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

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Series Solution For ODE

I am currently working on some introductory problems for series solutions for ODEs and am really struggling. The question is as follows: $$ (7+x)y' = y $$ Calculate the first five terms in the ...
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A hard problem on inverse laplace transform?

How can we find the inverse Laplace transform of: $[x]$ (floor function) ? \my question isn't laplace transform of floor function i asked the "inverse" laplace transform of floor function ...
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General solution for a parameter-dependent third order ODE

I'm having trouble with the solution of the following ordinary differential equation: $$ \begin{align} \begin{split} & A x^2 (x^2 + 1) F'''(x) - \left(3 A (\beta - 2) x^3 + \alpha x^2 +A (\beta - ...
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1answer
11 views

Matrix for Mixed Boundary Value Problem

My friend and I have been working on numerical solving the following equation $$-u'' = f$$ with $x \in [0,1] $ , $ u'(0) = 0$, $u(1) = 0$. Analytically, we found the eigenvalues and eigenfunctions ...
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20 views

Sketching the global phase portrait for a version of the Lotka-Volterra system

I'm trying to sketch the phase portrait for a version of Lotka-Volterra given by $$\begin{cases} \dot{x} = x(3-x-2y)\\ \dot{y} = y(2-x-y) \end{cases}.$$ I can sketch this just fine except for the ...
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1answer
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Solving for the particular solution of a system of differential equations

Consider the IVP $\vec{y}'= \begin{bmatrix}0 & -1\\-1 & 0\end{bmatrix}\vec{y} + \begin{bmatrix}t \\e^{2t}\end{bmatrix}$ $\vec{y}(0) = \begin{bmatrix}1 \\1\end{bmatrix}$ The complementary ...
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34 views

How to analyze ODE equilibrium stability with complex equilibria

Take this example: $y'=y^2+1$. There's no "real equilibrium", but is it right to say it has two "complex equilibria"? If so, what should be the conclusion of the derivative test? $$y'' = 2y \implies ...
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1answer
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“All phase plane solution points remain stationary as $t$ increases”?

Consider the linear system $y′(t)=A\vec{y}(t)$, where $A$ is a real $2\times2$ constant matrix with repeated eigenvalues. All phase plane solution points remain stationary as $t$ increases. I ...
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Defective eigenvalues problem - determining defect given a relation among elements, deducing third linearly independent eigenvector, etc.

Suppose I have the following matrix: $$\begin{bmatrix} 1 & 0 & 0 \\ 1 & 3 & 1 \\ -2 & -4 & -1 \\ \end{bmatrix}$$ The only eigenvalue of which ...
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1answer
38 views

“Simple” Linear ODE of order 100

I'm trying to solve the following ODE $$y^{(100)} + 100y = 0,$$ and I get the characteristic polynomial $\lambda^{100} + 100$, but do not know how to find its roots. Is there a way to find ...
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1answer
24 views

How to find the solution of non homogeneous differential equation [on hold]

I am solving attached question but it is not equal so its further solution is possible. Question
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24 views

cant solve the questions [on hold]

so my question is that i really can't solve the rest of the parts i have solved part a but i'm seeking help in the other parts matrix Laplace
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1answer
16 views

necessary and sufficient conditions for the existence of solutions.

Let $\phi$ and $\psi$ be two smooth functions on $\Omega$ open subset of $\mathbb{R}^2$. There exists a function $u=u(x,y)$ such that: $$\left\{\begin{array}{lll} \frac{\partial u}{\partial ...
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2answers
25 views

Euler Equation with a certain type of solution

I have the Euler equation in the following form $$x^2h''(x)+xh'(x)=b^2h(x)$$ with the condition $h(a)=0$. The general solution to this equation is $$h(x)=c_1x^{b}+c_2x^{-b}$$ Now, my question ...
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2answers
21 views

Given the family of solutions find the diff. eq.

Given the following 2-parameter family of solutions, find the differential equation which is satisfied by them. The family of solutions is: $$\log(y)=c_{1}x^{2}+c_{2}$$ and the resulting ...
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14 views

An identity related to Poisson's equation in 3D

$r'=\sqrt {(x'-x)^2+(y'-y)^2+(z'-z)^2}$ $\nabla = \frac {\partial}{\partial x'}+\frac {\partial}{\partial y'}+\frac {\partial}{\partial z'}$ I was studying Poisson's equation in 3D see this link ...
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1answer
34 views

how much time does a particle take to complete an arc of a circle with a given velocity under gravitational force of Earth?

How much time would a particle take to travel to a height of $5R/3$ from lowest point along an arc, if given velocity $v$ at the lowest point? (circle is in vertical plane)
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2D Poisson Equation With Mixed Boundary Conditions

I need to solve the Poisson equation with mixed boundary consitions analytically. There are complex maps such as (1+z)/(1-z), exp(z), or sin(z) which seem suitable for transformation of this geometry ...
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1answer
29 views

DSolve unable to solve a simple differential equation [on hold]

The Mathematica could not solve DSolve[{Derivative[1][x][t] == -(Exp[2 t a - 2 t b]/ C[1]) + x[t], x[0] == R}, x[t], t] I was wondering why this doesn't work.. ...
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Reduced Chebyshev approximation?

