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

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7 views

Evaluating a variation to Cahn–Hilliard equation

i need to find an $ \mathcal{O}({\varepsilon})$ accurate solution to the following : $$ \left(\partial_{xx}-W''\left(u\right)\right)\left(u_{xx}-W'\left(u\right)\right)=\varepsilon $$ \begin{cases} ...
1
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1answer
24 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
16 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
14 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
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
143 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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0answers
23 views

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
36 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
34 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
22 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
22 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
43 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
16 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
22 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
53 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
332 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
35 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
29 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 $\sin(\frac{\pi n}{a}x)$ ...
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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
52 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
20 views

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). ...
0
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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 ...
0
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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
15 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
47 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 ...
9
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2answers
99 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 $y=c_0+c_1x+c_2x^2$. ...
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1answer
28 views

A Problem That Involves Differential Equations, Implicit Differentiation, and Tangent Lines of Circles

Here is the Statement of the Problem: Consider the family $\mathbb F$ of circles given by $$ \mathbb F:x^2+(y-c)^2=c^2, c \in \mathbb R. $$ (a) Write down an ODE $y'=F(x,y)$ which defines the ...
1
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1answer
15 views

Constructing a linear first order ODE with convergent solutions.

I am studying for a test and cannot figure out for the life of me how to do this problem. I need to construct a first order linear ODE in the form of $y'+p(t)y=g(t)$ such that all of the solutions of ...
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2answers
11 views

Find the time that must elapse for the object to reach 98% of its limiting velocity?

I am given the initial value problem $$ \frac{dv}{dt} = 9.8 - (\frac v5) $$ and you are given $v(0) = 0$ I was looking at the solution to this problem. They first solved the differential ...
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3answers
43 views

How to solve $y' = -2x -y$

My thought: $\displaystyle\frac{dy}{dx}+x^0y=-2x$ Considering it as the form of linear equation, $\displaystyle\frac{dy}{dx}+P(x)y=Q(x)$ Multiplying $e^{\int1dx} = e^x$ on both sides, ...
2
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3answers
57 views

Hints on solving $y'=\frac{y}{3x-y^2}$

$$y'=\frac{y}{3x-y^2}$$ My attempt: $$\frac{dy}{dx}=\frac{y}{3x-y^2}$$ $$dy\cdot(3x-y^2)=dx\cdot y$$ $$dy\cdot3x-dy\cdot y^2=dx\cdot y$$ Any direction? I need hints please ...
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2answers
40 views

Am I solving these initial value problem correctly?

I was just hoping someone could check my work and tell me if I'm solving these types of problems correctly? (Large image version)
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1answer
41 views

Implicit equation. Can it be solved?

Is it possible to find a function $x:[0,T]\to [0,x_0]$ such that, for a fixed $0<\lambda<1$ we have: $$\dfrac{1}{1+\lambda}\left (1-\dfrac{x(t)}{x_0}\right )^{1+\lambda} +\dfrac{1}{1-\lambda} ...
2
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1answer
57 views

Advanced calculus: Solving quaternion differential equations

I have a system of two differential equations $$\frac{\partial X(t)}{\partial t}=a_1 A X(t)+a_2X(t) B+a_3 C Y(t)+a_4Y(t) D+a_5$$ $$\frac{\partial Y(t)}{\partial t}=b_1 E X(t)+b_2X(t) F+b_3 G ...
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0answers
11 views

Usage of Phase Portrait of a system of 2 linear first order ODEs

Let's say have a linear system $\frac{\mathrm{d}\underline{y}}{\mathrm{d}t} = A\cdot \underline{y}$, let say 2 dimensional, and I have $\lambda_1,\lambda_2$ eigenvalues of $A$ and ...
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2answers
46 views

How to show an ODE system has no global solution

Starting from any $(x_0,y_0,z_0)\in \mathbb{C}^3$, can the following ODE system have a solution for all real number? \begin{align} x'(t) &=3 y^2(t) \\ y'(t) &=2 x(t) z(t)-1 \\ z'(t) &=0 ...
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0answers
19 views

How much of the chemical will be in the pond after a very long time?

A pond initially containing 1000000 gal of water and an unknown amount of undesirable chemical. Water containing 0.01 gram of this chemical per gallon flows into the pond at a rate of 300 gal/hr. The ...
0
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2answers
18 views

Volume estimation with differential equations

The problem reads: "Using differential equations, estimate the volume necessary to build a tube that is 12m long and has an inner diameter of 25cm and an outer diameter of 25,2 cm." Unfortunately I ...
2
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0answers
23 views

Choose Scaling for t

My question is the last part of the d) part of the exercise 1.17 in Mark Holms' Introduction to Applied Mathematics. The exercise is given below, where I have emphasized the part of it that is my ...