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 have a question about solving this system.
I (naively, I think) initially did the following
Trial solution: ##\vec{x}=e^{\lambda t}\vec{a}##.
Sub this into the system (2)
$$\lambda^2 e^{\lambda t}\vec{a}=M^{-1}Ke^{\lambda t}\vec{a}\tag{3}$$
$$(M^{-1}K-\lambda^2I)\vec{a}=0\tag{4}$$
I then...
The equation of motions looks like
$$m\ddot{x}(t)+m\Gamma\dot{x}(t)=-K(x(t)-d_0\cos{\omega_d t})\tag{3}$$
Moving other end of the spring sinusoidally effectively produces a sinusoidally varying force on the mass.
Everything written above so far is as presented by the book "The Physics of...
Good evening,
unfortunately I'm pretty lost in this problem.
I tried to use the chain rule $$(\frac {\partial H} {\partial v})_P = (\frac {\partial H} {\partial T})_P (\frac {\partial T} {\partial v})_P$$ and using some Maxwell relations but it doesn't work very well.
I know that $$T = (\frac...
From the first equation we can write
$$y=\frac{x}{2}+\frac{x^2}{8}$$
Subbing into the rhs of the second equation and equating to zero we find (after some algebra) that
$$x(x-4)(x^2+12x+72)=0$$
This equation has roots ##0##, ##4##, and ##-6\pm 6i##.
Then, ##x=0\implies y=0## and...
The critical points are ##(0,0)## and ##(2,1)##.
The linearization of these equations is
$$\begin{bmatrix}x'\\y'\end{bmatrix}=\begin{bmatrix}-1+y_0&x_0\\y_0&x_0-2\end{bmatrix}\begin{bmatrix}x-x_0\\y-y_0\end{bmatrix}$$
At ##(0,0)## we have
$$\begin{bmatrix}x'\\...
Here are the notes.
We have the system
$$\begin{bmatrix} x'\\y' \end{bmatrix}=\begin{bmatrix}f(x,y)\\g(x,y)\end{bmatrix}\tag{1}$$
We eliminate ##t## by dividing one equation by the other
$$\frac{y'}{x'}=\frac{dy/dt}{dx/dt}=\frac{dy}{dx}=\frac{g(x,y)}{f(x,y)}\tag{2}$$...
HI,
consider the 4 Maxwell's equations in microscopic/vacuum formulation as for example described here Maxwell's equations (in the following one assumes charge density ##\rho## and current density ##J## as assigned -- i.e. they are not unknowns but are given as functions of space and time...
Hi guys
Please refer to the attached image.
It is really easy to derive a set of differential equations which present a spring on a mass system, however how can one consider a system where the mass and spring can decouple? The first image on the left shows a spring a rest with a mass which...
These problems are from a practice problem set from MIT OCW's 18.03 "Differential Equations.
Computing the convolutions ##t*1## and ##1*t## is straightforward. They both equal ##\frac{t^2}{2}##.
Then, ##(q*1)(t)=\int_0^t q(\tau)d\tau##
and ##(1*q)(t)=\int_0^t q(t-\tau)d\tau## which after a...
I find a solution in math.exchange site: https://math.stackexchange.com/questions/3100237/find-general-solution-given-two-particular-solutions
The way I thought about solving the problem is to somehow use the two particular solutions to generate a homogeneous solution, I couldn't figure out how...
This was a question from some past exam, I found online a solution but it uses Frobenius method which wasn't taught in the course.
I would approach the solution by attempting to find a solution to the homogeneuous DE ##\left(t^2-1\right) \ddot{y}-6 y=0##, but that by itself is quite tricky...
I initially solved this problem in quite a roundabout way by thinking about a mass-spring-dashpot system modeled by
$$m\ddot{x}+b\dot{x}+kx=f(t)$$
Since the response is constant at ##x=0## and has a jump at ##t=0## to ##x=1##, my reasoning was that there can be no spring otherwise there would...
As we noted above, stability is all about the solution to the homogeneous equation.
For the equation
$$y''+by'+ay=0\tag{3}$$
we have discriminant
$$\Delta = b^2-4a\tag{4}$$
and the roots are
$$r=\frac{-b\pm\sqrt{b^2-4a}}{2}\tag{5}$$
We have three cases.
Case 1 (Distinct Real Non-Complex...
If ##p^2-4q<0## then we know that the homogeneous equation has a general solution
$$y_g(x)=c_1\sin{kx}+c_2\cos{kx}\tag{3}$$
where
$$k=\frac{1}{2}\sqrt{-\Delta}=\frac{\sqrt{4q-p^2}}{2}\tag{4}$$
Suppose we guess at a solution ##y_p## to the non-homogeneous equation...
My question will be about item (c).
Part (a)
Note that for ##x\geq 0## we have ##f(x)=g(x)##.
