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

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2
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2answers
722 views

Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$.

Given the function $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} & x \neq 0 \\ 0 & x = 0 \end{array}\right. $$ show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$. So I have to show ...
2
votes
1answer
198 views

Help with Initial value problem : $y'= x^2+ xy^2, y(0) = 0$; Picard–Lindelöf Approximation.

i need solve this: $$y'=x^2+xy^2 , y(0)= y(t_0)= 0$$ a) Compute, starting from the constant function $u_0=0$ the successive approximations $u_1,u_2,u_3$ (in the sense of the theorem of ...
2
votes
1answer
188 views

Explain Dot product with Partial derivatives in Polar-coordinates

Related to page 819 prob 4 in this book. I am incorrectly calculating the left-hand-side (def. LHS), some newbie error with commutativity probably. Ideas? Errors? I propose ...
2
votes
1answer
828 views

Boundary conditions of an elastic bar

I was following some online lecture relating to an elastic bar with length $L$ that obey the differential equation $\displaystyle \frac{d^{2}u}{dx^{2}} = f(x)$, where $f(x)$ is its own weight or some ...
1
vote
2answers
92 views

Using laplace transforms to solve a piecewise defined function initial value problem

I want to use laplace transforms to solve the following: $$\frac{d^2 y}{dt^2}+16 y = f(t) = \left\{\begin{array} 1 1&t\lt\pi\\0&t\geq \pi\end{array}\right.\text{ with } y(0)=0 \text{ and } ...
1
vote
0answers
45 views

How to solve a system of two differential equations describing the concentration in a leaky tank?

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
1
vote
3answers
212 views

Differential equation $\sin \theta \frac{dr}{d \theta}+r\cos \theta =\tan \theta,0<\theta<\pi/2$ [closed]

This problem has been stumping me for over an hour how can I set it up, I think I have done it wrong over and over. Solving for $r$.
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3answers
3k views

How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system. I don't know how to use Mathematica/Matlab ... :( If anyone can make this portrait and post a print screen here, I would thank you ...
1
vote
1answer
99 views

Questions concerning the differential operator

Consider the differential equation:- $a \phi + (bD^3 - cD)w =0$, where $a, b$ and $c$ are constants, $D$ denotes the differential operator $\dfrac{d}{dx}$, and $w$ is a function of $x$. I'm ...
1
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3answers
276 views

General solution of second-order linear ODE

I am trying to look a bit deeper into the mathematics the equation of motion used in physics and engineering. I have some specific questions at the end, but please correct me if I make a mistake in ...
1
vote
1answer
301 views

How to show that the geodesics of a metric are the solutions to a second-order differential equation?

On $\mathbb R^n$, let $\rho: \mathbb R^n\to\mathbb R$ be a smooth function, and $g$ be the metric given by scaling the usual flat metric by $e^{2\rho}$. I want to know how to show that the geodesics ...
0
votes
1answer
66 views

van der pol equation

Consider the van der Pol equation below: $(x'')+a(x^2-1)(x')+(x)=0$ I need to : Find an equilibrium point and linearize this equation near it Find solutions of the linearized equation depending on ...
0
votes
0answers
75 views

Proof the first-order equation

$$ \begin{align} \tau\dot y + y &= KF(t) \tag{1}\\ y(t) &= C_0 + C_1 e^{-t/\tau} \tag{2} \end{align} $$ How these two equations can form $$y(t)=KA+(y_0-KA)e^{-t/\tau}?$$ Note ...
0
votes
2answers
416 views

Substitution $x=\sinh(\theta)$ and $y=\cosh(\theta)$ to $(1+x^{2})y'-2xy=(1+x^{2})^{2}$?

After this substitution I got to the point $$\cosh^6 (\theta)y'-\sinh(2\theta)-\cosh^4 (\theta)=0$$ and now let $$z=\cosh^2 (\theta)$$ so $$z^3 y'-z^2-\sinh(2\theta)=0$$ but then I ...
10
votes
3answers
395 views

Why is it legitimate to solve the differential equation $\frac{dy}{dx}=\frac{y}{x}$ by taking $\int \frac{1}{y}\ dy=\int \frac{1}{x}\ dx$?

Answers to this question Homogeneous differential equation $\frac{dy}{dx} = \frac{y}{x}$ solution? assert that to find a solution to the differential equation $$\dfrac{dy}{dx} = \dfrac{y}{x}$$ we may ...
7
votes
3answers
730 views

Why does the absolute value disappear when taking $e^{\ln|x|}$

I have noticed that if you have an equation (after integrating) such as $$\ln|y| = \ln|x| + c,$$ and you further simplify it using the law of exponents, you get $$e^{\ln|y|} = e^{\ln|x|+с},$$ which is ...
4
votes
2answers
132 views

A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$

Let: $f(x)$ and $g(x)$ be twice differentiable, non-decreasing functions. $f''(x) = g(x)$ and $g''(x) = f(x)$. $f(x) \cdot g(x)$ is a linear function. Then we have to show that $f(x) = g(x) = ...
4
votes
2answers
2k views

How can I solve this Initial Value Problem using the Euler method?

