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

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Help with Euler Equations

This is from my textbook. Can someone give me a better explanation of what to do here? What does part (a) mean, i.e., how am I supposed to write $x = ln(t)$ in terms of $\frac{dy}{dx}$ and ...
2
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1answer
30 views

When can you take the limit of a parameter before solving the differential equation?

Short example: consider the differential equation \begin{align*} f'(x)=\frac{k^2}{k^2+k+1}xf(x) \end{align*} where $k$ is a parameter. Wolfram Alpha tells me that the solution to this equation is ...
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0answers
7 views

Long-time behavior of semilinear autonomous systems

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain $f\in C^1(\overline\Omega\times\mathbb R)$ $u_0\in C^1(\overline\Omega)$ with $u_0=0$ on $\partial\Omega$ $u\in C^0(\overline\Omega\times ...
2
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3answers
30 views

Forming differential equation

I'm trying to get from: $$e^{\lambda t} (\frac{dN}{dt} + \lambda N) = re^{\lambda t} $$ To: $$ \frac {d}{dt}(Ne^{\lambda t}) = re^{\lambda t} $$ However I'm not sure what procedure to use to go ...
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2answers
25 views

question on second -> first order systems [duplicate]

I have heard that it is possible to write second order IVP as first order system. What are some strategies to writing $y''=xy^2$, $y(0)=1$, $y'(0)=2$ as a first order system $y'=f(y)$, $y(a)=y_0$? ...
1
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1answer
58 views

Units of ODE solution don't match

I have to solve the differential equation: $v\,'=g-cv$. Sorry in advance for lack of latex. I will learn it soon, please let me make a question using the common programming notation for my ...
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0answers
24 views

Separation of variables, Homogeneous or Exact Differential equations?

So I've just encountered these three, during exams of course they don't tell you which one is to use, if you need to use separation or homogeneous or exact. I was just wondering is there like a ...
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4answers
82 views

Solve the equation $(m^2-m-2)x=m^2+4m+3$

Here's how I solve it I think that m is the variable (am I right?). Then $$m^2x-mx-2x-m^2-4m-3=0$$ $$m^2(x-1)-m(x+4)-(2x+3)=0$$ $$D=x^2+8x+16+4(x-1)(2x+3)$$ $$=x^2+8x+16+4(2x^2-2x+3x-3)$$ ...
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0answers
15 views

The number of characteristic curves of a pde

When a partial differential equation is elliptic, $B^2-4AC\lt 0$ and eigenvalues are complex. does there exist any characteristic curves?
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0answers
19 views

Simple related rates derivative question

Rafael is walking away from a $12$-ft-tall lantern at a constant speed. If the tip of Rafael's shadow is moving twice as fast as he walks, how tall is Rafael? I'm confused on the step where $dL/dt = ...
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1answer
18 views

Finding a power series solution for a given differential equation and identifying the function represented by the power series.

Find a power series for the solution of the differential equation $y'(t)-2y(t)=0 ,\ y(0)=5$, and then identify the function represented by the power series. (I use the following information ...
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1answer
56 views

Flow of sum of non-commuting vector fields

Let $V,W\in\Gamma(M)$ be any two vector fields. Is there any "nice" expression for the flow of $V+W$ in terms of the flow of $V$ and the flow of $W$? It would be sufficient for me to have some sort of ...
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3answers
48 views

Lyapunov stability at origin with identically zero test function

At the origin, determine stability of $$x' = y \\ y' = -\tan(x)$$ If we use the test function $V(x,y) = 0.5y^2 + \int_0^x tan(s)ds$, we get $\dot{V}=x'\tan x +y'y = y\tan x -y\tan x = 0$, so the ...
1
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1answer
21 views

Norm bound on exponential matrix with eigenvalue negative real part, proof

If $A$ is $n \times n$ with negative real parts of all eigenvalues, then there exists positive $K,\alpha$ such that $$\|e^{At}\| \leq Ke^{-\alpha t}$$ Furthermore, if an eigenvalue has negative part ...
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2answers
41 views

Euler Cauchy equations, change of variables

To convert an euler cauchy: $x^{2}y''+pxy'+qy=0$ equation into a linear one we perfom the substitution $x = e^z$ from which we get: $$z=\log x$$ $$\frac{\mathrm{d} x}{\mathrm{d} z} = e^z =x $$ ...
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1answer
68 views

How to integrate $\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$?

