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

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PDE - Energy - Wave Equation

I dont know how solve a) and b), I'm read the book of Walter Strauss, but I have a lot of doubts, for the c) first, I tried estimate $(k(t)e^{-2at})'$ and integrate the inequal, but not unwind... :( ...
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RLC Circuit and 2nd order linear DE

An RLC circuit consists of a voltage source in series with a resistor, a capacitor, and an inductor. An inductor is a coil of wire. When the current passing through the coil changes, a magnetic field ...
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Validity of approximating a difference equation with a differential equation

Consider the following difference-differential equation defined for positive integer indices $k$ and $t$: $$ A_k(t+1)-A_k(t)=\beta \frac{(k-1)A_{k-1}(t)-kA_k(t)}{\alpha+2\beta t} + \delta_{k \beta} . ...
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Find K values that make a differential equation solution stable

Given some differential equations, ie. "a", or "b": a. $$Y'''+Y''+2Y'+KY=0$$ b. $$Y'''+KY''+3KY'+2Y=0$$ How do I get the $K$ values that make the solution stable? I know that for "a", it should be ...
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2answers
20 views

$0$ is an stable equilibrium of $x' = Ax$ iff $A$ is semisimple, given that all of its eigenvalues have real part 0.

$0$ is an stable equilibrium of $x' = Ax$ iff $A$ is semisimple, given that all of its eigenvalues have real part 0. I'm kind of confused here: I had understood that if all of the eigenvalues of $A$ ...
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22 views

Car traveling on a bumpy road

The suspension system of a car traveling on a bumpy road has a stiffness of $k = 5\times 10^6$ N/m and the effective mass of the car on the suspension is $m = 750$ kg. The road bumps can be considered ...
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1answer
26 views

System of differential equations, need help on correcting the answer I get.

I am solving this problem: $$ x'=z-y, y'=z, z'=z-x $$. The method I used involves eigenvectors. Eigenvalues that I found are: 1, i and -i, and the solution I get is x=0, y=ce^x, z=ce^x. Everything ...
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1answer
33 views

Questions about Lyapunov functions

I'm trying to find a Lyapunov function for the nonautonomous ODE $z'=g(z)$ with $z=(x,y)$ and $$ g(x,y):=(-2x-y^4,-y-x^2). $$ For the sake of analyzing the stability of the rest point $z=0$, one does ...
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18 views

Explanation of the Leibniz formula

I am reading the book Solving Ordinary Differential Equations I - Nonstiff Problems (1987) by Hairer et al. My question is from Section II, chapter 2 (Order conditions for RK methods), equation 2.4. ...
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Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
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17 views

how to find a function $f(x)$ such that $\lim_{x\to \infty} (\frac{y}{ f(x)})=1$

$$\frac{d^2w}{dx^2}+2\frac{dw}{dx}+5w=-5x^2-4x-2$$ Given that $$y=\frac{1}{w}$$ Where The particular solution (I found) is : $$w=e^{-x}(C \cos 2x+ E \sin 2x)-x^2$$ The general ...
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1answer
27 views

show that $A(t)\exp(\int_{t_0}^t A(s)\,ds )=\left(\exp(\int_{t_0}^t A(s)\,ds )\right)A(t)$, when $A(t)$ is symetric.

$A(t)$ is a symetric matrix for $t\in [t_0,a]$. show that $$A(t)\cdot \exp\left(\int_{t_0}^t A(s)ds \right)=\exp\left(\int_{t_0}^t A(s)ds \right)\cdot A(t)$$ it is easy but exhausting to show for ...
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1answer
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changing forms of constant of integration

In solving O.D.E in my book sometimes he changes the constant of integration in the form for example C=Sin(A) where C & A are constants obtain the general solution in an explicit form but how ...
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Solution of this nonlinear equation

How do I solve this equation for $y$? I can see there is a trivial solution $y=0$ but how do I get $y$ as a function of other variables? Does anyone know how to use MATLAB's fzero function to find ...
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29 views

Solve Bratu problem using Python

I am going crazy trying to solve the Bratu problem using Python: $y''(x)+ e^y(x) = 0$, $\lambda = 1$, $x\epsilon[0,1]$, $y(0) = y(1) = 0$ I have to solve this using the tridiagonal matrices. None of ...
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24 views

Differential inequality involving derivatives

I'm having trouble with a differential inequality. Consider a smooth function $f(x)$ defined for $x>0$ with $f'>0$. Given $0< a < b$, show that there exists smooth functions $g(x)$ and ...
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23 views

Writing solution to an arbitrary ODE with arbitrary initial values as the sum of a power series?

Let $f(t), g(t)$ be polynomials, and let $y$ be a function of $t$. Given the ODE $y'' + f(t) y' + g(t) y = 0$ with initial conditions $y(0) = \alpha$ and $y'(0) = \beta$, write $y$ as the sum of a ...
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27 views

Change of Variables in Cauchy-Euler equations

So I'm working on the change of variables in the Cauchy-Euler equation. And I understand everything except one step. It's the same one step skipped in the answer to the very related question here. I ...
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2answers
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Query about differential equations involving projectiles.

A projectile is launched at a speed $U$ at an angle $θ$ to the horizontal from $(x,y) = (0,0)$. Thereafter the projectile moves so that the second derivative of $X$ is $0$ and the second derivative of ...
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1answer
30 views

What is wrong with my general solution and how to find $f(x)$

Given that $$y=\frac{1}{w}$$ Here is my working: $$\frac{d^2w}{dx^2}+2\frac{dw}{dx}+5w=-5x^2-4x-2$$ Auxillary Equation: $$a^2+2a+5=0$$ $$a=-1+2i,-1-2i$$ C.F $$w=e^{-x}(C \cos 2x+ E ...
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1answer
29 views

Please help me understand the concept of variable, and differentiation of variables.

`I am in the first year of college and know mathematical analysis in a very rigorous context, from high school/ math olympiads Imo's etc. But the concept of $df$ seems totally weird and ...
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30 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 : ...
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1answer
32 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? $$yy''=y'-5y$$ $$y''=\frac{y'}{y}-5$$ $$\int{y''dx}=\int{(\frac{y'}{y}-5)dx}$$ ...
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1answer
25 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 ...
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1answer
21 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 $$y_p(t)\stackrel{!}{=}c(t)e^{-t}$$ ...
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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 ...
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1answer
35 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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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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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 ...
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1answer
33 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) ...
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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.
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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 ...
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21 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 ...
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1answer
20 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 $b^2>w^2$ ...
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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. ...
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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 ...
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1answer
15 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): ...
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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 ...
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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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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: ...
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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 ...
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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 ...
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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 ?
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1answer
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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 ...
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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 ...
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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..
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+150

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 ...
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1answer
24 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} ...
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2answers
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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 ...
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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 ...