Few days ago my teacher mentioned a method of approximation for a function. I think it was called "reduced chebyshev approximation", where you find the taylor series of degree $N$ then subtract from ...
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1answer
39 views

prove the result of a Laplace transformation

I have to prove the next problem $$\mathcal{L} \left(\int_{0}^{t}\frac{1-e^{-u}}{u}du,s\right) = \frac{1}{s}\log\left(1+\frac{1}{s}\right)$$ I'm quite new in the subject and I have troubles with ...
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1answer
28 views

Consider $\vec{y}'(t) = A\vec{y}$, find the matrix $A$

$\vec{y}'(t) = A\vec{y}$, where $A$ is a real $2 \times 2$ constant matrix with repeated eigenvalues. Phase plane solution trajectories have horizontal tangents on the line $y_2 = 2y_1$ and vertical ...
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28 views

Implicit system differential equations

I came across a system of differential equations in the form: $\newcommand{\D}[1]{\frac{\mathrm{d}#1}{\mathrm{d}x}}$ \begin{align} f_1(x,y,z)\D{y}+f_2(x,y,z)\D{z}&=f_3(x,y,z),\\ ...
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42 views

Solution to System of Complicated Differential Equations

I'm looking for a solution to this set of complicated differential equations: $$\begin{align} \dfrac{dθ}{ds} & = \dfrac{\cos θ}{r} − z\\ \dfrac{dz}{ds}& = − \cos θ \\ \dfrac{dr}{ds} &= ...
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Why Fuchs index $-1$ always there during singularity analysis for ODEs?

During singularity analysis of ODE/PDE I have seen that $-1$ always occur as default resonance, someone told me that this is actually Fuchs index and Fuchs index is always $-1$ for ODE/PDE. Can anyone ...
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19 views

sturm-liouville differential equations

I need method to find the values of $\lambda$ $$1.-y''+x^2 y=\lambda y$$ $$2. -y''+|x| y=\lambda y$$ $$ 3. -y''+(x^2 +x^4) y=\lambda y$$ with initial condition $y(0)=1,y'(0)=0$
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2answers
28 views

Finding the stable and unstable manifold of this system

Consider the system $$\begin{cases}\dot{x} = x \\ \dot{y} = -y + x^2\end{cases}$$ This has fixed point $\overline{X} = (0,0)$, which is a saddle point. The aim is to find the equation of the stable ...
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2answers
55 views

Find the diff. eq. whose solution is given by…

I have recently started to self-study the book "Ordinary Differential Equations" by M. Tenenbaum and H. Pollard. I have no previous knowledge in diff. eq. but I am pretty solid in the calculus of one ...
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1answer
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Statement verification - Stable and unstable manifold theorem

Let $\dot{X} = f(X)$ have hyperbolic fixed point $\overline{X}$ and linearisation $\dot{X} = Df(\overline{X})X$. Then there exists a stable manifold $W^s_{\overline{X}}$ of dimension $d_s$ and an ...
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On a linear non-homogeneous system of differential equations.

I rewrite my attempt at solving this system \begin{cases} x'(t) = 3x(t) + y(t) + e^{2t} \\y'(t) = - x(t) + y(t) + e^t\\ x(0) = 1 \\ y(0) = 0 \end{cases} I notice that the eigenvalue of the matrix ...
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2answers
43 views

Are all solutions to the ODE $ay''(t) + by'(t) + cy(t) = 0$ of the form $y(t)= \alpha e^{(\beta + i\gamma)t}$?

Let $a$ $b$ and $c$ be complex numbers. Consider the complex solution of the ODE $$ay''(t) + by'(t) + cy(t) = 0.$$ If there exist solutions to this, are they necessarily of the form $$y(t)= \alpha ...
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1answer
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Second order differential equation resolution.

I tried solving the following Cauchy problem \begin{cases} y''(t) = \frac{-4y'(t)}{y(t)^2} \\y(0) = 2 \\ y'(0) = 2 \end{cases} By setting $v(s) = u'(u^{-1}(s))$ where $u \in C^1(I,R)$ is a solution ...
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1answer
38 views

Solving the Euler-Lagrange equations for geodesics

I am trying to find geodesics on the following metric: $ds^2 = dx^2 + x^2 dy^2$ Setting $dx \rightarrow \dot{x}, dy \rightarrow \dot{y}$ in $ds^2$ i get following Lagrangian: $L = \dot{x}^2 + x^2 ...
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1answer
53 views

Use Power Series to solve system of differential equations

Problem: Hello, I wonder how you would use a Power Series to solve a system of differential equations. Lets say I have the system $$\begin{cases}(1)\text{ }\text{ }x_1'=2x_1+4x_2 \\ (2)\text{ ...
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2answers
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Variable Change In A Differential Equation

If I have the following differential equation: $\dfrac{dy}{dx} = \dfrac{y}{x} - (\dfrac{y}{x})^2$ And if I make the variable change: $\dfrac{y}{x} \rightarrow z$ I know have $\dfrac{dy}{dx} = ...
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1answer
32 views

Confusion about Poincaré-Bendixson Theorem

The two following theorems appear to be contradictory to me. I'm sure I must have overlooked something significant here. The Poincaré-Bendixson II(a) says that if $R$ is a Type I invariant region, ...
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1answer
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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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1answer
61 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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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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27 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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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, say, $ \binom{x}{y}= c_{1}\binom{1}{1}e^{-5t} + c_{2}\binom{1}{2}e^{-t} $, then it is ...
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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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Solve Turing's morphogenesis with other method than Euler's

I'm currently using following code to solve Turing's morphogenesis: ...
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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
83 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 = ...
4
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1answer
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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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1answer
38 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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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
41 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
36 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.