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

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12 views

Solving a simple Recurrence in summation form(very special case)

I have a bit confusing recursion form $\sum_{n=2}^{\infty}\{f(n)\frac{n}{n-1}\}=C, \tag 1$ $f(0)=b,f(1)= a,f(2)=c$ and $C$ are constants. Could you help me to solve this recursion or help me to ...
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2answers
24 views

Finding an ODE with specified solutions

So I have this math problem: Find functions $p(x)$ and $q(x)$ so that $y_1(x)=\sin x$ and $y_2=x*\sin x$ are solutions of the differential equation $y''+p(x)y'+q(x)y=0$. I'm just so lost as to what ...
3
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3answers
39 views

Differential Equation $\frac{dP}{dt} = kP(1-P)$

I have a question about solving this differential equation. So, the question is to solve it given that $P(0)=\frac23$ So this is what I've done so far $$\frac{dP}{dt} = kP(1-P)$$ $$ k\,dt = ...
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2answers
41 views

Exponential Growth Differential Equation

A population of buffalo grows exponentially (the rate of growth is determined by the population itself) but has a carrying capacity. Its population (in tens of thousands) at a time t ( in years ) is ...
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1answer
54 views

Existence of solution of ordinary differential equation

I am reading a proof of the existence of solutions for ordinary differential equations and I have some basic doubt. I'll copy the statement, the part of the proof I don't understand and my question: ...
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2answers
60 views

Solving a particular system of differential equations

The problem I'm trying to solve is this: $X'(t) \in \mathbb{R}^3 \,, \, \omega = (\omega_1,\omega_2,\omega_3) $ Find the general solution for $$X'(t) = \omega \times X(t)$$ After doing the cross ...
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0answers
12 views

Initial value problem test cases

I am working on some materials about numerical solution of initial value problem for ODEs. Are there any state of art test cases used to test properties of methods? I have found one in Wikipedia and ...
2
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1answer
73 views

Differential Equation $\frac{dy}{dt}$ = $y - t$

Given the differential equation $\dfrac{dy}{dt}$ = $y - t$ Is this equation separable? -> No it is impossible to separate this equation because we can't get $y$ alone with $dy$ and $-t$ alone with ...
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2answers
27 views

The set consisting of all solutions of a homogeneous linear differential equation of order n is a vector space.

The set $S$ consisting of all solutions of a homogeneous linear differential equation of order $n$ is a vector space.
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0answers
43 views

On a differential equation problem of international mathematical competition for university students

I am trying to solve problem 2 of this competition: http://www.imc-math.org.uk/imc2009/imc2009-day2-solutions.pdf I have other thought but i couldn't fill in the detail. Consider the initial value ...
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1answer
64 views

Solving the differential equation $y' \tan y = \frac1x$

Express the differential equation $$\tan y\,\frac{dy}{dx}=\frac{1}{x}$$ in a form not involving $\frac{dy}{dx}$. I undersand the concept of a differential equation (though, as a student, I am ...
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0answers
37 views

Solve the initial value problem 0f $x'=f(x),\quad x(0)=y$ [on hold]

Solve the initial value problem $$x'=f(x),\qquad x(0)=y$$ for $$f(x)=(x^2,x+x^{-1})^T$$ Denote the solution by $u(t,y)$ and compute $$Ф(t,y)=\frac{du}{dy}(t,y)$$ Compute the derivative $Df(x)$ for ...
2
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3answers
113 views

Separable differentiable equations

Which of the following is a solution to the separable differentiable equation: $$\frac{dy}{dx}=\frac{xy}{\ln y }$$ $A.\ \displaystyle e^{|x|}$ $B.\ \displaystyle e^{\sqrt{\frac{x^2}2}}$ $C.\ ...
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1answer
26 views

Finding the general solution of an ODE in matrix form using integration

so I have done this other times, but this seems to be a tricky case... The teacher also used a method that it is not in the book to show us another way of doing it, but I do not understand it :/ So ...
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0answers
77 views

need an equation [on hold]

I need an equation and I'm not very math oriented/dont have a lot of time to think so i figured, why not outsource? I am programming a pokemon battle simulator game. After a pokemon takes a blow, ...
-1
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1answer
35 views

Initial values are lost (diff eq to Transfer function)?

