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

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Solving differential equations with Bessel function solutions

In order for the question that I have to make any sense I must first include some background information as given in my textbook: The standard form of Bessel's differential equation is ...
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0answers
47 views

Representation of a real function through a Fourier Transformation

I 'm trying to do some calculations regarding some differential equations and I came across an interesting way to express a real function through a double integral of the form: ...
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1answer
37 views

For what value of $v_0$ is the solution periodic?

A solution of the second-order differential equation $$ x''+x-x^3=0 $$ satisfies the initial condition $x(0)=0$ and $x'(0)=v_0$. For what value of $v_0$ is the solution periodic? I have tried ...
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1answer
37 views

Why does the coordinate transformation from Cartesian coordinates leads to an additional term in the biharmonic operator in spherical coordinates

I am trying to solve a problem in physics where the biharmonic operator is involved. I think that the bihahmonic operator can be obtained by taking twice the Laplace operator, such that $\nabla^4 f = ...
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2answers
259 views

Solution to ODE Abel Equation

I aim to find the exact form solution to the this ODE: $$\frac{dS}{dw}S = \frac{a}{w}S^2 + \frac{b}{w}S - c$$ where S is a continuous differentiable function of w, real positive and a, b, c are ...
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4answers
229 views

Solution to the second order differential equation

Hello i have read in a book that second order diferential equation of this form ($\psi$ is a function of $x$): $$ \frac{d^2 \psi}{dx^2} = - k^2\, \psi $$ describes a simple harmonic oscilator and ...
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0answers
8 views

Solution of ode system (morphogenesis) has one value once large once small

I solve Turing's morphogenesis with code available in following question: Solve Turing's morphogenesis with other method than Euler's The problem is: the pictures are nice, however there ...
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1answer
18 views

The behavior of the trajectory of the phase portrait

For the plane autonomous system $$ x' = ax+by $$ $$ y' = cx+dy $$ If the solution to this system is, says, $ \binom{x}{y}= c_{1}\binom{1}{1}e^{-5t} + c_{2}\binom{1}{2}e^{-t} $, then it is ...
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1answer
19 views

characteristics in a simple region

For the system of equations describing a compressible fluid $\frac{\partial \rho}{\partial t} + u \frac{\partial \rho}{\partial x} + \rho \frac{\partial u}{\partial x} = 0\\ \frac{\partial ...
5
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1answer
89 views

Weird differential equation - Jacobi?

In class we had differential equations of the type $$y'=\frac{\left(Ax+By \right)y+ \alpha x + \beta y}{\left(Ax+By \right) x+ax+by},$$ where $A,\alpha,a,B,\beta,b$ are constants. The names of the ...
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23 views

Non-homogeneous solution to the second order differential equation

Given the DE of the form: $$(x+1)\frac{d^2y}{dx^2}+x\frac{dy}{dx}-y = (x+1)^2$$ How can one, without much guess work propose a potential solution to the non-homogeneous part of the equation? And if ...
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1answer
34 views

Solving a differential equation using symbolic computation

I have the initial value problem $$\ddot x (t) + k \sin(x(t)) = 0$$ with initial conditions $x(0) =: x_0 $ and $\dot x (0) =: v_0$. Using Maxima I should check that $$\frac{1}{2}(\dot x (t))^2 + k ...
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4answers
37 views

Find particular solution to nonhomogeneous DE $y'+y=x^2+\sin{x}+\cos{x}$

I'm new to nonhomogeneous DE's and I have come across this DE: $$y'+y=x^2+\sin{x}+\cos{x}$$ which I'm supposed to provide a general solution to. However, I get stuck with the particular solution. The ...
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2answers
60 views

Singular perturbation problem (ODE)

I have found the following singular perturbation problem, $\epsilon u_{xx} + |u_x|u_x + u = 0$, $x>0$; with initial conditions, $u(0) = \epsilon^2$, $u_x(0) = 0$, where $0 < \epsilon \ll 1$. ...
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0answers
10 views

1D Flows, Local bifurcation, method.

I am working through a problem sheet which consists of questions such as "Find the type of bifurcation which occurs in the 1D system defined by $\dot{x}= f(r,x):= rx - \sinh{x}$, and state the ...
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0answers
21 views

Differentiable function f(x)

Let $f(x)$ is a differentiable function satisfying $f'(x) + 100 f(x) ≤ 1 $ Then $f(x) -1/k$ is a non increasing function of $x$ , then we have to find the value of $k $ I tried , but at last ...
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2answers
33 views

Differential equation with absolute value $ax'+bx=|\sin(\omega t)|$

How could I solve the follow differential equation? $$a\cdot\frac{\text{d}x(t)}{\text{d}t}+b\cdot x(t)=|\sin(\omega t)|$$ Thank you for your time.
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1answer
33 views

A question about a route of a point that travels in a particular way through the plane

I don't know exactly how to classify this question. It is not from any homeworks, just something I've been wondering about. Let's say that in the beginning of an experiment ( the beginning is $t=0$ ...
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6answers
5k views

What's the difference between an initial value problem and a boundary value problem?

