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

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Recognizing and using hypergeometric function

Some expressions that interest me end up having something to do with hyper geometric function. I want to be able to derive such results myself. Where do I begin? For example, the equation $$ ...
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0answers
11 views

Solving a system of ODEs

I have three ordinary differential equations, with 3 dependent variables, $n_i$, $n_e$ and $\phi$, as follows: $$\nabla (-\frac{dn_i}{dx}-k_1n_i\frac{d\phi}{dx})=0$$ $$\nabla ...
2
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1answer
44 views

The real equation of a pendulum

In physics I never solve the equation $\ddot\theta = \sin(\theta)$. Instead, we used the approximation $\theta = \sin(\theta)$ for small angles and then it was easy to solve. I didn't do any physics ...
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0answers
37 views

A system of partial differential equations

I have 6 partial differential equations that in the first look they don't seem very difficult, but all my efforts for solving them were unsuccessful. $$\frac{\partial f(x,y,z)}{\partial ...
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0answers
22 views

Finding a Lyapunov Function for a system involving a trigonometric function

I'm dealing with determining if $(0,0)$ is stable or not for the following system via constructing a Lyapunov function. The system is $$ \begin{cases} x'(t)=(1-x)y+x^2\sin{(x)}& \\ ...
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0answers
19 views

Find the population [on hold]

Every year, the emigration rate from country A to B is 𝛼 (0 < 𝛼 < 1), whereas the emigration rate from country B to A is 𝛽 (0 < 𝛽 < 1). Note that the fluctuation in population of both ...
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3answers
36 views

differential equation with substituion

Solve for y: $y'tan(x+y)=1-tan(x+y)$ so far I have made the substituion $u=x+y$, which yields $\frac{du}{dx}=1+\frac{dy}{dx}$. However, I am not sure what to do from here.
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2answers
23 views

Is it possible to say that $L(f^n)=s^nL(f)$ when the differential equation is not in the rest condition?

Question Use the Laplace transform to solve the following equation: $y'+2y=\cos(3t)$ ; where $y(0)=1$ In class our teacher wrote that "When in rest condition: $L(f^n)=s^nL(f)$", but I want to use ...
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3answers
34 views

Existence/uniqueness of a Continuous Function

I ran across the following problem with a friend while we were studying for quals. Neither of us are really quite sure where to start. It feels like a differential equation. This is probably easy, ...
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3answers
38 views

Find the inverse Laplace transform of: $\frac{1}{(s^2+a^2)(s^2+b^2)}$

I'm having trouble doing this homework problem because I'm not sure how to deal with the $a$ and $b$. I did it the usual way we were taught - use partial fraction decomposition and then try to solve ...
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1answer
49 views

Non linear second order ODE up to $O(\epsilon) $ for $v_{xx}-\left(v^{3}-v\right)-\varepsilon\frac{1}{2}\left(1-v^{2}\right)=0$

I really need help solving this : $$v_{xx}-\left(v^{3}-v\right)-\varepsilon\frac{1}{2}\left(1-v^{2}\right)=0 $$ With boundary conditions : $$ v(\pm \infty )=-1+\frac{1}{4}\epsilon $$ I need to ...
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0answers
29 views

Taylor Series Methods with Quadrature. Local Truncation Error [on hold]

I don't know where to begin with this question. Advice would be helpful. Suppose that a differential equation is solved numerically on an interval $[a,b]$ and that the local truncation error is ...
-2
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3answers
37 views

Differentiation with respect to a constant variable? [on hold]

Let $y=f(x)$. If we are trying to find $f^{\prime}(x)$ and we know that in the domain we are trying to find $f^{\prime}(x)$ in, $x$ is constant , then what is $f^{\prime}(x)$? Is it zero?
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4answers
92 views

If $f(x)\ll1$ is it safe to assume that $f^{\prime}(x)\ll1$?

If $$f(x)\ll1$$ is it safe to assume that $$f^{\prime}(x)\ll1$$
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0answers
22 views

Proving one single solution for an ODE

let $f(y)$ be a continuous function in $R$. $f(y)=0$ only for $y=y_0$. as a result the integral: $\int_y^{y_0} \frac1{f(x)} dx $ diverges for every $y$. prove that the following problem: ...
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1answer
37 views

Wronskian of two differential equation solutions

Let $f$ and $g$ be the solutions of the homogeneous linear equation: $$y'' + p(x)y' + q(x)y = 0$$ and $p(x)$ and $q(x)$ are continuous in segment $I$. Is it true, that if the wronskian of $f$ and ...
2
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2answers
34 views

Differentiation with dependent variable

Let $$ F(x, y) = x^3 + 7 y^2 x^4 - (2 x - y)^3 $$ and let $y=f(x)=x^2+1$. Is it correct to write $$ \frac{\partial F}{\partial y}=\frac{\partial F}{\partial x}\frac{\partial x}{\partial y}? $$ ...
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1answer
52 views

Are there closed curves for which acceleration is orthogonal to position?