I read eternal Julius O. Smith III and he says that $$x_{n-m} = z^{-m}X(z)$$ Particularly, difference relation $$y_{n} = y_{n-1} + x_{n}$$ is solved by by $$Y = z^{-1}Y + X = {X \over ...
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0answers
39 views

Elliptical Coordinates PDE, wave equation and separation of variables

I need some help with this problem. I know how to use the method of separation of variables and that the constant lambda should give you trig functions with solutions at some interval of pi, which ...
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1answer
26 views

3D Cauchy Problem

I will answer the question myself but let me know what you think of my correctness. We have the Cauchy Problem $$ yu_x-xu_y+u_z=0 $$ with data $u(x,y,0) = x+y$.
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2answers
53 views

How to find the intersection points of lines that are normal to two curves?

Let I have two curves, \begin{gather} f(x)=\frac{x^3}{4}+1 \\ g(x)=\frac{(x-\tfrac{1}{2})^3}{7}+\tfrac{1}{2} \end{gather} There are zero or more lines that are normal to both curves. In other words, ...
0
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1answer
15 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-29

I completed near all problems om a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example u = ...
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1answer
14 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-28

I completed near all problems of a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example $u = ...
2
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2answers
70 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
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2answers
126 views

Solve nonlinear differential equation

Could you help me solve or give me some advice about following differential equation $$ 2(y')^2 + 3xy'y'' + 3yy'' = 0 $$
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1answer
24 views

Reduce the equation to a homogeneous equation by a change of variables

The equation is $$ (x+1)^2 y'= (x+y)^2 -(y-1)(x+1) $$ I've tried substituting for $z=x+y$, $z=x+1$, etc. but they don't seem to give me anything that's homogeneous after rearranging.
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2answers
44 views

Does local existence in every point imply global existence for an ODE?

Consider the following first order ODE: $y' = f(t,y)$ subject to $y(t_{0}) = y_{0}$. I would like to show that there exists a unique function $y(\cdot)$ that passes through ($t_{0}, y_{0}$) The ...
2
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1answer
40 views

Trying to find 2nd power series solution

For the equation $ xy'' + 2xy' + 6e^xy = 0 $, I need to find the first 3 nonzero terms in each of two linearly independent solutions about x=0. I changed this to the form of ...
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0answers
43 views

Long-time asymptotic behaviour of a system of two ODEs

We have the following nonlinear ODE: $$ f' = af-bg -(f+g)^k \bigl(f'(0) +g'(0)\bigr) + f'(0), $$ $$ \bigl(G-T(x)\bigr) g' = -af+bg - g'(0), $$ where $a,b,k,G$ are constants, $f'(0)$ and $g'(0)$ are ...
0
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1answer
23 views

Application of Ito's Lemma to a process

There is a function $S(X)=(A+1/b X_t)^b$, where $A$ and $b$ are constant I'll need to show how to get $dS = \frac13 S^{1/3} dX^2 + S^{2/3} dX$ and determine the value of $b$
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2answers
41 views

Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
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0answers
17 views

On the ODE $x^{\prime\prime}(t)+a(t)f(x(t))=0$

Problem. Let $a,f \in C^0(\mathbb R)$ with $a \ge 1$, $f \ge 0$ and suppose $\int_0^{+\infty} f = +\infty$. Let now $x$ be a solution of the ODE $$ x^{\prime\prime}(t) + a(t)f(x(t))=0. $$ Let ...
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3answers
46 views

Why does my derivation of $\mathcal{L(\frac{f(t)}{t})}$ lead to a wrong answer?