My Problem is this given Initial Value Problem: $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ I am looking for a way to solve this problem using the Euler method. I have a given Interval of $[1,2]$ and a ...
4
votes
1answer
170 views

Nonlinear equation (oscillon) comparison

Lagrangian for a spherically-symmetric, real scalar field in d spatial dimensions, $$L=c_d \int r^{d-1}dr\left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
4
votes
2answers
279 views

Formula for integration bounds of recursively defined polynomial sequence

We can recursively define a sequence of polynomials by $$P_0(x) := 1$$ and then with the definite integral $$P_n(x) := \int_{c_n}^x P_{n-1}(t) ~\mathrm dt$$ where the $c_n$ are to be chosen so ...
3
votes
1answer
83 views

Are Exponential and Trigonometric Functions the Only Non-Trivial Solutions to $F'(x)=F(x+a)$?

Are exponential & trigonometric functions the only non-trivial solutions to $F'(x)=F(x+a)$? $F(x)=0$ would be the trivial solution. Then, for $a=0$ (or $a=2\pi i$), we have $F(x)=e^x$, and ...
3
votes
2answers
197 views

Solving ODE with substitution

I have this as homework: $$(xy^2+y)dx+(x^2y-x)dy=0$$ I tried to solve it by substituting $z=xy+1$, but got the answer like $y=Cxe^{xy}$, which, I guess, is wrong. I tried to solve it couple of ...
3
votes
1answer
387 views

Two-Point boundary value problem

To solve ${d^2y \over dx^2} =f(x)$, $0<x<1$ with $y(0)=\alpha, y(1) = \beta$. We can get a finite difference approximation by taking $$\frac{y_{j+1}-2y_j+y_{j-1}}{h^2} =f_j \\\Rightarrow ...
3
votes
1answer
638 views

Why does acceleration = $v\frac{dv}{dx}$

If we define $x$ = displacement, $v$ = velocity and $a$ = acceleration then I am used to the ideas that $a= \frac{dv}{dt} = \frac{d^2x}{dt^2}$ However I also understand $a=v \frac{dv}{dx}$. Can ...
2
votes
1answer
36 views

Is the continuity of a vector field enough for the existence of the solution of a differential equation?

I've recently seen the existence-uniqueness theorem for ordinary differential equations from Arnold's book. I understand that the theorem as stated guarantees both existence and uniqueness if the ...
2
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0answers
68 views

Laplace Trouble to find solution

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 < t < \pi$ and $f(t)=0$ if $t > \pi$? Initial ...
2
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1answer
82 views

Finding a value a for topologically conjugacy between two flows

Let A be a hyperbolic matrix such that all solutions of $\overrightarrow x' = A \overrightarrow x $ tend to the origin at t goes to infinity, and suppose B = $\begin{bmatrix}a-3 & 5 \\ -2 & ...
2
votes
1answer
98 views

Eigenvectors Trajectories

I got stuck with a problem while studying for a control systems exam. It goes as following: "Look at the picture of trajectories of a linear, time-invariant system with the form: ...
2
votes
2answers
50 views

second order DE using reduction of order

Any Hints / details on how to find a second solution for $$x^2y'' + xy' -4y=0?$$ $$y_1 = x^2 y_2$$ I need to use reduction of order thanks
2
votes
1answer
106 views

Choice of the First Term in Legendre Polynomials

The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ...
2
votes
2answers
103 views

Differential operators confussion

I want to solve this PDE: $$u_t-6uu_x+u_{xxx} = 0\,(1)$$ with the Inverse Scattering Method. This method is based on showing that the above equation can be expressed as $$L_t=LB-BL,\,(2)$$ where $L$ ...
2
votes
2answers
117 views

Solve the pde $u_t(x,t)=u_{xx}(x,t)-bu(x,t)+q(t)$ for $u(x,t)$

I have the example pde $u_t(x,t)=u_{xx}(x,t)-b(t)u(x,t)+q_0$, where $b(t)$ is a function of only $t$ and $q_0$ is a constant, $0<x<\pi$, $t>0$. The subscripts denote derivatives. I also have ...
2
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1answer
142 views

