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

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0
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2answers
61 views

Simultaneous Total Differential Equations 2

To Solve : $\displaystyle \frac{dx}{x^2-y^2-z^2}=\frac{dy}{2xy}=\frac{dz}{2xz} $ Any hints?
2
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3answers
51 views

Find the solution of the initial value problem

Find the solution of the initial value problem: a) $x \, \dfrac{dy}{dx}-y=x+x^2, \ \ x>0 \ \text{ and } \ y(1)=2. $
2
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0answers
87 views

ODE with constraints

Given the ODE system $$\dot{x} = y \\ \dot{y} = \frac{1}{\alpha} (z - y)$$ where $\alpha > 0$ is a constant. How can I find a bound for $z$ depending on $x$ such that $\forall t ~x(t) \geq 0$ under ...
5
votes
3answers
76 views

How find this ODE solution $f''(x)=f(x)(1+2\tan^2{x})$

Question: Find the ODE solution: $$f''(x)=f(x)(1+2\tan^2{x})$$ such $f(0)=0$ My idea: let $y=f(x)$ then $$y''=y(1+2\tan^2{x})$$ $$\Longrightarrow \dfrac{y''}{y}=1+2\tan^2{x}$$ and I ...
0
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1answer
46 views

Why not shift the index of the derivative in Euler series?

I'm reading over solving linear differential equations with analytic coefficients, and finding the solutions that are near regular singular points. In the earlier section on solving similar equations ...
1
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1answer
68 views

Solving a first order nonlinear ODE

I have to solve $\dfrac{dy}{dt}=y^2+3y-3yt$? It is not a separable equation. Can we make a substitution so as to make it become separable?
1
vote
0answers
22 views

Partial differentiation for multi variables

A Candy company makes 2 types of candy A & B for which the average costs are 2 & 3euros per kg respectively. $Q_a$ & $Q_b$ (a & b subscript) are the kg that can be sold each week and ...
3
votes
1answer
53 views

How did they find this equilibrium condition?

I'm studying from a book titled "Mathematical Models in Population Biology and Epidemiology" and we're dealing with SIS models. In a chapter called "Infective Periods of Fixed Length", we get to this ...
1
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1answer
68 views

Partial derivatives-Why does this stand?

In my notes there is the following: $$u_{\xi \eta}=0 \Rightarrow \left\{\begin{matrix} u_{\xi}=0 \Rightarrow u=g(\eta)\\ u_{\eta}=0 \Rightarrow u=f(\xi) \end{matrix}\right.$$ I haven't understood ...
6
votes
1answer
111 views

Airy differential equation and Galois group

Consider the Airy equation $y^{(2)}=ry$ where $r \in \Bbb{C}(z)$ but not constant. How do you show that $G^0=G$, where $G$ is the galois group of the picard vessiot extension of solutions over ...
4
votes
1answer
669 views

What does the dot over $x$ or $y$ mean?

I am just starting to read up on differential equations. The problem is (in my materials) nowhere is explained what do these dots mean. Can anyone shed some light? $$\begin{align}&\begin{cases} ...
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3answers
59 views

How much money did i have from the beginning?

Ok, i will try to explain my problem i plain text: Every year i'll withdrawal $10 from the bank where i have a 2.5% interest rate. I have no money left after 10 year. How much did i have from the ...
1
vote
1answer
37 views

differential inequality of continuous functions

Let $u:[0,+\infty)\to (0,+\infty)$ be a continuous function such that $\int_0^\infty u(x)dx<\infty$. Suppose there exist $a,b>0$ such that $\frac{du}{dx}\leq u(a+bu)$. Prove that ...
3
votes
2answers
82 views

Provide an example of a function whose inverse is also it's derivative.

This is a question from a mathematics competition. I'm totally stumped with this one. If anyone could give an example, or even better, show working, that would be great.
1
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1answer
56 views

Euler's Numerical Method

Let $\eta(x;h)$ be the approximate solution furnished by Euler's method for the initial-value problem $y'=y, y(0)=1$. I proved that: $i) \eta(x;h)=(1+h)^{x/h}$; $ii) \eta(x;h)$ has the expansion ...
1
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1answer
44 views

Explaining the one-dimensional continuity equation with respect to density evolution

I've got a rather abstract question So the continuity equation for a one-dimensional continuum is: $$ \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x}(\rho v)=0 $$ and we can expand ...
1
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1answer
90 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
2
votes
2answers
443 views

Help with solving DE: $x^2y'' + 2xy' - 2y = 0$

Can anyone give me an advice that helps me to solve this kind of DE: $$ x^2 \cdot y'' + 2x \cdot y’ - 2y = 0 $$ knowing that $$ y_1=Ax+{B \over x^2} $$ is a solution. I've tried to solve it by ...
0
votes
1answer
39 views

Easy way to see general solution of PDE expressed in different form?

