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

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2
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1answer
18 views

How to get rid of $(\frac{dw}{dx})^2$ term in a differential equation

My try: $$y=w^{-1}$$ $$y'=-w^{-2} \frac{dw}{dx}$$ $$y''=\frac{2}{w^3} \frac{dw}{dx} - \frac{1}{w^2} \frac {d^2w}{dx^2}$$ Substituting these to the first expression : ...
0
votes
1answer
26 views

How do I solve $yy''=y'-5y$ given that $y(1)=1$ and $y'(1)=-1$?

How do I solve $yy''=y'-5y$ given that $y(1)=1$ and $y'(1)=-1$? Do I have to integrate both sides of ODE? $$yy''=y'-5y$$ $$y''=\frac{y'}{y}-5$$ $$\int{y''dx}=\int{(\frac{y'}{y}-5)dx}$$ ...
0
votes
1answer
18 views

Differential equation: general solution for formula

I have following formula and I need the general solution: $$ \frac{d^{2}\theta}{d\xi ^{2}}-\mu ^{2}\cdot \theta =0 $$ EDIT Following solution was given: $$ \theta(\xi )=C_{1}\cdot exp(\mu \xi ...
0
votes
1answer
20 views

If $y$ is the solution of $\left\{y'=-y+\sqrt{t},y(0)=y_0>0\right\}$, then $\lim_{t\to\infty}\frac{y(t)}{\sqrt{t}}=1$

The homogeneous equation $$y'=-y$$ has the solution $$y_h(t)=ce^{-t}\;\;\;\;\;(c,t\in\mathbb{R})$$ In order to find a particular solution we can take the approach $$y_p(t)\stackrel{!}{=}c(t)e^{-t}$$ ...
1
vote
0answers
11 views

How to solve this ode with absolute value

Let $y(t)$ be continuously differentiable, $y(0)>0$, and $$|y'(t)|=|y(t)|, t\geq 0.$$ Then how to show that $y(t)=y(0)e^{t}$ or $y(t)=y(0)e^{-t}$. The main difficulty lies in that I could not ...
2
votes
2answers
26 views

Solving a non linear second order differential equation [on hold]

How do I go about solving a nonlinear ordinary equation that is of second order? Such as $$y'' + ay^3 = 0$$ where $a$ is a constant.
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0answers
4 views

What are difference among natural boundary, exit boundary, regular boundary and killing boundary??

In the paper i'm reading, they used the terminologies, natural boundary, exit boundary, regular boundary and killing boundary. I can't find the difference of them and definition of them. Tell me ...
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0answers
13 views

Why does the set $\pi(C)\cap\pi(D)$ have $\gamma$-measure 1?

I have a question concerning the article Ergodic Theory and Linear Differential Equations by R.A. Johnson. My questions concerns the proof of Lemma 2.3 on page 27, namely the statement ...
1
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1answer
27 views

How to solve system of Differential Equations with 1 independent and 3 dependent variables

How can one solve this set of three differential equations in one independent variable "t" and three dependent variables A, B and F, which are functions of only t? $$ \frac{F(t) B''(t)+B'(t) ...
1
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1answer
25 views

Existence and uniqueness for the ODE $y''-y^{1/3}=0$

For the ode $ y'' - y^{1/3}=0 $, is there any way to check the existence and uniqueness of the solution? I know the Picard's Theorem, but it can only be used for the first order ode.
0
votes
1answer
18 views

Differential equation solution inconsistency

When solving the differential equation: $dy/dx = y^2$, with $y(0) = 1$ I've found $y = 1/(1-x)$ as the solution. The problem asks then for an explanation to why $x=3/2$ is an invalid point to ...
0
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1answer
20 views

Differentials word problem

The Questions Use differentials to find the approximate amount of copper in the four sides and bottom of a rectangular tank that is 6 feet long, 4 feet wide, and 3 feet deep inside, if the copper is ...
0
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1answer
19 views

Overdamped and critically damped

Consider $y''+2by'+w^2y=0$. Show that as the limit of $b\to w$, the overdamped solution is equal to the critically damped solution. The roots are $D=-b\pm\sqrt{b^2-w^2}$. Thus, if $b^2>w^2$ ...
1
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0answers
5 views

Using MATLAB to solve a system of 2nd order non linear ODEs

I have 2 coupled non linear 2nd order ODEs which describe a particle's trajectory in space, subject to an initial horizontal and vertical velocity, and also to gravitational and aerodynamic forces. ...
3
votes
2answers
13 views

The volume of a spherical balloon (constant rate)

The volume of a spherical balloon is increasing at a rate of $3$ cubic inches per second. After you find the rate of change of the balloon's radius at the time when the radius is $8$ inches explain ...
0
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1answer
14 views

Help identifying differential equation extraneous solutions

For the following differential equation: dy/dx = (xˆ2)/y, with initial condition y(0)=10 I've found the solution: (½)(yˆ2) = (1/3)(xˆ3) + 50. However, I've found two answers for y(5): ...
0
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0answers
11 views

Positive Characteristic exponent, then $\lVert x(t)\rVert\to 0$?

