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

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Explicit conjugacy on 2 linear systems involving flow.

We need to find an explicit conjugacy between the flows of these 2 systems 1st system $X'$ = $AX$ and second system $Y'$ = $BY$ A = $$\begin{bmatrix} -1 & 1 \\ 0 &2\end{bmatrix}$$ B= ...
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87 views

How to solve this 2nd order differential equation

Find the general solution of $$\frac{{{\partial }^{2}}W}{\partial {{\xi }^{2}}}=\left( W+\frac{1}{W}\cdot \frac{\partial W}{\partial \xi }-\frac{\lambda }{W} \right)\frac{\partial W}{\partial \xi ...
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101 views

non-linear ordinary differential equation

Studying some Newtonian mechanics, I've encountered this differential equation : $y'+a y^2=b$ where $a,b$ are constants. how could we solve it ? (I trying to get an algebraic solution)
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If $f(cx, cy) = f(x, y)$ can we always find $g$ such that $g(\frac{x}{y}) = f(x, y)$?

Motivation: A differential equation $\frac{dy}{dx} = f(x, y)$ is said to be homogeneous if we can find $g$ such that $g(\frac{x}{y}) = f(x, y)$, which allows us to solve the equation using the ...
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118 views

Self-adjointness of $D=\frac{d^2}{dx^2}-1$ with boundary conditions $u'(0) = 0 = u'(a)$ on $[0,a]$.

Im trying to show that $$D=\frac{d^2}{dx^2}-1$$ is self adjoint on $[0,a]$ subject to $u'(0)=u'(a)=0$. I think I need to use integration by parts but I'm not sure how to do that.
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solving this second order ode

Consider the second order ODE where $ (k-x)^2 y''+6(k-x)y'+12y=F(x) $ where $k$ is some constant. I want to compute the real valued general solution. progress: guess $(k-x)^{m}$ to be the solution ...
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103 views

How can I find two independent solution for this ODE?

Please help me find two independent solutions for $$3x(x+1)y''+2(x+1)y'+4y=0$$ Thanks from a new beginner into ODE's.
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74 views

eigen value of the gradient operator

Eigen value of the following differential equation $$\nabla \phi (\vec r) = a \vec {k} \phi(\vec{r})$$ is $$ \phi(\vec{r}) = e^{a \vec{k}.\vec{r}}$$ How can i derive this result?
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82 views

How to know if the equation is linear?

According to my maths book an equations is linear if, there are no products of the function and neither the function or its derivatives occur to any power other than the first power. It should be ...
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1k views

Clairaut's Equation Singular and General Solutions

I want to know how one how one would prove that the singular solutions to Clairaut's equation are tangent to the General solutions. so I have here: $$y(x) = xy' - e ^{y'}$$ Differentiating $$y' = y' ...
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Nonhomogeneous second order ODE constant external force

I've been trying to solve this equation $$-\mu u'' + \beta u' = 1$$ where $u(0) = 0$, $u'(1) = 1$. So far the result I have is $$u = \frac{(exp(\frac{\beta x}{\mu})-1)\mu}{\beta ...
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740 views

What does it mean for a phase portrait to have “limit cycle behavior?”

Consider a system: $dx/dt = x(1-x)-\frac{kxy}{kx+1}$ $dy/dt = ry(1-\frac{y}{x})$ For values of r as 0.15, 0.11, and 0.05, which of the corresponding phase portraits displays limit cycle behavior? ...
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562 views

Using Octave to solve systems of two non-linear ODEs

How to solve following system of ordinary differential equations using Octave? $$\frac{dx}{dt} = - [x(t)]^2 - x(t)y(t)$$ $$\frac{dy}{dt} = - [y(t)]^2 - x(t)y(t)$$ Update: initial conditions: ...
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Find $dy$ and evaluate $dy$ given $y=e^\frac{x}{10}$ , $x = 0$ and $dx = 0.1$

I have a question below but I missed this day of class maybe someone can show me how to approach? Find $dy$ and evaluate $dy$ for the given values of $x$ and $dx$ $\displaystyle y=e^\frac{x}{10}$ ...
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87 views

2 ways to solve an ODE, different solutions…

Given the ODE $$y'= \left( \frac{x+2y+1}{2x-6} \right)$$ I have tried two methods to solve this equation. One method is seperate variables: $$(2x-6 )dy = (x+2y+1)dx $$ integrating gives ...
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Find the fixed points of the two systems

What are the fixed points of the two non linear systems below? \begin{align} x(t)& = x(3-x-2y)\\ y(t)& = y(2-x-y) \end{align} I know that $(0,0)$, $(0,2)$, $(3,0)$, $(1,1)$ are the fixed ...
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64 views

Can you help me solve this ODE?

