Differential equations Definition and 999 Threads

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...

I have the following differential equation, which is the general Sturm-Liouville problem, $$ \dfrac{d}{dx} \left[ p(x) \dfrac{d\varphi}{dx} \right] + \left[ \lambda w(x) - q(x) \right] \varphi(x) = 0\ , $$ and I want to perform the change of variable $$ x \rightarrow y = \int_a^x \sqrt{\lambda...

hi, i am going through differential equations which are nonlinear and singular - like Lane-Emden equation etc. my question is how to tackle singularity - while coding. regards

I am reading on this part; and i realize that i get confused with the 'lettering' used... i will use my own approach because in that way i am able to work on the pde's at ease and most importantly i understand the concept on separation of variables and therefore would not want to keep on second...

An exact gravitational plane wave solution to Einstein's field equation has the line metric $$\mathrm{d}s^2=-2\mathrm{d}u\mathrm{d}v+a^2(u)\mathrm{d}^2x+b^2(u)\mathrm{d}^2y.$$ I have calculated the non-vanishing Christoffel symbols and Ricci curvature components and used the vacuum Einstein...

Hi, Is the answer: y(x) _homogenous =v(x) y(x) _private =u(x)v(x) ? Or they refer to something else? I don't know how to approach to it

So I am a sophomore physics major at a university near my hometown. I have always been fascinated by the way studying physics makes me think about the world, and I have struggled with but enjoyed my other undergraduate physics and math classes. This semester, however, I am taking multivariable...

I am given this system of differential equations; $$ x_1'=2t^2x_1+3t^2x_2+t^5 $$ $$ x_2' =-2t^2x_1-3t^2x_2 +t^2 $$ Now the first question states the following; Find a fundamental matrix of the corresponding homogeneous system and explain exactly how you arrive at independent solutions And the...

Problem statement: Second order linear differential equation in standard from Reasoning:

Summary:: A nitric acid solution enters at a constant rate of 6 liters / minute into a large tank that originally contained 200 liters of a 0.5% nitric acid solution. The solution inside the tank is kept well stirred and leaves the tank at a rate of 8 liters / minute. If the solution entering...

The number of organisms in a population at time t is denoted by x. Treating x as a continuous variable, the differential equation satisfied by x and t is dx/dt= xe^-1/(k+e^-1), where k is a positive constant.. Given that x =10 when t=0 solve the differential equation, obtaining a relation...

Consider the second order linear ODE with parameters ##a, b##: $$ xy'' + (b-x)y' - ay = 0 $$ By considering the series solution ##y=\sum c_mx^m##, I have obtained two solutions of the following form: $$ \begin{aligned} y_1 &= M(x, a, b) \\ y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\ \end{aligned} $$...

Hello. After a lot of researching, I am still not clear how the subject of differential equations is really any different from derivatives and integrals which are learned in the main part of calculus. For example: "Population growth of rabbits: N = the population of rabbits at any time t r=...

I am looking at this and i would like some clarity... at the step where "he let" ##μ_y##=0" Could we also use the approach, ##μ_x##=0"? so that we now have, ##μ_y##M=μ(##N_{x} -M_{y})##... and so on, is this also correct?

I am currently pursuing a Bachelors in Physics. With my current work experience, that degree will eventually allow me to reach an engineering position in Non Destructive Testing. While I enjoy the career field I believe I could do more with my degree. I personally would like to work at LHC or...

One thing that bothers me regarding the phase portraits, if I plot a phase portrait, then all my possible solutions (for different initial conditions) are included in the diagram? In other words, a phase portrait of a system of ODE's is its characteristic diagram?

Hi guys, I have just started studying DEs on my own, so pardonne moi in advance for the probably silly question :) Via Newton's second law of motion: $$x''=\frac{F}{m} \ [1]$$ Which is a second-order differential equation. But, from here, how do I get the good old equation of motion...

Hi guys, how are you doing? My maths teacher asked me to work on and deliver an engaging insight-oriented "lesson" to my class, about physical/engineering and real-world applications of differential equations, in order to better get the meaning of operating with such mathematical objects. Of...

Hello there, I need some advice here. I am currently studying intro physics together with calculus. I am currently on intro to oscillatory motion and waves (physics-wise) and parametric curves (calc/math-wise). I noticed that in the oscillatory motion section, I need differential equation...