Can we find $\vec{f} : \mathbb{R}\rightarrow \mathbb{R}^3 $ such that $\vec{f}(t) \cdot \frac{d^2 \vec{f}(t)}{dt^2} =0$ and $\vec{f}(0) = \vec{f}(T)$ for some $T >0$ ? Exclude the trivial cases. I ...
0
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2answers
53 views

Solve the following initial value problem: $2y''+y'-y=e^{3t}$

$$ 2y''+y'-y=e^{3t}; \text{ with } y(0)=2,\ y'(0)=0 $$ I got to this point: $$ L(y)=\frac{1}{(s-3)^2}\cdot\frac{1}{(2s-1)(s+1)} $$ but now I'm not sure what to do with these polynomials. I know ...
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1answer
28 views

solving equation in terms of $w_1$ and $w_2$

I have a a physics problem involves the following equation $$\tan(\alpha) = \frac{(w_1 + w_2)^{1/2}}{w_3}$$ from a certain set of equations that I use I derive the following equation: ...
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1answer
31 views

Why can we subtract two terms and use this as a way to simplify the overall expression?

Our professor was doing a Laplacian transform example in class. Original problem: $$ y''+4y=\sin t $$ He was working on the problem and got to this step: $$ ...
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1answer
14 views

ODE boundary condition and integer values?

When separating the variables of the 3d wave equation we generate (amongst others) the following ODE. \begin{equation} \frac{d^2\Phi}{d\phi^2}=-m^2\Phi \end{equation} Where $m$ is the separation ...
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1answer
26 views

ODE, Picard approximation of a second order equation: How do I make sure that this is correct.

I have the following problem: $$\ddot{x} + \dot{x}^2-2x=0$$ and I.V are: $x(0)=1 \qquad$ $\dot{x}(0) = 0$. and I need to find two first "Picard" approximations. I first arranged it in the form ...
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3answers
101 views

Books various maths subjects [on hold]

I am a Civil Engineering student and i am planning on following physics in my career.I want to be ready for the advanced undergraduate courses that i will attend to,so i need to learn Differential ...
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2answers
55 views

Practical use for negative $dt.$

I am writing a section of notes for Calculus 1 on related rates. In the section where I discuss differentials, I write that the quantity $dt$ must be nonnegative. I imagined the only reason it would ...
2
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2answers
39 views

Finding the kernel, eigenvalues, and eigenvectors of the operator $L(x) := x'' + 3 x' + 4 x$

I want to find the kernel, eigenvalues and eigenvectors of the differential operator: $$L(x)=x''+3x'-4x$$ on the $\Bbb C \space \space \text{vectorspace} \space \space C^{\infty}(\Bbb R)$ as well ...
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1answer
25 views

Differential Equation with a Motorboat

A motorboat and its load weigh 2150N. Assuming the propeller force is constant and equal to 110 newtons and water resistance is equal numerically to 6.7V Newton where V is the velocity at any instant ...
2
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2answers
35 views

stability of equilibria for $n$-dimensional nonlinear systems of differential equations: examples

I'm currently self-studying dynamical systems. I'm trying to summarize what can be said about the stability of equilibrium points for an $n$-dimensional non-linear system of differential equations: ...
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1answer
28 views

How close should be the boundary value of $x$ and $y$ to ensure that $|x(t)-y(t)|<0.1$

I was given the following differential equation: $$y' = \sin y\cdot \sin t+y\cos t$$ Say that $x(t)$ and $y(t)$ solve this equation, and that $x(t_0) = x_0$ , $y(t_0)=y_0$. Find $\varepsilon$ small ...
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0answers
22 views

Name of numerical methods for second-order differential equation

Numerical methods that try to solve first-order differential equations of the form: $$ \frac{\partial}{\partial t} y = f(y,t) $$ are often Runge-Kutta methods, and there is a whole family of ...
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17 views

instability of a differential equation

I have the following simple differential equation ; $$\frac{dz}{dt}=\left(\alpha_{1}+\alpha_{2}\right)-q_{t}\left(z_{t}-1\right)$$ I know that $\alpha_{1}$ and $\alpha_{2}$ are positive constants. ...
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1answer
11 views

What's the difference between wave equation in PDE form and wave equation in normal form?