For ##x<0## we have ##f(x)=-g(x)##.
Since ##f## is a constant times ##g## then one column of the matrix in the Wronskian is a constant times the other column. Thus, the Wronskian is zero, Note that...
a) We can use reduction of order
$$p=y'\tag{1}$$
$$p'=y''\tag{2}$$
The DE becomes
$$p'+p^2=0\tag{3}$$
$$\frac{1}{p^2}p'=-1\tag{4}$$
This last step contains the assumption that ##p^2=y'^2\neq 0##.
$$-\left (\frac{1}{p(x)}-\frac{1}{p(x_i)}\right )=-(x-x_i)\tag{5}$$...
The Wronskian of these two solutions is also a function of ##x##.
$$W=y_1y_2'-y_1'y_2$$
$$W'=y_1y_2''+y_1'y_2'-y_1'y_2'-y_1''y_2$$
$$=y_1y_2''-y_1''y_2$$
The two solutions satisfy
$$y_1''+Py_1'+Qy_1=0$$
$$y_2''+Py_2'+Qy_2=0$$
Multiply the first by ##y_2## and the second by ##y_1## and...
Looking at the wronskian applications- came across this;
Okay, i noted that one can also have this approach(just differentiate directly). Sharing just incase one has more insight.
##-18c \sin 2x -4k\cos x \sin x - 4k\sin x\cos x =0##
##-18c\sin 2x-2k\sin2x-2k\sin 2x=0##
##-18c\sin 2x =...
TL;DR Summary: Variation of parameter VS Undetermined Coefficients
Hi all,
Suppose we want to solve the following ODE
2y''+y'-y=x+7
with two different methods: undetermined coefficients and variation of parameters.
The solutions to the homogeneous problem are given by y_1(x)=exp(-x) and...
I am trying to solve this system of differential equations using elimination method, but I am stuck.
$$\begin{cases}
y'_1 = y_2, \\
y'_2 = -y_1 + \frac{1}{\cos x}
\end{cases}$$
Here's what I tried:
I've been suggested to differentiate the ##y_1'= y_2## again to get ##y_1''= y_2'=...
I have not been able to solve this. Here is what I tried to do.
##z=y'-4y##
##z'=y''-4y'##
Thus, the second order equation in ##y## becomes ##z'+x^2z=0##, a first order equation in ##z##, the solution to which is ##z(x)=ke^{-\frac{x^3}{3}}## with ##k>0##.
Thus...
I am going through this,
I noted that, i shall have a separation of variables, that leads to
$$\left[\int \dfrac{1}{y(y-1)} dy\right]= \int \dfrac {1}{6} dt$$
and using partial fraction, i then have,
$$\left[\int -\dfrac{1}{y} dy - \int \dfrac{1}{y-1} dy\right] = \int \dfrac {1}{6} dt$$...
The characteristic equation has a zero discriminant and the sole root of ##-1##.
The general solution to the associated homogeneous equation is thus
$$y_h(x)=e^{-x}(c_1+c_2x)\tag{1}$$
Now we only need to find one particular solution of the non-homogeneous equation.
The righthand side of the...
Here is my solution to this problem. Unfortunately, I can't check it because it is not contained in the solution manual.
$$\frac{dv}{dt}=\frac{dv}{ds}\frac{ds}{dt}=v\frac{dv}{ds}$$
$$\frac{ds}{dv}=\frac{v}{v'}=\frac{v}{ge^{-kt/m}}$$
$$=\frac{\frac{m}{k}v}{\frac{gm}{k}e^{-kt/m}}$$...
a) Observe that ## \frac{\partial}{\partial z}F(y, z)=y^{n-1}\cdot \frac{2z}{2\sqrt{y^2+z^2}}=\frac{zy^{n-1}}{\sqrt{y^2+z^2}} ##.
This means ## G(y, z)=\frac{z^2\cdot y^{n-1}}{\sqrt{y^2+z^2}}-y^{n-1}\cdot \sqrt{y^2+z^2}=\frac{z^2\cdot...
Lowly engineer here. I am struggling - I think like many - to develop intuition on DEs.
From looking at the history and applications of DEs, general themes that come to mind are, conservation, energy (eg. isochrone problem), causality, feedback (control systems), etc.
However, I can't seem to...
Hello
May I begin by saying I do not exactly know what I am asking, but here goes...
In the Finite Element Method (as used in Solid Mechanics), we convert the differential equations of continuum mechanics into integral form. Here, I am thinking of the more pragmatic Principle of Virtual Work...
Hi,
unfortunately I have several problems with the following task:
I have problems with the tasks a, d and e
Unfortunately, the Green function and solving differential equations with the Green function is completely new to me
In task b, I got the following for ##f_h(t)=e^{-at}##.Task a...