I am trying to self study Ordinary Differential Equations and am totally fed up of "cookbook style ODEs". I have recently finished Hubbard's Multivariable Calculus Book and Strang's Linear algebra book. I would like a rigorous and Comprehensive book on ODEs. I have shortlisted a few books below...

>10. Let a family of curves be integral curves of a differential equation ##y^{\prime}=f(x, y) .## Let a second family have the property that at each point ##P=(x, y)## the angle from the curve of the first family through ##P## to the curve of the second family through ##P## is ##\alpha .## Show...

I have my set of differential equations which is dx/dt = -2x, dy/dt=-y+x2, with the initial conditions x(0)=x0 and y(0)=y0. I'm a little confused about how to approach this problem. I thought at first I would differentiate both sides of dx/dt = -2x in order to get d2x/dt2 = -2, and then I would...

Summary:: We want to find explicit functions ##g(y,t)## and ##f(y,t)## satisfying the following system of differential equations. I attached a very similar solved example. Given the following system of differential equations (assuming ##y \neq 0##) \begin{equation*} -y\partial_t \left(...

I would to know if I'm solving system differential equation by elimination correctly. Could somebody check my sample task and tell if something is wrong?

I got to know of this book through Freeman Dyson's obituary. Just wondering, is it useful in studying Physics (it seems to cover everything), do people even use it these days? I understand differential equations are basically half of Physics. By the way, this book is really old, are there any...

Hello, I would like to is it possible to solve such a differential equation (I would like to know the z(x) function): \displaystyle{ \frac{z}{z+dz}= \frac{(x+dx)d(x+dx)}{xdx}} I separated variables z,x to integrate it some way. Then I would get this z(x) function. My idea is to find such...

I have three differential equations with three unknowns ##p##, ##q## and ##r##: $$\displaystyle {\frac {\partial }{\partial p}}\sum _{k=1}^{5}f_{{k}}\ln \left( P \left( X=k \right) \right) =0$$, $$\displaystyle {\frac {\partial }{\partial q}}\sum _{k=1}^{5}f_{{k}}\ln \left( P \left( X=k...

The first equation leads to x = ae^2t + be^t and the second equation leads to y=[1/(ln(sint+pi/2)+c)] this corresponds to the system a+b=1/c 2a+b=1 which has infinitely many solutions. what am I missing here?

I identified the root 1 with multiplicity 1 and the root 2 with multiplicity 1. So The characteristic equation is ((m-1)^2)*(m-2)=0. Simplifying and substituting with y I found: y'''-4y''+5y'-2y=0. So now I've realized that this is actually describing y(t)=(C1)*(e^t)+(C2)*(e^t)+(C3)*(e^2t) and...

I know the solution to the equation (1) below can be written in terms of exponential functions or sin and cos as in (2). But I can't remember exactly how to get there using separation of variables. If I separate the quotient on the left and bring a Psi across, aka separation of variables (as I...

I was wondering if anyone could help me clarify which null cline solutions are correct for this question I've got: I've got two differential equations: \[ du/dt =u(1-u)(a+u)-uv \] \[ dv/dt = buv-cv \] where a, b and c are constants. I know to find the u null clines you set du/dt to 0. \[...

My History -------------- I attended Oregon State U. and majored 3 years in Electrical Engineering. Then I switched to a Math major for my final years and graduated with a B.S. in Math (1967). Developed several Apps for Engineers & Scientists. My first job was with Lockheed Aircraft Co...

I use the operator method here: (D^2 + D+3)y = 5cos(2x+3) ## y = \frac{1}{D^2+D+3} 5cos(2x+3) ## ## \Rightarrow y= \frac{5}{-(2)^2+D+3}cos(2x+3) ## ## \Rightarrow y= \frac{5}{-4+D+3}cos(2x+3) ## ## \Rightarrow y= \frac{5}{D-1}cos(2x+3) ## At this, if I revert back to write: (D-1)y = 5cos(2x+3)...