How to integrate: $$\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$$ I don't really have a clue? Do I need to simplify it first somehow?
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2answers
30 views

Solving $\frac{b}{a-b}e^{at}=\frac{x(t)}{a-x(t)}$ for $x(t)$

I`ve been trying to solve the differential equation $x(t)'=x(t)(a-x(t)), x(0)=b, t\in [0, \infty]$. Using the technique of seperation of variables, I get $\frac{b}{a-b}e^{at}=\frac{x(t)}{a-x(t)}$. Now ...
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2answers
33 views

What makes a differential equation, linear or non-linear?

Among these differential equations why one is linear while other is non-linear? What is criteria to find out whether a differential equation is linear or non-linear?
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1answer
26 views

Ordinary differential equation . [on hold]

The roots of the auxiliary equation for a homogeneous linear differential equation with real constant coefficients that has $ y= 4 + 2x^2 - e ^{-3x}$ as a particular solution are : 1) $ m= 0 , 0 , ...
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0answers
34 views

Proving this is the unique solution to this simple system of diff equations.

So the set of equations are these $\frac{d \omega_x}{dt}+\Omega \omega_y =0$ $\frac{d \omega_y}{dt} - \Omega \omega_x =0$ You can easily differentiate again, get two second order linear diff ...
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0answers
33 views

how to rearrange matrix equation to have unknown in vector form

I am looking for the name/type of following equations: $$\dot{\theta}\dot{J} = \ddot{x} - J\ddot{\theta}$$ here the unknown is $J \in R^{m \times n}$, $x \in R^{m \times 1}$, $\theta \in R^{n \times ...
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1answer
19 views

Question about assumptions for Picard-Lindelof Theorem in Zeidler's functional analysis text

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelof Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
5
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1answer
74 views

How to solve this DE?

Consider the ordinary differential equation $$y''=xyy'$$ I'm pretty stumped, so any tips on how to proceed? It seems fairly simple but I'm drawing a blank.
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3answers
62 views

Initial Value Problem $dy/dx = (y+1)^{1/3}$

Consider the differential equation $$\frac{dy}{dx} = (y+1)^{1/3}$$ (a) State the region of the $xy$-plane in which the conditions of the existence and uniqueness theorem are satisfied (using any ...
2
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2answers
29 views

Coupled second-order differential equations

I am trying to solve the following system of coupled ODEs: \begin{align} -x^2 f'' - 3xf' + (1-2a)f - (a+1)x^2g'' + (2-4a)xg' + (4a-2)g &= 0,\\ (a-1)x^2 f'' + (4a+2)xf' + (12-6a)f + 12xg' + ...
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3answers
221 views

Solution of Differential equation

Question: Find solution of differential equation $$ 3e^{4x} \frac{dy}{dx} = -16\frac{x}{y^2} $$ which satisfies the initial condition y(0)=1 Solution: I know that I have to bring it in the general ...
3
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1answer
32 views

homogeneous first order differential equation

is there a method to solve $$\dfrac{dy}{dx} = f(x,y)$$, where $f(x,y)$ is a homogeneous function. I found some examples like $f(x,y)=(x+y)^2$ where it can be solved after converting it to Ricatti's ...
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1answer
37 views

graphing the solution of $y'=x^2-3$

I have a Ordinary Differential Equation(ODE) and I got the solution as ​ Now I want to draw graph? How can I do that? I think: ...
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3answers
180 views

Given the differential equation, how to solve the y function with x as the independent variable?

$y\frac{dy}{dx} = x(y^4 + 2y^2 + 1)$ $y = 1$ when $x = 4$ I tired to integrate by substitution, but it doesn't seem to work out.
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0answers
17 views

Dimensional reduction of system of ODEs

Given a nonlinear system of eight autonomous differential equations with all variables and parameters living in the positive octant of real numbers: $$dX_1/dt = \ldots\\ dX_2/dt = \ldots \\ \ldots ...
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3answers
48 views

Solving the differential equation $dr=(r\cos\theta +r\sin\theta)d\theta$

$dr=(r\cos\theta +r\sin\theta)d\theta$ In my book this is under separation of variables then i tried to factor out r and divide both sides then integrate both sides but where can i find my $C_1$? I ...
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0answers
50 views

Trapping region for Nonlinear ODE system?