I don't really see the difference, because in both case we need to determine y and the values of the constants. The only difference is that we give the value of y and y' in the former and the value of ...
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0answers
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What is the classification of this equilibrium point?

Let $(x^*,y^*)$ be an equilibrium point of a nonlinear planar autonomous ODE. Suppose, in the linearization, that $(x^*,y^*)$ gives the zero matrix for the Jacobian. What kind of point is $(x^*,y^*)$? ...
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1answer
40 views

Best way to go about solving specific tricky 2.ODE

I've been working on this equation for a while now. Find the particular solution of: $$y''-4y'+y=te^t+t$$ My first instinct was to use the method of undetermined coefficients, solving for $te^t$ and ...
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2answers
185 views

Solve first order nonlinear differential equations

I want to solve this nonlinear 1-st order ODE, $$\frac{1}{1+x}=(\frac{1}{x-y}-\frac{1}{y})\frac{dy}{dx}$$ I find it non-separable, and Wolfram Alpha does not give me a closed form solution, but the ...
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2answers
59 views

None exact first order ODE

i have to solve the following $1^{st}$ order differential equation $(xy+1)dx+(2y-x)dy=0$ i am in the elementary differential class,and have not learned multivariate functions, the equation below is ...
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2answers
53 views

y'=y^2-y/x-1/x^2, solve ODE

Try to find a solution of the following differential equation: $y'=y^2-y/x - 1/x^2$ I see that it is of Riccati type: $ y'=f(x)*y+g(x)*y^2+h(x)$ with $g(x)=1, f(x)=-1/2, h(x)=-1/x^2 $ In my lecture ...
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4answers
58 views

Solution of a differential equation involving $3$ variables $x,y,t$

QUESTION: Solve the differential equation- $$\frac{dx}{dt}+\frac{dy}{dt}+2x+y=0$$ $$\frac{dy}{dt}+5x+3y=0$$ I am unable to progress in solving these equations. For by any manipulation, I am ...
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0answers
21 views

Isothermal coordinates [duplicate]

Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E∗(du^2+dv^2)$ and where $E>0$ is an harmonic function? I know that this metric is a special kind ...
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0answers
25 views

Parametric Equation of Elliptical Cycloidal Sine Curve

I am trying to find the parametric equations of a cycloidal curve, which, instead of using a circle, uses an ellipse to oscillate around a base circle. Below are equations of the standard, circular ...
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1answer
49 views

Can you solve $y=\frac{a}{2}x^2\left(y'-\frac{1}{y'}\right)^2+x\left(y'-\frac{1}{y'}\right)+ax^2+c$?

I've recently come across this differential equation, but I am having trouble proceeding toward a solution. $y=\frac{a}{2}x^2\left(y'-\frac{1}{y'}\right)^2+x\left(y'-\frac{1}{y'}\right)+ax^2+c$ ...
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1answer
17 views

Solving ODE with essential singularity

I would like to solve the following linear ODE. $y''(x) -\frac{2}{x} \frac{1-3x^4 +2x^3}{1+3x^4-4x^3} y'(x)+\frac{\omega^2}{(1+3x^4-4x^3)^2} y(x) = 0$ Here $x$ is a dimensionless variable which runs ...
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0answers
23 views

Problem solving a PDE using separation of variables and Fourier expansion

I am trying to solve the heat equation: $$\frac{\partial \theta}{\partial t}=\frac{\partial^2 \theta}{\partial x^2}$$ The boundary conditions are: $$\frac{\partial \theta}{\partial x}(x=0)=0$$ ...
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3answers
28 views

How to model a checking account with continuous-time compounding?

"Say you have a bank account in which your invested money yields 3% every year, continuously compounded. Also, you have estimated that you spend $1000 every month to pay your bills, that are ...
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Linear stability of an ODE $\frac{dN}{dt}=-\mu N(t)+\mu N(t-T)\left(1+q\left(1-\left[\frac{N(t-T)}{K}[\right]^z\right)\right)$

This is a part of exercise: Consider the following equation: $$ \frac{dN}{dt}=-\mu N(t)+\mu N(t-T)\left(1+q\left(1-\left[\frac{N(t-T)}{K}[\right]^z\right)\right) $$ where all involved constants are ...
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0answers
8 views

$\exists$ primitive for $a \partial^n(b) $ in $\mathbb{C}[u_0,u_1,u_2, \dots]$?

