Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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How to I find $y(t)$ when the flow rate in is $=r$ and concentration of chemical $Y$ coming in is $=X$ grams per liter?

A tank initially contains $1$ liter of water and $628$ grams of chemical $Y$. A solution containing $X$ grams per liter of chemical $Y$ flows into the tank at the rate of r liter/hour. The mixture ...
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30 views

Uniqueness of solutions to ODEs for functions of complex domain

Given the ivp $$ \tag{1} \frac{df}{dz}=F(z,f), \hspace{1cm} f(z_0)=f_0 \hspace{1.5cm} f:\Bbb{C}\rightarrow\Bbb{C} $$ where $f$ and $F$ are complex valued and analytic, then does it folow that $(1)$ ...
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Poisson differential equation

I'm stuck on an old exam question: Let $\Omega = \{(x,y) \in R^2 : 1 < x^2 + y^2 < a \}$. Determine the unique solution for the following boundary condition problem: $\Delta u = 1$ for $(x,y) ...
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76 views

Solving a system of linear ODEs

Based on my previous post, I have been stuck on this for a few hours now. I want to solve for $x$ and $y$ from the equation $$\frac{dx}{dt} + \frac{dy}{dt}=a-(b+c+d)y-bx.$$ The original two equations ...
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157 views

Where can I find the TOC of “Calculus and analytic geometry” by George B. Thomas 4th ed?

I am currently following the course Calculus Revisited, by Prof. Herbert Gross. In his lecture notes he makes references to the book mentioned in the title, by section number, so far I found a copy ...
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16 views

Implementing Equation on current data

I am trying to implement Personality, Gender, and Age in the Language of Social Media equation. I have 5 patterns and one list of 100 text = 900 words. The result of find a Match in the 900 to the ...
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203 views

Stability of a critical point

I have a question about the stability of a critical point of the system: $$\frac{dx}{dt} = \begin{bmatrix} 4 & -26 \\ 1 & -6 \end{bmatrix}x + \begin{bmatrix} 2 \\ 0 \end{bmatrix}.$$ I have ...
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Linear homogeneous ODE system of first order

Good afternoon. I recently encountered the following problem to which I couldn't find a solution anywhere so far: Given $A:D\to\mathbb C^{2\times 2}$, $D\subset\mathbb C$ open, with holomorphic ...
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23 views

Partial differentiation for multi variables

A Candy company makes 2 types of candy A & B for which the average costs are 2 & 3euros per kg respectively. $Q_a$ & $Q_b$ (a & b subscript) are the kg that can be sold each week and ...
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26 views

What is the function $f$ verifying : $f(\frac{x}{2}+\frac{x}{2}\cos(\frac{v\pi}{x}))=\frac{x}{2}\sin(\frac{v\pi}{x})$

What are the solutions to the functional equality (for a constant $v$): $$ \forall\, x > 0, \ \ \ \ ...
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49 views

Function with bounded derivative as ODE

Given a function $x(t)$, I am looking for a function $y(t)$ which closely follows $x(t)$ except that its derivative must be bounded by a constant $c$, i.e. $\dot{y} \leq c$. Is there a way to describe ...
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230 views

Newton's Law of Cooling Application

A thermometer that has been stored indoor where the temperature is 22 degrees Celsius, is taken outdoor. After 5 minutes it reads 18 degrees. After 15 minutes it reads 15 degrees. What is the outdoor ...
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77 views

General solution to $f(x,y,z) = g(p,q,z)$ for all $x,y,z,p,q$ by method of inexact differential

I have a book that says that if $$ f(x,y,z) = g(p,q,z) $$ and $$ h(x,y,p,q) = 0 $$ then f and g both have the form (for instance): $$ \phi(x,y)\zeta(z) + \eta(z) $$ I'm guessing the proof has to ...
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How to solve a $ay''(x) + (b-x)y'(x)+ c y(x) = 0$?

Does anyone know if there exists an explicit solution to the following differential equation: $a y'' + (b-x)y'+ c y = 0$ Here, $a$, $b$ and $c$ are constants. The trivial solution is $y(x)=0$. I ...
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35 views

System of differential equations with references to each other

For system of differential equation as follows:\begin{align} \frac{\partial}{\partial t} \begin{pmatrix}\rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{00}\end{pmatrix} &= -\tau i ...
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15 views

Finding normal components of a vector

If \begin{align} \notag A_{1}=c_{1}y_{1}^{2}y_{2}^{2}u(u+2t)(u+y_{2}^{2})\frac{\partial}{\partial x_{2}} \end{align} and \begin{align} \notag A=-c_{11}\frac{\partial}{\partial ...
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56 views

what can you conclude about the following IVP from the Existence and Uniqueness Theorem for first oder nonlinear ODE?