I need to solve this differential equation. YWhat I'm looking for is a way to simplify this equation. Can anybody give me hints/tricks to understand the following equation better: ...
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218 views

Is the initial value problem of an ODE considered as a dynamic system?

Is the initial value problem of an ODE considered as a dynamic system? A dynamic system is defined as In the most general sense, a dynamical system is a tuple (T, M, Φ) where T is a monoid, ...
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274 views

Differential operators: elliptic vs strongly elliptic

This morning a collegue of mine came to me with the following question: does there exist any elliptic operator of order $2m$ with real (variable) coefficients that is not strongly elliptic? After ...
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How to create a simple differential equation

I am doing numerical analysis where we work with differential equations but I have never had any classes on differential equations. It seems you can get by in an introductory numerical analysis course ...
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Find the value of $f(t)$ for which $f ' (t) = 0$

$$f(t) = 8t^{1/2} + 6t^{-1/2}$$ Somehow I think the question is related to the previous parts, where I did: a) Find an expression for $f'(t)$. $f'(t) = 4t^{-1/2} - 3t^{-3/2}$ b) Find the ...
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462 views

Want to understand differential equations…how to solve this basic differential equation?

So I put the differential equation $y' = xy$ into Wolfram Alpha and it tells me the solution is $$y(x) = c_1\,e^\frac{x^2}{2}$$ Can anybody explain to me how to sub this back into the original ...
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465 views

Comparable Graduate ODE Text Suggestions

First off, I'm very sorry if this sort of question is not allowed here. I've seen a couple similar questions on the OverFlow site, but I think discussions of basic material should be kept to this ...
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101 views

Second Order Homogeneous Differential Equation from Optimal Controls

Please excuse my moment of ignorance while I reboot my education in math. I am taking an optimal controls course and it has been quite some time since I've worked with calculus. On to my question... ...
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371 views

Advanced integration problem

The solution to Schrodinger's equation are wave functions $\Psi (r,\theta ,\phi )$ of the form, $\Psi (r,\theta ,\phi )= R(r)\Theta(\theta)\Phi(\phi)$ Where, the probability of finding an electron in ...
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110 views

Differential Equation dependent or independent

I am asked to state whether the variable is independent or dependent for the equation $$ \frac{d}{dx}\left(x\frac{df}{dx}\right)+f=0 $$ How do you know which are dependent and independent? What about ...
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91 views

Estimate on the interval of definition as parameter varies

Given $\alpha\in\mathbb R$ can we give an estimate on the interval of definition of the solution of $$x'=x^2(\alpha+\sin(x)),\quad x(0)=1?$$
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68 views

The necessary and sufficient conditions for the solution of the equation $\frac{dy}{dx} = f(y)$ is locally unique.

$$\frac{\mathrm{d}y}{\mathrm{d}x} = f(y)$$ where $f(y)$ is continuous on $|y-a|\leq \epsilon$,and $f(y)=0$ iff $y=a$. To Proof : For the initial value point on $y=a$,the equation has local unique ...
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Find a continuous solution of the initial-value problem

This question is from DE book by Braun(Pg no 10, Q no 17), Find a continuous solution of the initial-value problem $y'+y= g(t), y(0)= 0$ where $g(t)=\begin{cases}2, &0 \leq t\leq 1, \\0, ...
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Find out the differential equation of the following families of curves.

Find out the differential equation of the following two families of curves : Straight lines having slope and $x$-intercept equal in magnitude. Straight lines at a fixed distance $p$ from the origin. ...
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538 views

Parametric equations and line segments

I am not sure how to do this one at all, I can't even start it. I am suppose to show that $$x = x_1+(x_2-x_1)t \\ y= y_1+(y_2-y_1)t$$where t is between zero and one describes the line segment that ...
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261 views

How do I numerically calculate a function from its gradient?

I know the gradient of a function t on a cartesian grid: $\vec{g}(xi,yj,zk)=\nabla t(xi,yj,zk)$. I know t for the center pillar: $\ t(xc,yc,zk)$. For each node in the cartesian grid I want to ...
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266 views

Finding Green’s function

I tried to solve my homework, but unfortunately got stuck at some point. Could you please help me to get through it? For a given boundary value problem: $$ Ly =y'''+y''\\ y(0)+y'(0) = y''(0)=y(1) = ...
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364 views

Differential equations: Connection between repeated roots of characteristic equation and generalized eigenvectors

My question is about homogeneous linear equations with constant coefficients: $ay''+by'+cy=0$. When you solve this equation via a characteristic equation (see e.g. here) problems arise when you ...
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246 views

diff.eq.: linear stability of ODE

I'm stuck with a little exercise and cannot find out where I'm wrong. Maybe you can help me. So we have a differential equation (modified logistic growth): $$\frac{dN}{dt}=k \ N ...
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solving Differential Equation

For equation below: $$(t+1) \, dx=4(x+4) \, dt$$ After separation I ended up with: $$(x+4)dx = \frac 4{t+1}dt $$ Resulting in: $$\int x+4 \,dx = 4 \int \frac 1{t+1} \,dt$$ So: $$\frac 12 x^2 ...
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What is 'imbedding' with Sobolev space and $ L^2 $ space?