I vaguely (strong word there because I can no longer remember the source, but the idea sticks in my head for 30 years now) recall reading (somewhere long forgotten) that method of separation of variables is possible in only 11 coordinate systems.
I list them below:
1.Cartesian coordinates...
Hello,
can someone help me to solve the following differential equation analitically:
$$\frac{2 y''}{y'} - \frac{y'}{y} = \frac{x'}{x}$$
where ##y = y(t)## and ##x = x(t)##
br
Santiago
I will have to teach a first course in differential equations. What should I cover in this module? For example, in most books they have Laplace Transforms which is fine but I would not use LT to solve differential equations.
I want to write a course that it motivates students and has an impact...
Hello,
I want to model a thermal battery based on phase change materials (PCM). It is a plate heat exchanger immersed in a PCM bath. The diagram is given in the attached file.
I want to determine the temperature at each moment and from everywhere in the battery. The hypotheses are the...
Lecture 20 by Prof. Mattuck is simply great. I don’t think there can be any better communication than that, it seemed as if what was in his mind simply got teleported in to mine, and that’s why, it seems now, Aristotle gave so much importance (and almost considered it as a divine power) to the...
I'm learning Differential Equations from Prof. Mattuck's lectures. The lectures are absolutely incredible. But there are a few topics in Tenenbaum's book and my syllabus which he doesn't seem to teach (I have reached upto lecture 14, but in future lectures too the following topics are not...
I made this exercise up to acquire more skill with polar coordinates. The idea is you're given the acceleration vector and have to find the position vector corresponding to it, working in reverse of the image.
My attempts are the following, I proceed using 3 "independent" methods just as you...
I ordered Differential Equations and Boundary Value Problem ( Computing and Modelling) by Edwards and Penney. There are several things in the book which I don't like
Too much focus is given to modelling, almost every topic is explained not from mathematical point of view but from application...
[Mentor Note -- This thread start is by a new member from the recent MHB forum merger]
Hello Guys, I want to find a friend with whom I can discuss differential equations!
I would like to do that via WhatsApp or zoom applications.
I am interested in applications of differential equations ( for...
Hello, there. I am trying to solve the differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##. However, ##A(t)## and ##B(t)## can not be simultaneous diagonalized. I do not know is there any method that can apprixmately solve the equation.
I suppose I could write the...
Hello everyone. I hope anyone can help me with this problem. I will greatly appreciate it. Willing to compensate anybody to answer this problem correctly for me.
I am trying to compute the Peebles equation as found here:
I am doing so in Python and the following is my attempt:
However, I'm unable to solve it. Either my solver is not enough, or I have wrongly done the function for calculating the Equation.
# imports
from scipy.optimize import fsolve...
My question i am trying to solve:
I have successfully done first order equations before but this one has got me a little stuck. My attempt at the general solution below:
$${5} \frac{\text{d}\theta}{\text{d}t}=-6\theta$$
$${5} \frac{\text{d}\theta}{\text{d}t} =\frac{\text{-6}\theta}{5}$$...
I am going through this page again...just out of curiosity, how did they arrive at the given transforms?, ...i think i get it...very confusing...
in general,
##U_{xx} = ξ_{xx} =ξ_{x}ξ_{x}= ξ^2_{x}## . Also we may have
##U_{xy} =ξ_{xy} =ξ_{x}ξ_{y}.## the other transforms follow in a similar manner.
*** MENTOR NOTE: This thread was moved from another forum to this forum hence no homework template.
Summary:: Trying to find transfer functions to design a block diagram on simulink with a PID controller and transfer functions for a water tank system.
----EDIT---
The variables and parameters...
Summary:: There seems to be a mismatch, in the "Maxwell's" equations, between the number of equations and number of variables.
I was trying to play around with the equations for Electromagnetism and noticed something unusual. When expanded, there are 8 equations, 6 unknown variables, and 4...
I chose to set the upwards direction to be positive and dM/dt = R = 190 kg/s, so I can solve the problem in variable form and plug in. With the only external force being gravity, this gives
M(t) * dv/dt = -M(t) * g + v_rel * R
where M(t) is the remaining mass of the rocket. Rearranging this...
I'd like a good set of notes or a textbook recommendation on how to approach vector differential equations. I'm looking for something that isn't specific to one type of application like E&M, fluid dynamics, etc., but draws heavily from those and other fields for examples.
I'd strongly prefer a...
Hello.
Considering this DE;
$$ x^7 x' = (x^8-300)t^6 $$ with inital value x(0) = -2
Now the solution for the initial value should be
C = -44;
And for x(t) I get ;
$$x(t) = (-44 e^{\frac{8}{7} t^7} + 300)^{\frac{1}{8}}$$
Now to get the biggest domain of definition I did this;
$$ -44...
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...