Let $x(t)$ be any solution to the ODE $x'=a(t)x$. Say that $x(t)$ has strong characteristic exponent $\beta$ as $t\to\infty$ ($t\to-\infty$) if $\lim_{t\to\infty}\frac{1}{t}\ln\lVert ...
1
vote
1answer
23 views

Elliptic linear ODE

I have a rather short question: What does "elliptic" mean in the the context of linear ODE? Only found "elliptic" in the context of partial differential equations.
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0answers
17 views

Estimating upper bound

Let the following Cauchy Problem be $\displaystyle\cases{ y'(t)=f(t,y(t)) & \cr y(0)=\eta }$ for $t\in[0,T]$ Define the approximation $y_n$ of $y(t_n)$ as: ...
0
votes
1answer
20 views

Linear Second order ODE with oscillating solutions

I encountered the following second-order ODE while tutoring recently, and struggled with the proper approach: $x^2y''+2xy'+\alpha y = 0$ The problem is: for which values of $\alpha$ do solutions ...
2
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0answers
11 views

Derivation of higher order bessel function in terms of lower order functions

I am really stuck trying to prove this.. ((x^-p)Jp(x))’ = -(x^-p)Jp+1(x) ---(1) Can someone please help how to actually prove this step by step, because whichever notes i see, they prove ...
0
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0answers
17 views

why equilibrium points are important in ODE theory

Why equilibrium points are important for the study of differential equations $\dot{x(t)} = h(x(t)$? There can be arbitrary sets which are stable, why stable "equilibrium point"s are important ?
0
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1answer
15 views

A basic question on equilibrium point of coupled differential equation

The system of ordinary differential equations given by $$ \dot{x_1}(t)= k + \sin(x_1 + x_2) + x_1$$ $$ \dot{x_2}(t)= k + \sin(x_1 + x_2) - x_1$$ do not have any equilibrium point for $k >1$. Why ...
0
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0answers
10 views

Why choosing small values of Alpha and Beta gives inefficient Runge Kuttan method

I have been trying my hand at a past exam paper and one of the questions is as follows: The second-order Runge-Kutta method to solve the equation $ \frac {dy} {dx} = f(x,y)$, $y(x_0)=y_0$ at the ...
0
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0answers
18 views

Solving Duffing equation by Matlab ode23

How can I use Matlab to solve numerically this duffing equation with known $\kappa, \Gamma, \omega$..thanks.. $$x'' +\kappa x' +x -x^3 =\Gamma \cos\omega t$$ I have only few knowledge of Matlab..
2
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0answers
35 views

Constructing Lyapunov function for system of ODEs

Background: I have been working on this problem for my research for months now, and I am in dire need of help. That is why I have come here to seek help. I have a system of nine ODEs that describe ...
0
votes
1answer
22 views

how to show this manipulation in the integral

Let we have: $$G(t)=y_1(t)\int y_2(s)ds$$ when we take the limits as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds$$ then is it possible to write it as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds=\int^t_{t_0} ...
1
vote
2answers
12 views

Question in undetermined coefficient method for ODE

How should I formulate particular solution of this ODE? I want to use method of undetermined coefficients. $$ y'' - y = e^x \\ y_H = C_1 e^x +C_2 x e^x $$ $y_H, y_P$ are homogeneous and particular ...
0
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0answers
17 views

On Nonlinear Autonomous system of two equations if the eigenvalues of the Jacobian matrix are 0.

Suppose we have a non-linear autonomous system of two equations: $$\begin{cases} x'(t) = F(x,y) \\ y'(t) = G(x,y) \end{cases} $$ and we obtain a fixed point for this equation, but the eigenvalues of ...
0
votes
1answer
17 views

Solution of an ODE (Show equation)

Here are the preliminaries of my question: Let $\Omega$ be a compact metric space, and suppose a flow $(\Omega,\mathbb{R})$ is given. Let $A\colon\Omega\to M^2$ be continuous. Here $M^2$ is ...
0
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0answers
22 views

Wronskian, Linear Independence

Show that the following functions are linearly independent: $$e^t\begin{bmatrix}1\\1\\0\end{bmatrix}$$ $$e^{2t}\begin{bmatrix}0\\2\\1\end{bmatrix}$$ $$e^{-t}\begin{bmatrix}1\\0\\1\end{bmatrix}$$ ...
0
votes
0answers
10 views

Showing that a collection of m solutions is linearly independant

Show that a collection $ \Phi_1 .. \Phi_m $ : I-->R of continuous functions satisfying $ \\ $ $ \int_I(\Phi_J(t)\Phi_k(t)dt $ =1 when j=k , 0 when j$\neq$k $ \\ $ is linearly independent. Multiply the ...
1
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3answers
38 views

How do you solve ODE $x' = x^2$? [on hold]

How do you find all the solutions for that ODE?
1
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1answer
42 views

Explicit formula for the implicit Euler method

Given the problem; $\displaystyle\cases{ y'(t)=y^2(t) & \cr y(0)=1 }$ for $t\in[0,1]$ Using the implicit euler method, find an explicit formula to get $y_{n+1}$ HINT: The ...
1
vote
2answers
39 views

On a linear 3x3 system of differential equations with repeated eigenvalues.