What's the difference between "wave equation in partial derivative form" and "wave equation in y(x,t) form" ? Are they both same? And why "wave equation in in y(x,t) form" is the solution of "wave ...
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1answer
48 views

Differential equations, chemical reactions

A chemical substance A changes into substance B at rate $\alpha$ times the amount of A present. Substance B changes into C at rate $\beta$ times the amount of B present. If initially only substance A ...
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3answers
148 views

Transforming to a homogeneous equation

Consider the equation $$\frac{dy}{dx}=F(\frac{ax+by+c}{dx+ey+f})$$ Show that if $ae \neq bd$ then there exists constants $h \; , \; k$ such that the substitution $x=z-h$ and $y=w-k$ converts the ...
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27 views

Finding differential equation satisfied by the following families of curves [on hold]

How to find differential equation satisfied: $$y=e^{Cx}$$
3
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3answers
47 views

How to solve the following problem?

How to solve the following ODE? $$y′ − y = 2x − 3;\ y(0) = 1$$
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2answers
41 views

Finding solutions to the ODE [on hold]

Please help to find general and particular solutions which satisfy the given additional condition: $y′ = e^{x+y};\ y′(1) = 1$
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1answer
25 views

Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Let $\Omega\subseteq\mathbb{R}^n$ be a domain with a smooth boundary $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $p>1$ such that $$p<\begin{cases}\infty&\text{, if ...
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2answers
23 views

ordinary differential equations-morphogen gradient

I am reading a paper by Merkin and Sleeman (2005) Find the approximation solution of $(u')^2=\frac{2}{k}(u-\frac{1}{k}\ln(1+ku)); ~~u(0)=1$ for $k$ sufficiently small. they gave the following ...
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0answers
48 views

Two ODEs, why is one solution the solution of the other?

Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t. $$u'(t) = F(t,u(t))$$ $$u(0) = u_0$$ given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ Caratheodory, and we know that if it has a solution, it ...
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1answer
23 views

How do I find invariant lines for a system of differential equations?

How do I find invariant lines for the following system of differential equations: $$x' = 2x - xy + x^3$$ $$y' = y - xy$$
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1answer
31 views

Second order differential equation with multiple bessel functions

I have an differential equation which is $af(R)=\frac{1}{R}\frac{\partial}{\partial R} \sqrt{R}\frac{\partial}{\partial R}\left(f(R) 3\nu\sqrt{R}+g(R)cR^2\right)$ where $c,\nu, a$ are all constants. ...
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1answer
15 views

System of separable diff. eqns, explicit solution and curves, Lotka-Volterra model

In the book on p.68 is a system of differential equations for a Predator-Prey model (Lotka-Volterra) given as: $$ \dot x=x(\alpha-c\gamma) \\ \dot y=y(\gamma x -\delta) $$ On the next page, it is ...
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1answer
56 views

Operator theory curiosity

I'm not an expert in operator theory... but i was wandering if there's some practical applications. For example (the first one i came up with) compared to normal calculus techniques that usually the ...
1
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2answers
333 views

why the standard deviation is not as the same as online calculator

I need to calculate the standard deviation for these numbrs: -12 -3 0 -13 8 -6 0 -22 -1 7 -7 1 -2 -13 -4 0 -6 -4 -10 3 I did everything, but still my answer is ...
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2answers
52 views

Second order nonlinear differential equation $x''+Hx =A(1-J/(2x^2))$ [on hold]

I have arrived at a differential equation and I need to solve for $x$. $$\frac{d^2x}{dE^2}+Hx =A\left(1-\frac{J}{2x^2}\right)$$ where H,A and J are constants. I know that I can use elliptic ...
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32 views

Solutions of the following differential equation [on hold]

$$\frac{-2q}{k}+z^2+2zp-2zN+(p-N)^2=0$$ What is the solution of this differential equation? Where $N$ is a constant and $p$ and $q$ are the usual notations.
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1answer
30 views

Connecting a mathematical solution to a differential equation with it's physical solution

I have seen this question in a neuroscience course: It is given after the lecture with these and these slides. I have no background in physics. However, I do know how to solve a differential ...
2
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1answer
24 views

First Eigenfunction of Simple Equation

Consider the interval $[-a,a]$ and the following problem: $$\phi'' + \lambda\phi=0$$ $$ \phi(\pm a) = 0. $$ The obvious sequence of orthogonal eigenfunctions seems to be $\sin(\frac{\pi n}{a}x)$ ...
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1answer
9 views

Need help plotting this direction field in Maple: vars must be declared as list [on hold]

I'm having trouble trying to plot this ODE's direction field in Maple. dv/dt=9,8-(v/5) I'm running ...