I'm trying to prove that $$\mathcal{L(\frac{f(t)}{t})(s)} = \int_s^{\infty}\mathcal{L(f(t))}(u)du$$ Here's my attempt: $$\mathcal{L(\frac{f(t)}{t})}(s)=\int_{0}^{\infty} \frac{f(t)}{t}e^{-st}dt$$ ...
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0answers
25 views

Finding alternating series for Power series

Given data and conditions I have a power series, $PS(x) = \sum_{n=0}^\infty R_nx^n$. I have a infinite GP,something like G(x) = $\sum_{k=0}^\infty ax^k = \frac{a}{1-x} $ . Never take G(x),such ...
2
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0answers
28 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
1
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3answers
29 views

Linearize a first order differential equation

The system described by $x'=2x^2-8$ is linearized about the equilibrium point -2. What is the resulting linearized equation? Answer is $x'=-8x-16$. How? I have no idea how it went from the first ...
2
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1answer
58 views
+50

Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
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1answer
21 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < ...
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1answer
38 views

Ricatti differential equation solution

I attempting to solve some Riccati differential equations. It has been a while since I have worked with differential equations so I am rusty. I would appreciate if someone would show me how to do ...
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2answers
34 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
2
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1answer
46 views

Solving $xy''-(1-x)y'+y=0$

$$xy''-(1-x)y'+y=0$$ So I know how to solve this via power series. Recently, a friend of mine was asking me how one could solve this without using series. I've got no real idea how to answer this ...
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1answer
40 views

Prove two solutions of differential equation are the same

In a recent work I had to solve the following differential equation: $$ r x''(r)+r x'(r)^2+x'(r)-\frac{4}{r}=0~~. $$ To do so I used two methods and I got, using each, two solutions with different ...
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0answers
38 views

Is my function singular at these two points?

My function $S(x,y,t)$ satisfies the following PDE $$\frac{\partial S(x,y,t)}{\partial t}=-H(x,y)$$ where the known function $H$ is singular when $x=\alpha$ and/or $x=\beta$, hence my question: ...
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1answer
54 views

Find an expression of the direction field

I have a directions vector field which I got empirically using quiver in Matlab. I want to find some analytical expressions that might work at least in part of the direction field. How can I ...
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0answers
17 views

Accuracy of a finite-difference method for numerically solve a PDE or BVP

When solving the Poisson Equation $$-u''(x)=f(x)$$ with Dirichlet-Neuman boundary conditions $$u(0)=0, u'(1)=0$$ using a finite difference 2-order centered scheme and a 2-order upwind ...
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1answer
17 views

Potential equation in rectangle with boundary values

I'm running into problem with the boundary conditions for u(x). I get u(x) = sin((n*pi*x)/a) based on u(0,y)=0, but that doesn't agree with du/dx(a,y)=0 unless the whole function u(x)=0. Is that the ...
0
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2answers
19 views

General Solution - Differential Equation

Question asks to find the general solution of the differential equation. $$\frac{1}{r}\left(\frac{d}{dr}\left(r\frac{dw}{dr}\right)\right)-\frac{\lambda^2}{r^2}w=0.$$ The answer given is ...
0
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1answer
28 views

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the green's function. I have to solve the given differential equation using Green's function method $\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta ...
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1answer
17 views

Second order D.E. - general solution

If $y=y(x)$, and we have the differential equation $y''=-k^2y$ for some constant $k>0$, then wolfram alpha gives the general solution as $y=A\cos(kx)+B\sin(kx)$. I've also seen this result used ...
2
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2answers
65 views

How to prove that solution to ODE in spherical coordinate is equivalent to the ODE in cartesian coordinates if it is a thin shell

Solving a diffusion-type ODE across a spherical shell, the equation is: $$\frac{d}{dr}\left(r^2\frac{df}{dr}\right)=0\tag{1}$$ with boundary conditions $f(r_1)=f_1$ and $f(r_2)=f_2$. The solution is: ...
2
votes
3answers
34 views

Differential equations with missing variable

$$y\cdot y'' + (y')^2 = 0$$ I'll make $V=\frac{\mathrm{d}y}{\mathrm{d}t}$, $y''=\frac{\mathrm{d}v}{\mathrm{d}t}=V\cdot\frac{\mathrm{d}v}{\mathrm{d}y}$ $$\Rightarrow ...
2
votes
2answers
95 views

Asymptote of solution of a differential equation without solving it

Consider the following differential equation (domain $\mathbb{R}$): $$ u(x) = 1 - u'(x) $$ and suppose $u(0) = 0$. How can one prove that $u(x) \to 1$ for $x \to \infty$ without solving the ...