I think this question should be easy and shouldn't require me to solve the entire PDE for a general solution. Basically, how would you see immediately that the general solution of: $$y^2 u_{xx}-2xy ...
1
vote
1answer
37 views

Trouble with ordinary differential equation

I am having a bit of trouble with this differential equation and could use some help. I am trying to solve : $x^3 y''-(x^2+xy)y'+(y^2+xy)=0$ with the conditions y(1)=k and y'(1)=k(k+1). After much ...
0
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1answer
30 views

Notation confusing my understanding of a homework problem

Probably ultra simple, but asking google about notation is non-trivial in a case like this. The text is Oksendal's Stochastic Diff Eq and, very simply, the question is as follows: Let $B_t$ ...
1
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0answers
93 views

Can a third degree'Riccati' like differential equation be re-written as a third order linear ODE

Consider the equation $$ y' = a_0(x) + a_1(x)y + a_2(x)y^2 + a_3(x)y^3 $$ Does there exist a substition $y = f(a_0, a_1, a_2, a_3, u, u', u'')$ such that after simplifying from this subsitution we ...
3
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0answers
65 views

Finding the exact solution of a differential equation

Let $y=f(x)$. Is it possible to find an exact solution of the following differential equation?: \begin{equation} \ddot y+2\dot y-5xy=e^{-2x}\nonumber \end{equation} Many thanks in advance, -- Cesar ...
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2answers
66 views

Questions about the Laplace's equation in polar coordinates

The Laplace's equation in polar coordinates at a cyclic disk: $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta \theta}, \ \ \ 0 \leq r \leq a, \ \ \ 0 \leq \theta \leq 2 \pi$$ $$u(a,\theta)=h(\theta), \ ...
4
votes
3answers
448 views

The differential equation: $ \arctan (y) = \arctan(x)+C .$

I solved the equation and stalled. Help with decision please. $$(1+y^2)\,dx=(1+y^2)\,dy \iff \int \frac{dx}{1+x^2} = \int\frac{dy}{1+y^2} $$ Transformed expression for the table of integrals. $$ ...
0
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0answers
64 views

Why is the kinetic energy an elliptic quadratic form? (double pendulum)

Here is one question concerning the kinetic energy of the double pendulum I've used the coordinates $$ x_1=L_1\sin\theta_1,~~x_2=L_1\cos\theta_1 $$ for the first bob with mass $m_1$ and $$ ...
1
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2answers
47 views

Differential equation. $(1+y^2)dx=(1+x^2) dy$.

$$(1+y^2)dx=(1+x^2) dy$$ Help please. I'm new to the solution of such equations. Thanks.
0
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0answers
60 views

Is this a first integral for the system (double pendulum)?

Consider the system $$\dot{\theta}_1=w_1$$ $$\dot{\theta}_2=w_2$$ $$\dot{w}_1=-L_1\sin\theta_1+\frac{m_2}{m_1}\cos\theta_2\sin(\theta_2-\theta_1)$$ ...
5
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1answer
69 views

Finding out properties of this ODE system knowing only partial informations about it.

I was doing some preparation for my exam, and I found this interesting exercise (text is in italian). Let's consider the planar system \begin{cases} \dot{x} &= P(x,y) \\ \dot{y} &=Q(x,y) ...
1
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1answer
52 views

Sturm-Louiville Problem

Sturm-Louiville problem $y''+ \lambda y=0 , y'(0)=0, y(2) =0$ I need to find the eigenvalues, $\lambda_n$ and I thought I'd found it to be $\frac{2n\pi-\pi}{4}$ but apparently this is wrong. With ...
8
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0answers
330 views

How to solve a time-dependent Schrodinger equation in periodic Dirac delta potential

I'm trying to solve a 1D time-dependent Schrodinger equation: $$ i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)*x\right]\psi(x,t) $$ where $V(x)$ ...
2
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0answers
187 views

Solving an infinite non autonomous system of differential equations.

For all $\lambda\in\mathbb{R}$, let $J(\lambda)$ be the infinite matrix where $(J(\lambda))_{nn}=\lambda$, $(J(\lambda))_{n,n+1}=1$ for all $n\in\mathbb{N}$, and all other entries are $0$. This matrix ...
3
votes
2answers
95 views

Solving the differential equation $x^3y''+2x^2y-6xy = 0$

First question on here, so I hope I'm doing this right. I've been reading up on differential equations lately and have now stumbled upon one that I have no idea how to solve. $x^3y''+2x^2y-6xy = 0$ ...
4
votes
1answer
390 views

ODE with additional term

In an application I encountered the ODE $$ \left( x^2-1 \right) \frac {{\rm d}^{2}}{{\rm d} x^2} f ( x ) +x \left( \frac {\rm d}{{\rm d}x} f (x) \right) ( 8x^2-7 ) -4 (C+1) f( x ) =0. $$ I don't ...
1
vote
1answer
57 views

Satisfaction of Bessel equation by any other function.