Let $\mathcal{A} = \mathbb{C}[u_0,u_1,u_2, \dots]$ be the algebra of finite polynomial expressions in $u, u_1, \dots$ We define the derivation $\partial : \mathcal{A} \rightarrow \mathcal{A}$ as the ...
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0answers
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Solve the EDO $p'={\alpha}p^a+{\beta}p^b,\quad t>0,$

Fix $\alpha , \beta \in (0,\infty)$ . Use Osgood's criterion to show that the equation $$p'={\alpha}p^a+{\beta}p^b,\quad t>0,$$ has at most one nonnegative solution if $a,b \ge 1$. Also, prove ...
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2answers
89 views

Solving a Second Order PDE

I'm trying to solve the equation $u_t = \alpha^2 U_{yy}$ given $u(y,t)$ bounded $y \rightarrow\infty$ and $u(0,t) = U_o e^{iw_ot}$. Initial is $u(y,0) = 0$. I have gotten both separations as $Y'' - ...
3
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0answers
25 views

Elementary properties of gradient systems

Consider $x_0\in\mathbb{R}^n$ and a $C^{1,1}$ function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ (that is, a differentiable function whose gradient is Lipschitz function). Consider the system $$ ...
3
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1answer
22 views

References for gradient systems

I am interested in the gradient system $$\dot{x}(t)=-\nabla f(x(t))$$ where $f:\mathbb{R}^n \to \mathbb{R}$ is a $C^{1,1}$ function (that is, a differentiable function whose gradient is Lipschitz ...
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0answers
15 views

Find all the equilibrium solutions of $x' = \cos(x^2)$ and determine stability.

One can determine equilibrium solutions of autonomous EDO's setting the derivative equal to zero. So, in this case, all the equilibrium solutions take the form $x^* = \pm \sqrt{\frac{\pi}{2} + k\pi}$, ...
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nonlinear spreading of sound waves [on hold]

I want to get application examples, something like using spreading of sound waves in medicine branch or for musical instruments.
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17 views

Nonlinear differential equations

How to solve the following set of differential equations analytically $$\begin{align} a_1 \ddot{\phi} + b_1 \dot{\phi} + c_1 \phi & = d_1 \dot{\theta}\dot{\psi} \\ a_2 \ddot{\theta} + b_2 ...
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2answers
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+50

Initial Value Problem for $(\cos x -x\sin x +y^2)dx + 2xy\,dy =0$, $y(\pi )=1$

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following question. If it's any indication of difficulty, the exercise is only Part 1-B of the sheet, ...
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0answers
11 views

Floquet Boundedness for Floquet multiplier $|\lambda_i|=1$

The statement: Consider the system $x'=A(t)x$, where $A(t)$ is a periodic matrix with period T. If $|\lambda_i|=1$ then the corresponding Jordan block to $e^{TR}$ is diagonal. The constant matrix R ...
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0answers
25 views

Integrate by parts $\int_0^\infty w' \bar w$; any nice expression for $w$ complex-valued?

Let $w$ be a complex-valued function of $t \in [0,\infty)$. At $t \to \infty$, it decays to zero. And $w_t(0)$ is prescribed. Is there any nice expression for the integral $$\int_0^\infty w' \bar w$$ ...
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4answers
154 views

How to solve $f'(x)=f'(\frac{x}{2})$

How do we solve this given $f'(0)=-1$. It does not look separable. I can integrate both sides but end up with a functional equation with is not helpful.
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0answers
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I'm comparing two different methods for solving the Navier Stokes equations. Why are my velocity results so different?

I want to use a code for modeling 2-D fluid flow out of a tank to understand a chemical process. The code has never been used for pressure boundary conditions, so I want to check that it works as it ...
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1answer
27 views

How to solve this Nonhomogeneous ODE problem of beam deflection and find particular solution

Problem states that the load on the beam having length L and fixed on both end is; $\omega(x)=w_0\frac{x}{L}$ Function of the deflection of the beam is given as; ...
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2answers
29 views

Decouple a system of two second order differential equations

I have a system of second-order differential equations that I want to decouple. they are, $\ddot{x} = \frac{\omega_1^2}{2} x + \omega_2 \dot{y}$ and $\ddot{y} = \frac{\omega_1^2}{2} y - \omega_2 ...
3
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1answer
70 views
+200

Solving or knowing something about a non-linear PDE which is “almost” linear?

Let $a>0$ be fixed. I have the following PDE: $u=u(t,x)$, $t\in [0,1]$, $x\in \mathbb{R}$, $$-\partial_t u = |\partial_x u| + \frac{1}{2}\partial_x^2 u, \quad ...