I want to know that the meaning of the following. $$ W^{n,1}\textrm{ is continuously imbedded into }L^2$$ Here, $W^{n,1}$ is a Sobolev space.
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solution to a differential equation

I have given the following differential equation: $x'= - y$ and $y' = x$ How can I solve them? Thanks for helping! Greetings
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459 views

Solving $f''+f=\sin x$ with Fourier series

Does there exist a twice differentiable periodic function $f$ such that $f''(x) + f(x) =\sin(x)$ for all $x \in [-\pi, \pi]$? How to solve this differential equation using Fourier series? I ...
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71 views

Differentiate respect to $x$

$(x^2+2x+1)^3$ let u=$x^2+2x+1$ $\frac{du}{dx} = 2x$ or $2x+2$ $\frac{dy}{dx}=3u^2 $ if $\frac{du}{dx} = 2x$ then $3(x^2+2x+1)^2 (2x)$ answer is $6x(x^2+2x+1) $ Or if $\frac{du}{dx} = 2x+2$ ...
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Why this is not a differential equation?

On the exam I was asked the question about Transcritical bifurcation. I gave the equation $$ \dot x = rx - x^2 $$ Then I was asked why it is not a differential equation and I couldn't answer. I ...
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418 views

Differential equation with Bessel function-like solution

I have the following differential equation: $$ r^2f''(r)+2rf'(r)-2f(r)=0 $$ I think a solution has something to do with Bessel functions but I can't figure out how. Could somebody help ...
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Solve $\frac{dy}{dx} - \frac{dx}{dy} = \frac{y}{x} - \frac{x}{y}$

Solve: $$\frac{dy}{dx} - \frac{dx}{dy} = \frac{y}{x} - \frac{x}{y}$$ What I have done till now: $$\left(\frac{dy}{dx}\right)^2 -1= \frac{dy}{dx}\left(\frac{y}{x} - \frac{x}{y}\right)$$ ...
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Which ODEs guarantee that their solutions don't go through $x(t)=0$?

For $\ x(0)\equiv x_0>0\ $ and a system governed by $$\dot x(t)=-k\ x(t),$$ I find that $$x(t)>0\ \ \ \forall\ t.$$ (Because the solution is $x(t)=e^{-kt}x_0$.) For which $f$ and $$\dot ...
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286 views

Differential equations - exact differential equations

I am self studying ode from boyce diprima book and while doing exercises on chapter 2.6. I couldnt undestand how question 17 is derived I checked the solution manual and I found the same answer which ...
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156 views

The Ambiguous Use of Differential in Solving Differential Equations?

Is there any theoretical basis for employing differential methods like separation of variables in solving differential equations? As we all know, differential is formally defined as a linear ...
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275 views

Weak lower semicontinuity of a functional on Hilbert space?

Let $H:=\left\{u\in L^2(R^N):\nabla u \in L^2(R^N)\right\}$ and a functional $$f(u)=\int_{R^N} |\nabla u|^2dx+\left(\int_{R^N} |\nabla u|^2dx\right)^2.$$ If $\{u_n\}\subset H$ is a sequence such that ...
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Solving the differential equation $y' - \frac{1}{x} y = x^2\sqrt{y} $

Which technique should I use for solving the follwoing DE? $$ y' - \frac{1}{x} y = x^2\sqrt{y} $$ I have tried some algebraic manipulations but I could not recognize any pattern.
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100 views

Differential Equation Breaks Euler Method

Solving ${dy\over dx} = 2y^2$, $y(0)=2$ analytically yields $y(8)= -2/31$, but from using Euler's method and looking at the slope field, we see that $y(8)$ should be a really large positive answer. ...
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If the Wronskian W of $f$ and $g$ is $3e^{4t}$, and if $f(t) = e^{2t}$ how may I find $g(t)$?

I'm struggling with a seemingly simple problem in differential equations. If the Wronskian W of $f$ and $g$ is $3e^{4t}$, and if $f(t) = e^{2t}$, find $g(t)$. So from that I made a first order ...