric type differential equations of the type:
can be solved e.g. by means of Mellin transforms when σ(s) is at most a 2nd-degree polynomial and τ(s) is at most 1st-degree, and λ is a constant. I'm trying to reproduce the method for the case where λ is not constant...
I have a system with three inductors connected together at a common point.
The unconnected ends of each inductor is connected to an independent voltage source.
Basically I want to get three expressions for the dynamics of the currents with V1, V2 and V3 as inputs.
i.e. i need to eliminate the...
1. The problem statement, all variables, and given/known data
Task requires you to solve a partial differential equation $$u_{xy}=2yu_x$$ for ##u(x,y)##. A hint is given that a partial differential equation can be solved in terms of ordinary differential equations.
According to the solution...
We have a population of y = 1000 at year 1980 (call it year 0).
Every year the population growth rate is 5% per year.
y' shows the growth rate of the y (population).
Since the population grows by 5% every year, the growth rate is:
y' = 0.05y.
This is a simple differential equation.
When y(0)...
Homework Statement
$$y'=-\frac{1}{10}y+(cos t)y^2$$
when doing substitute for ##z=\frac{1}{y}##
I understand this is ##z(t)=\frac{1}{y(t)}##
I know t is independent variable and y is dependent variable
but I want to know what is z role here, is it change the dependent variable?
when...
The Problem
So, I know the basic, the basic differential equation of movement in terms of the angle it is formed, which has the form: 0 = g⋅sin[α(t)] + α''(t)⋅l. However, I decided to consider the friction force of the air, which is always contrary to the movement, and proportional to the...
Hello,
I am studying control theory. And I have encountered something I have never considered or thought about.
Consider a system with y as the output differential equation and u as the input.
any(n) + ... + a1y(1) + a0y = bmu(m) + ... + b1u(1) + b0u
Here, the subscripts indicate...
Homework Statement
I need to come up with an equation that would model the motion of a non-linear pendulum with air resistance. [/B]
Homework Equations
Fc=mgsintheta
Fdrag=(1/2)p(v^2)CA
The Attempt at a Solution
I started with mgsintheta-(1/2)p(v^2)CA=ma
After substituting v=r*omega and...
I'm a bit lost in all the numerous methods for solving differential equations and I would be very grateful if someone could point me to some direction.
I want to solve the following boundary conditioned differential equation:
$$a_1+a_2\nabla f(x,y)+a_3\nabla f(x,y)\cdot \nabla^2...
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...
JKC
Thread
differentialequation
frobenius
power series
second order
singular points
Hi! I was wondering how I could come up with a differential equation for projectile motion on a 2D plane when air resistance is not negligible. I'm trying to guess the position of a projected ball at a certain time period by approximating the coordinates using the Euler's method.
Here, I would...
ChanYoung Park
Thread
air resistance
differentialequation
projectile motion
Homework Statement
A projectile is launched at an angle to the horizontal and rises upwards to a peak while moving horizontally. What is the angle of maximum range and how it is dependent on initial velocity if we include air resistance and if the wind is blowing in the horizontal direction of...
Homework Statement
We can treat the following coupled system of differential equations as an eigenvalue
problem:
## 2 \frac{dy_1}{dt} = 2f_1 - 3y_1 + y_2 ##
## 2\frac{dy_2}{dt} = 2f_2 + y_1 -3y_2 ##
## \frac{dy_3}{dt} = f_3 - 4y_3 ##
where f1, f2 and f3 is a set of time-dependent sources, and...
I am beginning to learn about differential equations and I saw in an explanatory video a solution to this separable differential equation:
##\frac {dy} {dx} = \frac {-x} {y{e^{x^2}}}##
from there through simple steps the equation changed to ##y⋅dy=-xe^{-x^2}⋅dx
##.
Then the video did an...
Homework Statement
A particle of mass m is subject to a force F (x) = -kx. The initial position is
zero, and the initial speed is v0. Find x(t).
Homework Equations
F = m*v*dv/dx = -kx
v = dx/dt
The Attempt at a Solution
I'm new to differential equations, so please excuse me if I make any...
I need a translation of "Differentialgleichungen : Losungsmethoden und Losugen", I guess it is written in German. This book was referenced in Shepley L. Ross' book on ODE.
If the English translation is not unavailable, I am fine with a book that contains a "list" of special differential...
Homework Statement
Solve the differential equation ##(2x+1)^2y'' + (4x+2)y' - 4y = x^2##
Can someone verify whether my solution is correct?
Homework Equations
The Attempt at a Solution
We perform the substitution ##t = \ln|2x+1|##. Then, ##e^t = |2x+1|## and ##x = \pm(e^t -1)/2##
Without...
Homework Statement
A beam is supported at one end, as shown in the diagram (PROBLEM 11 page 281 of Lea, 159 of the course pack). A block of mass M and length l is placed on the beam, as shown. Write down the known conditions at x = 0. Use the Laplace transform to solve for the beam...
Homework Statement
Solve ## \frac{d^2y}{dt^2} + \omega^2y = 2te^{-t}##
and find the amplitude of the resulting oscillation when ##t \rightarrow \infty ## given ##y=dy/dt=0## at ##t=0##.
Homework Equations
The Attempt at a Solution
I have found the homogenious solution to be:
##y_h =...