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

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4
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3answers
954 views

Find the time span of snow plow operation, given that its speed is inversely proportional to the height of the snow

One day snow began to fall before dawn and continued to fall at a constant rate. At midday a snowplot set out to clear a road. At 2pm it turned back, arriving to the starting point at 3pm. If we ...
0
votes
1answer
54 views

Ball motion with air resistance coupled differential equation for fourth-order Runge-Kutta

I've created a MATLAB function for solving coupled differential equation with the fourth-order Runge-Kutta method based on what is provided here (Simultaneous Equations of First Order). Here the ...
-1
votes
0answers
23 views

What is the inverse Laplace transform of $\lfloor s \rfloor$?

How can we find the inverse Laplace transform of: $[x]$ (floor function) ? My question isn't LLaplace transform of floor function i asked the "inverse" laplace transform of floor function ...
3
votes
1answer
107 views
+50

Convolution: How to construct it for a given function?

While working on my thesis my advisor handed me an unfinished paper which states the following: First, define the operators \begin{align*} A_i &:= -\operatorname{div}(\sigma_i\nabla) \\ A_e ...
0
votes
1answer
20 views

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 ...
0
votes
1answer
35 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)
0
votes
0answers
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 ...
0
votes
0answers
9 views

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 - ...
1
vote
2answers
48 views

Rewriting solutions in the standard form: simple harmonic motion

The differential equation: $y''+\omega^2 y=0$ has as a general solution: $$y=A\cos{(\omega t)}+B\sin{(\omega t)}$$ By taking: $$A=R\cos{(\omega t_0)}$$ and $$B=R\sin{(\omega t_0)}$$ We can rewrite ...
0
votes
1answer
12 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 ...
5
votes
0answers
24 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 ...
0
votes
1answer
14 views

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 ...
2
votes
1answer
114 views

How to solve second order differential equations? [summary]

As I do my engineering studies, I find more and more ways to solve differential equations, especially the second order ones. With more and more ways to solve these equations, I am loosing my overview ...
1
vote
1answer
14 views

“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 ...
2
votes
0answers
18 views

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 ...
1
vote
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 ...
-1
votes
1answer
25 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
0
votes
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 ...
-3
votes
0answers
25 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
17
votes
3answers
553 views

Does Tom catch Jerry?

Tom has Jerry backed against a wall. Tom is distance 1 away (perpendicularly). At time t=0, Jerry runs along the wall. Tom runs directly towards Jerry. Tom always runs directly towards Jerry. Tom and ...
1
vote
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 ...
0
votes
2answers
26 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 ...
3
votes
1answer
56 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 $$ ...
0
votes
0answers
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 ...
0
votes
1answer
37 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, say, $ \binom{x}{y}= c_{1}\binom{1}{1}e^{-5t} + c_{2}\binom{1}{2}e^{-t} $, then it is ...
1
vote
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 ...
0
votes
0answers
13 views

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 ...
1
vote
2answers
563 views

How can i convert nonhomogeneous ode to homogeneous ?

I have an equation system $$y'(t) = M(t)y(t)+h(t)$$ where $[M(t)]_{2\times2}$ square matrix and $[h(t)]_{2 \times1}$ is the nonhomogeneous part of the system. I can solve numerically homogeneous ...
0
votes
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 ...
0
votes
0answers
14 views

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 ...
0
votes
3answers
129 views

Resolvent Kernel of Volterra Integral Equation

The resolvent kernal $R(x,t,\lambda)$ for the Volterra integral equation $$\phi(x)=x+\lambda\int\limits_a^x\phi(s)ds$$ is $\begin{array}1 1. e^{\lambda(x+t)} && 2. e^{\lambda(x-t)} ...
2
votes
0answers
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),\\ ...
0
votes
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.. ...
1
vote
0answers
16 views

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 ...
3
votes
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 ...
0
votes
1answer
40 views

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 ...
4
votes
2answers
86 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 = ...
0
votes
0answers
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} &= ...
1
vote
1answer
39 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 ...
0
votes
0answers
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 $y(0)=1,y'(0)=0$
1
vote
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 ...
0
votes
1answer
22 views

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 ...
1
vote
1answer
14 views

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 ...
1
vote
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 ...
2
votes
1answer
502 views

Show $\nabla^2g=-f$ almost everywhere

let a continuous function $f(x,y,z)$ be absolutely continuous over every bounded region and let it be in $L^1$ that is $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} ...
5
votes
1answer
105 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 ...
0
votes
1answer
29 views

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 ...
1
vote
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{ ...
1
vote
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 ...
0
votes
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, ...