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

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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: ...
2
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1answer
30 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
30 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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0answers
20 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 ...
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1answer
43 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 ...
0
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1answer
26 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
30 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: $$ ...
0
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1answer
12 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
25 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
93 views

Books various maths subjects

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
54 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
36 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
24 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
33 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: ...
1
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1answer
26 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
20 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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0answers
16 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. ...
0
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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
146 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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Finding differential equation satisfied by the following families of curves [on hold]

How to find differential equation satisfied: $$y=e^{Cx}$$
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3answers
41 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
38 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
23 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 ...
2
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1answer
17 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
27 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. ...
1
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1answer
14 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 ...
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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 ...
0
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2answers
40 views

Second Order Non linear Differential Equation [on hold]

I have arrived at a differential equation and I need to solve for $x$. $$\frac{d^2x}{dE^2}+Hx =a\left(1+\frac{J}{x^4} -\frac{1}{2x^2}\right)$$ Thank you
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0answers
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
23 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 $\sin(\frac{\pi n}{a}x)$ ...
0
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1answer
8 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 ...
0
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1answer
59 views

Initial conditions for second order ODE with complex stiffness

I'm trying to find initial conditions to ensure systems of the form stay bounded $\ddot{x}_i+\sum_{j=1}^N k_{ij} x_j = 0, \quad k_{ij} \in \mathbb{C}$. For simplicity let's say the $k_{ij}$ lie in ...
2
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2answers
55 views

Solving the differential equation $x^2y''+xy'-y=x^2$

$$x^2y''+xy'-y=x^2$$ My attempt: Divided by $x^2$: $$y''+\frac{y'}{x}-\frac{y}{x^2}=1$$ Now to solve the homogenous equation using Euler's method $$y''+\frac{y'}{x}-\frac{y}{x^2}=0$$ To ...
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1answer
51 views

Python vs Matlab? [on hold]

I've the problem that I have to transform a function (a rosenbrock-wanner method of 2 order) which is written in python to a matlab-function. Unfortunately I've never done anything with Python and ...
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0answers
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Find the greens function of the following non homogeneous problem:

The problem is 100(\left(\fracdy^2dx^2)\right) + y =f(x) with Boundary conditions of y(0)=y'(10pi)=0. \left(\frac12\right) As worked out the general solution is y(x) = Acos(x/10) + Bsin(x/10). ...
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1answer
25 views

Find two linearly independent solutions of a Legendre equation about $x=0.$

Here is the statement of the problem: Consider the Legendre Equation $$ (*)\qquad (1-x^2)y''-2xy'+2y=0 $$ (a) Find two linearly independent solutions about $x=0$, solving completely any relevant ...
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3answers
30 views

find the general solution to the following homogeneous differential equation.

$$100\frac{dy^2}{dx^2} + y = 0$$ Is this worked out by using the auxillary equation such that: $$100m^2 + 1 = 0$$ so $m = \pm i\sqrt{1/100}$ ? So the general solution would be $y(x) = A cos ...
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1answer
26 views

How would you integrate this homogeneous equation?

I am solving a homogeneous equation $\frac{dy}{dx}= \frac{x^2+xy+y^2}{x^2}$ and have come to this step and I'm stuck now with the integration. I could really use some helpful hints to help me $$ ...
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2answers
16 views

Determining the interval where the solution is valid

I am given the initial value problem $$ y' = \frac{1+3x^2}{3y^2-6y} $$ given y(0)= 1 I have solved this and I got $y^3-3y^2 -x-x^3=-2$. How would I got about finding the interval in which the ...
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3answers
32 views

function bounded by an exponential has a bounded derivative?

here's the question. I want to be sure of that. Let $v:[0,\infty) \rightarrow \mathbb{R}_+$ a positive function satisfying $$\forall t \ge 0,\qquad v(t)\le kv(0) e^{-c t}$$ for some positive constants ...
2
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4answers
80 views

Hints on solving $y''-\frac{x}{x-1}y'+\frac{1}{x-1}y=0$

$$y''-\frac{x}{x-1}y'+\frac{1}{x-1}y=0$$ Is there any simple method to solve this equation? I need hints please $\color{red}{not}$ a full answer
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1answer
49 views

Prove that a second order diff. eq. has only two linearly independent solutions.

Let $p(t)$ and $q(t)$ be two continuous functions. Prove that the second order linear equation $$y'' + p(t)y' + q(t)y = 0$$ has two, and only two linearly independent solutions. $\textbf{Sketch of ...
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1answer
27 views

How to prove that the BVP has only the trivial solution? [on hold]

How to prove that the BVP $$x''+f(t)x'+g(t)x=0, t\in[0,1],$$ $$a_1x(0)+b_1x'(0)=0,$$ $$a_2x(1)+b_2x'(1)=0,$$ where $f,g\in C[0,1]$ and $a^2_i+b^2_i>0, i=1,2,$ are constants has only the trivial ...
2
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0answers
48 views

Preparations to finals, validation needed

I have an exam in a few days from now and I'm very nervous. I tried to tackle this one with all I got, but I'm not sure if I'm correct. If anyone can direct me, and tell me if and where I'm doing ...
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2answers
100 views

what would a planetary orbit look like if gravity had constant magnitude?

Consider a unit-mass particle that is always experiencing a single unit-magnitude force towards the origin. This is a central force, but it is not one of the familiar ones, e.g. gravity whose ...
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3answers
28 views

Solving an ODE by inspection

I am trying to solve the following ODE by inspection $$(x-1)y''-xy'+y=0$$ So that method that is recommended is to guess the general form. EDIT : If you guess the general form $y=c_0+c_1x+c_2x^2$. ...