what can you conclude about the following IVP from the Existence and Uniqueness Theorem for first oder nonlinear ODE?  here is the theorem: the Existence and Uniqueness Theorem I could only get ...
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293 views

Calculate the rate of change of volume, $V$ (in $m^{3}$), of water in a tank

$V=t^3-5t^2+6t+2$, where $t$ is the time in hours measured from a particular instant and $0 \leq t \leq 4$. Find the rate of change of volume with respect to time after: (i) $0.5$ hours My attempt: ...
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54 views

Stability conditions

Below is a problem about stability conditions that I have been struggling with it during an exam: Find the stability conditions for $$A\left ( \frac{\partial^2 u(x,\, y,\,t)}{\partial x^2} + ...
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How to establish a lower bound on this difference operator?

If I define the approximation of the second derivative as $$\delta^2_xV_{i}=\dfrac{D^+_xV_{i}-D^-_xV_{i}}{(x_{i+1}-x_{i-1})/2}$$ where $$D^+_xV_{i}=\dfrac{V_{i+1}-V_i}{x_{i+1}-x_i}, ...
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29 views

Stochastic differential equation of a falling body

It's well known the motion of a falling body in a constant gravity model, for high speed is given by: $$m\ddot{x}(t)=g-\beta\dot{x}(t)^2$$ where $\beta$ is he drag coefficient. In a turbolent flow we ...
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291 views

Differential Equations Text (Math GRE)?

I am currently looking for a Differential Equations text from which to study for the Math GRE Subject Test. (As a side note, I currently own a copy of Stewart's Calculus.) Any suggestions of ...
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30 views

characterising attractors for master equations

I have a master equation for $(x,y,z)$ with the constraint $x+y+z=N$. $x$ can be regarded as the number of animal of a certain species in the whole system. In other words, I have a differential ...
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Solution to differential equation is defined on all of $\mathbb R$

Let $f$ and $g$ be real, continuous functions and $i,j > 0$ constants, with $$0 < g(x) \le i|x| + j$$ for every $x$ $\in \mathbb R$. Prove that $$y'=f(x)g(y)$$ $$y(x_0)=y_0$$ has a solution ...
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66 views

Eigenvalue equation under neumann boundary condition

Let $c(x)\in L^{\infty}(0,a)$, a>0. We consider the problem $-u''+c(x)u=\lambda u(x),\quad x\in (0,a),$ under the boundary conditions $u'(0)=u'(a)=0$. Show that $u>0 $ in $[0,a]$ and that ...
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35 views

Solving Differential Equations to satisfy a condition

I have two differential equations $$\frac{dX}{dt} = -\frac{d\cdot N(0)\cdot X}{m+X}$$ and $$\frac{dY}{dt} = \frac{d\cdot N(0)\cdot X}{m+X}$$ with initial conditions $X(0) = X_0$ and $Y(0) = 0$. ...
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Is there any relation between positive definite operator and an operator that satisfies maximum principle?

Suppose $L$ is a self adjoint differential operator which satisfies maximum principle. Maximum principle: Assume that $u(x)$ satisfies $u(0)\geq 0$ and $u(1)\geq 0$. Now $L$ is said to satisfy ...
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Derivation of an integral identity from the kdv equation.

The stationary KdV equation given by \begin{equation} 6u(x)u_{x}−u_{xxx}=0 \quad \quad \quad(1) \end{equation} It has a solution given by $$\bar{u}(x)=−2\mathrm{sech}^{2}(x)+\frac{2}{3} \quad \quad ...
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To be Lyapunov stable solution and don't be stable solution asymptotically

Find the parameters $ a, b \in R $ of equation $$ y''' + ay'' + by' + y = 0 $$ to the function $ y = e^{-x} $ be Lyapunov stable solution and don't be stable solution asymptotically? Am I correct? ...
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2 by 2 Linear Homogenous System with Complex Eigenvalue. Boyce, p409, Question 7.6.4

I don't know how to align multiple equations, hence I post this screenshot. If someone can show me how, thanks. I think understand everything above the red line, but please inform me about any ...
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107 views

Brachistochrone Problem to find out the path by which a bead travels in least time

The question is to find the shape of the curve down whcih a bead sliding from rest and accelerated by gravity will slip(without friction) from one point to another in the least time. So I proceeded ...
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Finding another solution in certain form (confluent hypergeometric series)

Consider the differential equation $$xf''(x)+(\alpha+1-x)f'(x)+nf(x)=0 ..........(*)$$, we know that the Laguerre Polynomial ...
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Checking a solution of differential equation

I am trying to prove that the Laguerre polynomial $$L=L_n^{\alpha}(x)=\frac{x^{-\alpha}e^{-x}}{n!}\frac{d^n}{dx^n}(e^{-x}x^{n+\alpha})= ...
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Best way of learning dealing with DEs?