Is it possible that any function $y(x)$ other than Bessel group of functions, satisfy Bessel's equation? $$x^2 \dfrac{d^2 y}{d x^2} + x \dfrac{d y}{d x} + (1-n^2/x^2) y = 0.$$
1
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1answer
55 views

Solution of differential equation: domain vs integral of reciprocal of defining function

Let $g:I\to \mathbb R$, $I\subset \mathbb R$, be the unique maximal solution of $$y'=f(y)$$ $$y(a)=b$$ with $f:\mathbb R \to \mathbb R^+$ continuous. Now I want to prove that $I=\mathbb R$ iff ...
1
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1answer
303 views

General solution of a system of linear differential equations with multiple generalized eigenvectors

I am looking for general solutions for the linear sODE's $$\textbf{x}'(t) = A\textbf{x}(t)$$ with $t \geq 0$ and $A \in \mathbb{R}^{n \times n}$ Let focus on just real eigenvalues and eigenvectors. ...
1
vote
1answer
135 views

Calculating a double pendulum

consider the following situation of a double pendulum. We found the moving equations as $$ \ddot{\theta_1}=-L_1\sin\theta_1 + \frac{m_2}{m_1}\cos\theta_2\sin(\theta_2-\theta_1),\\ ...
3
votes
1answer
42 views

Linear Differential Equations of higher order

I am studying the basics of linear differential equations. $$\displaystyle \frac{d^ny}{dx^n}+k_1\frac{d^{n-1}y}{dx^{n-1}}+k_2\frac{d^{n-2}y}{dx^{n-2}}+\cdots+k_ny=X$$ First the complementary function ...
0
votes
2answers
61 views

Higher Order Differential Equation Solution

The differential equation$$y′′′′+ay′′′+by′′+cy′+dy=0$$ has solution $$ y=−3te^{2t}+2e^{−2t}\sin(5t)$$ Find $a, b, c$ and $d$. I've tried looking online for problems similar to this but had no ...
3
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1answer
197 views

Motivation and Derivation of the Riccati Equation Transformation

Given a Riccati Equation which is differential equation of the form: $$ \frac{dy}{dx} = a_0 (x) + a_1 (x)y + a_2 (x)y^2 $$ It is well known that the transformation: $$ y = -\frac{1}{a_2(x)} ...
-1
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1answer
133 views

Creating a model for a drag racing car

We would like to create a model for a drag racing car. The horizontal distance of the car from the start line is given by x(t) and is measured in meters. Therefore the speed of the car is given by ...
6
votes
1answer
109 views

Help needed in solving a system of DE

The system of DE is: $$\frac{dI}{db}=-\frac{b}{c}\frac{dJ}{db}-\frac{2ab+1}{2c}J$$ $$\frac{dJ}{db}=\frac{b}{c}\frac{dI}{db}-\frac{2ab-1}{2c}I$$ Assume that $a$ and $c$ are constants and both $I$ and ...
6
votes
1answer
195 views

How can I choose the best algorithm to integrate ODE's numerically?

I have studied in a course several algorithms to integrate ODE's numerical: Runge-Kutta, Predictor-Corrector methods, Taylor... However the teacher failed to show which is the best for every ...
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1answer
38 views

A class of solutions to $\frac{d^2x}{dt^2}+2k\frac{dx}{dt}+(k^2+n^2)x=0$

Question: Show that, the solution $x = \exp(-kt)(a\cos(nt)+b\sin(nt))$, for all $a$ and $b$, always satisfies the differential equation $\dfrac{d^2x}{dt^2}+2k\dfrac{dx}{dt}+(k^2+n^2)x=0$. My ...
1
vote
0answers
26 views

What is the function $f$ verifying : $f(\frac{x}{2}+\frac{x}{2}\cos(\frac{v\pi}{x}))=\frac{x}{2}\sin(\frac{v\pi}{x})$

What are the solutions to the functional equality (for a constant $v$): $$ \forall\, x > 0, \ \ \ \ ...
3
votes
1answer
118 views

Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
0
votes
1answer
68 views

Solving 2nd-order ODE for SHO

In physics for a Simple Harmonic Oscillator, we have the differential equation $$ {\frac {d^2x}{dt^2}} + \frac kmx = 0 $$ from the balance of forces, which has a solution $$ x(t) = {x_o}\cos(\omega ...
1
vote
0answers
43 views

Function with bounded derivative as ODE

Given a function $x(t)$, I am looking for a function $y(t)$ which closely follows $x(t)$ except that its derivative must be bounded by a constant $c$, i.e. $\dot{y} \leq c$. Is there a way to describe ...
0
votes
2answers
49 views

Differential operator equation

I have to solve this equation: $\left( \frac{d}{dx}+ a\right)^{n+1} Z(x)=0$ with the following initial conditions: $Z(0)=0$ $\hspace{10mm}\vdots$ $\frac{d^{n-1}}{dx^{n-1}}Z(x)_{|x=0}=0$ ...