I need to find a trapping region for $u'=-u+vu^2$ $ v'=b-vu^2$ with $b>0$. I don't know what theory to use or in wich book I can find some examples to find optimal trapping regions. Thank you ...
0
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1answer
24 views

Existence of the first weak eigenvalue of the Laplacian in a bounded domain

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $$H:=W_0^{1,2}(\Omega):=\left\{u\in L^2(\Omega):\nabla u\in L^2(\Omega)\right\}$$ be the Sobolev space, where $\nabla u$ denotes the weak ...
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0answers
26 views

how to solve complex differential equation [on hold]

how to solve this differential equation $$ a_1\cdot \phi'(x)^+ + a_2 \cdot \phi'(x)^-=c_2\cdot g(x) $$ where $\phi(x)$ is a complex analytic function thanks
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3answers
43 views

Finding the Correct Function that fits the Scenario

i have been trying to find a function that fits the following scenario: $$ f'(c) = 1^0 $$ $$ f''(c) = 2^1 $$ $$ f^{(3)}(c) = 3^2 $$ $$ f^{(4)}(c) = 4^3 $$ and so on, the purpose is to derive a way to ...
6
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0answers
186 views

How to classify/ solve this PDE?

I am searching how to solve the PDE below but I can not seem to find a decent example online. My major did not focus much in solving PDEs so I feel very deficient. I know how to solve for the steady ...
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0answers
34 views
+50

Jacobi field geodesic and calculus of variations.

How can we show that the second order variation to a geodesic is given by the Jacobi differential equation? In essence, \begin{equation} \frac{D^2}{dt^2}J(t)+R(J(t),\dot \gamma (t))\dot \gamma ...
3
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2answers
33 views

LaSalle invariance, Lyapunov stability

Trying to understand the LaSalle invariance principle. Consider the system $x' = y \\ y' = -y-6x-3x^2$ a) Using the test function $V(x,y) = 0.5y^2+3x^2+x^3$, show that the origin is asymptotically ...
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0answers
11 views

Question about Spring mass damping system ?? [on hold]

I have a spring mass damping system with mass = 6 gram, spring constant = 157000 N/m, damping co efficient = 6.7 N/m, input y(t) = 20 um. is it necessary that doubling mass from 6 to 12 gram would ...
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2answers
18 views

Domain of existence for this ODE.

I think this is some pre-calculus concept that I've forgotten. I am supposed to solve this initial value problem and determine how the interval in which the solution exists depends on $a$. $$yy' + x ...
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1answer
11 views

How to compute the solution of a differential equation involving Brownian local time

My problem is to compute numerically a function F. F is known to be convex and have kinks. It's also known to satisfy a "second order differential equation". Since the function is not everywhere ...
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0answers
14 views

Find functions that satisfy a given differential relation

If I have a relation between two sets of functions $A_{i}(x,y,...,z)$ and $B_{k}(x,y,...,z)$ of the form $$ A_i = F_i(B_k, \partial B_m/\partial x_n) \tag{1} $$ that is: $A$s are functions of ...
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1answer
22 views

Hamiltonian flow local diffeomorphism?

I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point. The background So he wants to show that any symplectic form is ...
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2answers
59 views

differential equations of second order [closed]

How may I solve this differential equations: $$y''+4y=12x^2-16x\cos(2x)?$$
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0answers
6 views

Radial trajectory equation solution (large trajectories/$g$ is not constant)

Well for a simple radial trajectory one could create the following equation ($s$ being the distance from the origin $G$ being newton's gravitational constant, and $m$ the mass): $$\ddot{s} = ...
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0answers
14 views

dominant balance for coupled differential equations

I have been trying to solve following set of nonlinear differential equations: $\frac{dy(t)}{dt}=k z(t) - 3 k y(t) - y(t)^2 + \epsilon_1 (M-z(t))^2$ $\epsilon_2 \frac{dz(t)}{dt}=Mz(t) - z(t) y(t) - ...
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solve the question by charpit's method please [closed]

i need this question to be solved by charpit's method, it is urgently needed as i am doing my preparations for the annual exam. i shall be thankful. px+qy=y
1
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1answer
28 views

Stability of a system of differential equations of the form $\dot x = y, \dot y = g(x)$

Let $g: \mathbb{R} \to \mathbb{R}$ be a locally Lipschitz-continuous function with $g(0) = 0$ and $xg(x) < 0$ for all $x \neq 0$. Consider the differential equation $\dot x = y, \dot y = g(x)$. I ...
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0answers
9 views

strong minima and maxima condition in calculus of variation

I am going through the topic CALCULUS OF VARIATION. There are not many examples on the topic strong/weak maxima minima. Can anybody provide the link of the source or book name where this topic is ...
1
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1answer
35 views

$y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$

Let , $y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$ , where, $$f(x)=\begin{cases}1 & \text{ if } 0\le x\le 1\\0 & \text{ if } ...