I'm from a cognitive science/computer science background. As I'm currently dealing with neuroscience a lot, I want to get better at dealing with DEs. My primary goal is to learn the tools required for ...
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30 views

Finding Characteristic Exponents for $x^2 (x-1)^2 y'' + 4 (x-1)y' - 4x^2 y = 0$

I've found that the only regular singular point of this differential equation: $$x^2 (x-1)^2 y'' + 4 (x-1)y' - 4x^2 y = 0$$ is $x = 1$. How do I determine the characteristic exponents for it?
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Can a linear solution make a non-linear (functional) differential equation linear?

I was inspired by this question Does a non-trivial solution exist for $f'(x)=f(f(x))$? And tried coming up with similar problems, one interesting case I found was $f'(x) +f(x)=f(f(x))$ which has ...
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Eigenvalues for the double-well potential

I am trying to find eigen-values, $E$, for the following differential operator: $$\left[ -\frac{1}{2}\frac{d^2}{dx^2} +L\left(x^2-a^2\right)^2\right]y(x) = E\,y(x) $$ where $L,a$ are two positive ...
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help with exercise of differential equations(Rafael Iório)

maybe someone could help me prove the uniqueness of the solution of the following problem: \begin{equation*} \left\lbrace \begin{array}{l} \partial_x^2u-u=0, \ (x,y)\in \mbox{R}^{2}\qquad ...
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Existence of particular solution of an ODE with some topological hypothesis

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a continuous map. Assume the existence of two solutions $\varphi_1,\varphi_2: [0,1] \to \mathbb{R}$ of $x'=f(t,x)$ such that: Graph$(\varphi_1)$ $\cap$ ...
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system of non-linear differential equations

I have the system $$x'=\sin(z-y), y'=\sin(x-z), z'=\sin(y-x)$$ and I'm not sure how to approach it. Mathematica can't seem to spit out a closed form solution. Any ideas?
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Solving a non-exact differential

I started off solving the differential equation $$(xy^2 + 3e^{x-3})dx - x^2ydy = 0$$ It's a non-exact first order equation whose integrating factor is $1/x^4$. Finally I got to the equation where I ...
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Proof of the uniqueness of the solution of the problem: $v=0$

The proof of the uniqueness of the solution of the problem $$\Delta u=f, \text{ at } D$$ $$u=h, \text{ at } \partial{D}$$ is the following: Let $u_1, u_2$ solution of the problem above. ...
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136 views

Frobenius Method For System of Differential Equations

I have a system of ODEs. Can you explain how to solve a system of ODEs using the method Frobenius expansions ? There are 5 ODEs which are coupled and 5 variables. $\omega\hat\rho + i\alpha V_z ...
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36 views

Using test functions

Let F(x) be the fundamental solution for the ODE y''-y=g, i.e. formally F satisfies $F''-F=\delta$. Using test functions, explain that this means $\int_{- \infty}^{\infty} F(\varphi'' - \varphi)= ...
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70 views

Periodic solutions to Riccati equations

Suppose $\alpha, L>0.$ Under what conditions (between $\alpha, L$) the Riccati equation $d\Phi/dz=2i[\Phi(z)^2+\alpha Cos(2\pi z/L)\Phi(z)+1]$ can have a periodic solution with period $L$ (under ...
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45 views

One-Sided Limit Evaluation

I am currently struggling with a homework problem. Actually, it is a part of the problem since it is divided into sections. My problem is the following: Consider the following boundary value problem ...
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96 views

Proof of Uniqueness Theorem

Let $x \in D \subset R^m, t \ge 0$ I would like to prove the theorem of uniqueness of the solution $\frac {\partial u}{\partial t} - \Delta u + F(u,x,t) = f(x, t)$$ u \Bigg|_{\partial D} = \mu(x, ...
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Making a change of variable to transform an equation

$\displaystyle m\frac {dv} {dt} = mg - kv^2$ $\displaystyle\frac {dV}{dT} = 1 - V^2$ Make a change of variable $v=aV$ and $t=bT$, show that for suitable choices of the parameters $a>0$ and ...
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50 views

A question about isoclines.

Let $f$ continuous in $\Omega$ open in $\mathbb{R}^2$. Suppose that an isocline of the differential equation $$y'=f(x,y)$$ satisfies the equation. Then we have to prove that, this isocline have to be ...
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27 views

MultiEquations (with fractions)

Can you please help me solve these equations i don't understand how to solve them with fractions. 1=n-2/15 151/20 =2a+1 3/4 -3/5 -2 1/5k = - 26/25