Good Morning Recently, I asked why there must be two possible solutions to a second order differential equation. I was very happy with the discussion and learned a lot -- thank you. In it, someone wrote: " It is a theorem in mathematics that the set of all functions that are solutions of a...

\begin{equation} y_{1}{}'=y_1{}+y_{2} \end{equation} \begin{equation} y_{2}{}'=y_2{}+u \end{equation} build a control \begin{equation} u \epsilon L^{2} (0,1) \end{equation} for the care of the appropriate system solution \begin{equation} y_{1}(0)=y_{2}(0)=0 \end{equation} satisfy...

To be able to build a control y_{1}{}'=y_1{}+y_{2} y_{2}{}'=y_2{}+u u \epsilon L^{2} (0,1) for the care of the appropriate system solution y_{1}(0)=y_{2}(0)=0 satisfy y_{1}(1)=1 ,y_{2}(1)=0 Please kindly if you can help me Discipline is Optimal ControlHELP! i need to find...

Hello, I need help deciding on whether to take ODE (MAP2302) and Calc III during the summer. Would it be wise to take ODE along with Calc III in the same semester? Some people have told me to take Calc III first because there are a few things in ODE that are taught in Calc III, but others have...

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Hi, I have an experimental setup where we are taking certain different types of metals of varying shapes and sizes, weighing them, taking approximate measurements, and then blowing it off of a table of a fixed height with an Air Nozzle. The data taken down in experiment is the PSI at which the...

Can someone list to me (and whoever is going to view this thread) what topics in differential equations should be studied so that we can have a decent knowledge of the general physical theories in which they occur? (And I believe, they appear in all theories.) So far, I believe the two most...

Good evening, I have been wrestling with the following and thought I would ask for help. I am trying to come up with the equations of motion and energy stored in individual suspension components when a wheel is fired towards the car but, there is a twist! I am assuming a quarter car type...

I got a C last semester in elementary differential equations. It was an online class using ProctorU and I always had technical difficulties, so while my homework category was a 95% my test category was around a 60%. I am a spring-semester sophomore right now and my GPA is 3.611. If I retake...

I'm having quite a bit of a problem with this one. I've managed to figure out that ##T_0 = 0##. However, not knowing what ##q(t)## is bothers me, although it seems that I could theoretically solve the problem without knowing it. For ##t>t_1##, integration by parts gives me ##T = Ce^{-t/10}##...

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v(t)=600+(9-6)t =600+3t 1500=600+3t therefore t=300 hrs when tank is full Cin=1/5(1 + cost) ds/dt=Rate in - rate out = CinRin - Cout*S(t)/V(t) =1/5(1 + cost)*9 - 6*S(t)/(600+3t) S(0)=5 ib Solving the first order linear ODE we get: https://www.desmos.com/calculator/l7iixzgyll therefore...

Solve the boundary value problem Given u_{t}=u_{xx} u(0, t) = u(\pi ,t)=0 u(x, 0) = f(x) f(x)=\left\{\begin{matrix} x; 0 < x < \frac{\pi}{2}\\ \pi-x; \frac{\pi}{2} < x < \pi \end{matrix}\right. L is π - 0=π λ = α2 since 0 and -α lead to trivial solutions Let u = XT X{T}'={X}''T...

In genaral relativity, how to solve differential equations is seldom be discussed. I want to know how to sole the differential equations like this: $$\partial_kv^i(x)+\Gamma^i_{jk}(x)v^j(x)=\partial_kA^i(x)$$ Here ##\Gamma^i_{jk}(x)## is connection field on a manifod and ##A^i(x)## is a vector...

In order to obtain equation (3), I think I have to do the Fourier transform in the x direction: \begin{equation} \tilde{G}(k,y,x_0,y_0) = \int_{- \infty}^{\infty} G(x,y,x_0,y_0) e^{-i k x} dx \end{equation} So I have: \begin{equation} -k ^2 \tilde{G}(k,y,x_0,y_0) + \frac{\partial^2...

I'm fine with this up to a certain point, but I'm not certain if I'm using the substitution correctly. After finding the homogeneous solution do I plug in x= e^t in the original equation and then divide by e^2t to put it in standard form before applying variation of parameters so f=1, or do I...

In order to solve for ##x##, I need to re-write the equation for ##dx## so it is independent of ##y## and ##dy##. However, I am having some issues with this. Can someone give me a push in the right direction?