Legendre Equation Properties

Is there a nice way to derive, starting from the Legendre differential equation, the generating function, the recurrence relation, the Rodrigues differential form & the Schlafli integral form ...
2
votes
0answers
533 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
2
votes
0answers
119 views

Calculate half life of esters

I'm trying to calculate the level of testosterone released from different testosterone esters. Here are some graphs of testosterone levels after single injections of 250mg of each ester. Testo U ...
2
votes
3answers
533 views

Express differential equations as system of first order equations

Express the differential equation $$y'''-6y''-y'+6y=0$$ as a system of first order equations i.e. a matrix equation of the form $$A(\vec x)'=0$$ where $$\vec x\text{ is the vector }\left[ ...
2
votes
2answers
386 views

Considering the linear system $Y'=AY$

What would be an equation that I can use when I compute the eigenpairs for the coefficient matrix $A.$
2
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0answers
69 views

Differential Equation - $y'=5|y|^{4/5}, y(0)=0$

in the spirit of this question I ask about this one. $y'=5|y|^{4/5}, y(0)=0$ If $y> 0$ then $$y'=5|y|^{4/5}\iff y'=5^{-1}y^{4/5}\iff 5^{-1}y'y^{-4/5}=1\iff y^{1/5}=x+C\\ \iff ...
2
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1answer
64 views

Differential Equation - $y'=|y|+1, y(0)=0$

The equation is $y'=|y|+1, y(0)=0$. Suppose $y$ is a solution on an interval $I$. Let $x\in I$. If $y(x)\ge 0$ then $$y'(x)=|y(x)|+1\iff y'(x)=y(x)+1\iff \frac{y'(x)}{y(x)+1}=1\\ \iff \ln ...
2
votes
1answer
445 views

butcher tableau runge kutta methods

Hi I have had a go at this question- am i heading in the right direction? it would be much appreciated if someone could me Write the Butcher Tableau for the 1-stage $\theta$ method: $$U^n ...
2
votes
3answers
1k views

Simple Harmonic Oscillator Solution

In Physics, the Simple Harmonic Oscillator is represented by the equation $d^2x/dt^2=-\omega^2x$ . By using the characteristic polynomial, you get solutions of the form $x(t)=Ae^{i\omega t} + ...
1
vote
2answers
47 views

Differentiation - simple case

In the book calculus made easy, page 22 the case of the negative power for $y=x^{-2}$ $$\begin{align} y+dy & =(x+dx)^{-2}\tag{1}\\ \\ & = x^{-2}\left(1+\frac{dx}{x}\right)^{-2}\tag{2} ...
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0answers
57 views

Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
1
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2answers
157 views

How can I solve these pde's?

Three different problem I got: 1.. $xu_x+2x^2u_y-u=x^2e^x$ and $u(x,x^2+x)=xe^x+x^2$ 2.. $yu_{xx}+(x+y)u_{xy}+xu_{yy}=0, \quad x\neq y$ 3.. $(y+xu)u_x+(x+yu)u_y=u^2-1$ Couldnt even start. Could ...
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3answers
100 views

Deducing the exact solution of a ODE

In page 53 of Arieh Iserles's A first course in the numerical analysis of differential equations, he presents the following ODE: $(\vec{y})'=\Gamma\cdot\vec{y}$, $\vec{y}(0)=\vec{y_0}$ Using the ...
1
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1answer
94 views

Consider the following ordinary differential equation

Consider the following ordinary differential equation $\frac{dy}{dx} = xy^2$ given $y(0)=1$ (i) Find the analytical solution to this problem. (ii) Given that $y(1.4) = 50$, use the modified ...
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0answers
126 views

Differential equation with random variable

How can I derive analytically or compute numerically the solution to following differential equation $$ dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X $$ where X is a random variable (e.g. from a ...
1
vote
5answers
243 views

I.V.P $y'=\sin(e^{y}), y(0)=a$

Is the I.V.P: $$\begin{cases} \dfrac{dy}{dx}=\sin(e^{y})\\[8pt] y(0)=a \end{cases} \text{ where } a\in \mathbb{R}$$ a) Find the values ​​of $a$ for which $y(x, a)=0$ b) Prove that if $a=0$ then ...
1
vote
1answer
153 views

First order non linear Ordinary differential equations

Consider the first order differential equation $\displaystyle\frac{dy}{dt} = f(t,y)= -16t^{3}y^{2}$, with the inital condition $y(0)=1$ Estimate the lipschitz derivative for the differential ...
1
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1answer
2k views

How to Solve the Coupled Differential Equations?

I came across the set of following coupled equations while studying cycloid motion in Griffiths' Intro to ED $\ddot{y}=\omega \dot{z}$ $\ddot{z}=\omega (\frac{E}{B}-\dot{y})$ I am at a loss as to ...