I have the following system: $$\begin{cases} x'= 2x + 2y -3z \\ y' = 5x + 1y -5z \\ z' = -3x + 4y \end{cases} $$ $$\det(A - \lambda I)= -(\lambda - 1)^3$$ the eigenvector for my single eigenvalue ...
1
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0answers
27 views

Convert time derivative to a function of time

Physics: I am asking for help to derive a general expression for the total amount of energy lost as a function of time from a radiating object. I'll simplify my problem like this: Say for example ...
0
votes
1answer
30 views

First-order nonlinear ordinary differential eqauation

Can someone help me to solve the equation $y'=\dfrac{y}{x}\left(\dfrac{xy + 1}{xy - 1}\right)?$ I have been trying a few methods. Thanks. $P=xy^2+y,$ $Q=-(x^2y-x)$ I tried to make it exact ...
0
votes
1answer
21 views

polynomial solution of second order differential equation

Find the polynomial solution $$u_n(x) = x^n + a_1x^{n-1}+...+a_n$$ of the differential equation $$u_n'' + xu_n' - nu_n = 0$$ satisfied by u_n(x). Note that this is entry-level calculus, so in my ...
1
vote
1answer
32 views

Analytic solution for a type of PDE systems

Peace be upon you, I have the following system of partial differential equations \begin{align*} \begin{cases} \frac{\partial}{\partial a}S(a,b,c,d)=f_1(a)\\ \frac{\partial}{\partial ...
0
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0answers
33 views

Duffing equation of forced spring motion

The motion of a forced spring is described by the equation $$x'' + \kappa x' +x-x^3 =\Gamma \cos(\omega t)$$ We wish to investigate the stability of solutions of this equation having the forcing ...
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0answers
9 views

ODE -parabolic cylinder functions

How do we solve $\frac{d^2f}{dz^2} + \left(Az^2+Bz+C\right)f=0 \tag 1$ where $f(z),A,B,C$ are matrices of order $3 \times 3$.
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0answers
15 views

bibliography for weak solutions of ODE's

Some one could recommend to me some bibliography about weak solutions of ODE's, and solutions of ODE's that are not lipschitz??
0
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2answers
40 views

solving a second-order non-linear differential equation

Good day. Could you help me to solve the DE $$ y''=\sqrt{1+y'^2} $$ I have tried to write the equation in terms of $y'=z, y''=z'$, which results in the new DE $$ z'=\sqrt{1+z^2} $$ but then I got ...
1
vote
1answer
23 views

Show that the function is positive

Let $f:\mathbb R\to\mathbb R$ be a Lipschitz continuous, monotone increasing function, with $f(0)=0$, if a function $\phi$ satisfies; $\displaystyle\cases{ \phi'(t)=f(-\phi(t))-f(\phi(t)) ...
1
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0answers
21 views

Get a special form of an linear System of ODE (using polar form)

In this post Converting an ODE in polar form it is shown that a linear system of ODE $$ x'=\begin{pmatrix}a(t) & b(t)\\c(t) & d(t)\end{pmatrix}x $$ can be written in polar coordinates ...
0
votes
1answer
30 views

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$ I integrated it once to get, $2x + \sin x + C$, $C$ being a constant. Then I integrated it a second time to get $x^2 - \cos x ...
1
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1answer
18 views

Converting an ODE in polar form

Convert the ODE system $$ \dot{x}=\begin{pmatrix}a(t) & b(t)\\c(t) & d(t)\end{pmatrix}x $$ into polar form. You should get two equations $$ \frac{d}{dt}\Phi(t)=...\\ ...
0
votes
0answers
7 views

What are some planes (spaces) akin to the trace-determinant plane in other disciplines?

When studying basic differential equations, I found the trace-determinant plane incredibly illuminating. Similarly, I find it very helpful to see different kinds of conics as slices of a cone. What ...
1
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1answer
20 views

System of Linear differential equations with variable coefficients

Could someone please suggest a technique for solving the following linear system of ODEs: $$ \begin{array}{l} i\alpha \frac{{dx(q)}}{{dq}} = \left( {\beta - 2c\cos (q)} \right)x(q) - ig\,y(q)\\ ...
1
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0answers
18 views

Prove $x\to 0$ as $t\to \infty$ if we consider the system of equations $x'=(A+B(t))x$ where $B(t)\to 0$ and $A$ has negative eigenvalues.

Consider a matrix $A$ such that all of its eigenvalues are negative. Consider $B(t)$ where $B(t)\to 0$ as $t\to\infty$. Then consider the system of equations $$ x'=(A+B(t))x$$ Prove that any ...