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

learn more… | top users | synonyms (1)

2
votes
0answers
106 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
2answers
285 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
votes
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
votes
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
366 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
2answers
562 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
326 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 ...
2
votes
2answers
4k views

Find the differential equation of all circles of radius a

Can someone please post a detailed step-by-step procedure. Given the circle with a radius a, what is the differential equation of the circle.
1
vote
2answers
146 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 ...
1
vote
1answer
93 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 ...
1
vote
0answers
90 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
242 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
2answers
135 views

How to know if a point is analytics or not?

So I have the equation y" + [(x-x^3)/x]y' +[(sinx)/x]y = 0 My x nought it equal to 0. I know this is a singular point because my denominator is equal to ...
1
vote
1answer
66 views

Uniqueness and Existence problem

I just need a bit of help with this question. If I know that $dg/dx = g^2$, and that $g(0) = g_0$, then I can solve: $$ dg/dx = g^2\\ \frac{1}{g^2} dg = dx \\ -\frac{1}{g} = x + \hat c \\ -g = ...
1
vote
1answer
127 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
vote
2answers
259 views

The Decision of three methods of the solutions $dx/P=dy/Q=dz/R$

Question: (A) $$\frac{adx}{(b-c)yz}= \frac{bdy}{(c-a)xz}=\frac{cdz}{(a-b)xy}$$ (B) $$\frac{dx}{xz-y}=\frac{dy}{yz-x}=\frac{dz}{1-z^2}$$ These are simultaneous diff eq. of the first order and the ...
1
vote
1answer
1k 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 ...
1
vote
1answer
101 views

dropping a particle into a vector field

I'm independently studying Colley's Vector Calculus and am on the section on line integrals. I understand that the line integral gives the amount of work done on a vector field for a predetermined ...
1
vote
2answers
66 views

Laplace question - help needed

I am currently studying the Laplace transformation and came across this question: I have no idea of how to start this and am completely lost. If anyone could help I would be really grateful. ...
1
vote
2answers
221 views

how did he conclude that?integral

So the question is : Find all continuous functions such that $\displaystyle \int_{0}^{x} f(t) \, dt= ((f(x)^2)+C$. Now in the solution, it starts with this, clearly $f^2$ is differentiable at every ...
1
vote
1answer
157 views

Multistep ODE Solvers

Write both a fourth order Adams Bashforth and Adams Moulton procedure to solve $$x'(t) = x(t)-y(t)-\exp(t);$$ $$y'(t) = x(t)+y(t)+2\exp(t)$$ with initial values $x(0) = -1, y(0) =- 1$ on the ...
1
vote
1answer
66 views

Solutions and attraction regions of following odes?

Assume a mapping $X: \mathbb{R} \to \mathbb{R}^d$. We know that the solution to ode $$ d X_t = (\mu - X_t) dt $$ is $X_t = (X_0-\mu) e^{- t} + \mu$, which indicates that $X_t$ converges to $\mu$ as ...
1
vote
1answer
542 views

Existence and Uniqueness Theorem

I had a question about how to do one of these problems. So here's the question: Given this equation $y'=\frac{-\cos(t)y(t)}{(t+2)(t-1)}+t$, find if the initial conditions $y(0)=10, y(2)=-1, y(-10)=5$ ...
1
vote
2answers
125 views

solving this second order ode

Consider the second order ODE where $ (k-x)^2 y''+6(k-x)y'+12y=F(x) $ where $k$ is some constant. I want to compute the real valued general solution. progress: guess $(k-x)^{m}$ to be the solution ...
1
vote
2answers
408 views

Using Octave to solve systems of two non-linear ODEs

How to solve following system of ordinary differential equations using Octave? $$\frac{dx}{dt} = - [x(t)]^2 - x(t)y(t)$$ $$\frac{dy}{dt} = - [y(t)]^2 - x(t)y(t)$$ Update: initial conditions: ...
0
votes
1answer
26 views

Poincare-Bendixson theorem contradiction help

Lets suppose p is asymptotically stable but not a singularity for the planar differential equation dx/dt=f(x), then for points x sufficiently closed to p we must have x(t) tends to p. so the limit set ...
0
votes
2answers
83 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } ...
0
votes
1answer
44 views

Derivative of a differential equation help??

Please can someone explain this to me in detail: if $y''+4y'+3y=14\cos(2x)$ and $z'''+4z''+3z'=-28\sin(2x)$ show that the $z=y+c$ where $c$ is a constant I know the second is the integral of the first ...
0
votes
1answer
68 views

Find the solution for a boundary value problem

Please, how can we find the solution of this second order boundary value problem $$-(e^{-2x}u')'-\ln(x^2+2)u= 2 e^ {-2x} - x \ln(x^2+2),\,\, x\in ]0,1[, u(0)=0,u(1)=1?$$ Or more generally, What's the ...
0
votes
1answer
34 views

Solve a differential equation and evaluate the solution at a particular value of independent variable

If $\frac{dy(x)}{dx}=(2-3i)y(x)$ where $i=\sqrt{-1}$, what is the value of $y(\pi)$?
0
votes
1answer
236 views

Wronskian-Differential Equations

The equations below are matrices: Consider the vectors $y^{(1)} (t)$=$\begin{pmatrix}t \\1 \end{pmatrix}$ and $y^{(2)}$ (t)=$\begin{pmatrix}t^2 \\2t \end{pmatrix}$ (a) Compute the Wronskian of ...
0
votes
1answer
79 views

Let $f$ be a field with only one singularity in the origin. Show that the phase diagram of the field $f$ has exactly three distinct orbits

Let $f:\mathbb R\to \mathbb R $ be a field with only one singularity in the origin. Show that the phase diagram of the field $f$ has exactly three distinct orbits which are the following: I need ...
0
votes
2answers
217 views

Solve separable DE with integrating factor and homogeneous substitution

I just came out of test which asked to solve $$\frac{dy}{dx}=\frac{y}{x}$$ with $x,y>0$ in three ways: by separating the variables, using the substitution $y=vx$ and using an integrating factor. ...
0
votes
2answers
372 views

Find a particular solution of the differential equation $-3y''-2y'+y=3xe^x$

Using the method of undetermined coefficients. Guess $(Ax+B)e^x$ Plug into diff eq: $-3[(Ax+B)e^x]'' - 2[(Ax+B)e^x]' + (Ax+B)e^x = 3xe^x$ Wolfram alpha simplifies this to: $A(x-2)=e^x(4B+3x)$. ...
0
votes
0answers
112 views

eigenvalue problem

let be the 2 eignevalue problems $-y''(x)+f(x)y(x)=E_{n}y(x) $ with the boundary condition $ y(0)=0= y(L) $ $-y''(x)+g(x)y(x)=\lambda _{n}y(x) $ with the boundary condition $ y(0)=0= ...
-4
votes
1answer
249 views

$\frac{dy}{dx} = 3y^{2/3}$ general solution?

What's the general solution of $\frac{dy}{dx} = 3y^{2/3}$ ? Im pretty sure this is a separable equation, but I'm not sure how to go forward? Just multiply by $dx$ and $\frac{1}{3y^{2/3}}$ well then I ...
14
votes
3answers
4k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
21
votes
8answers
2k views

What's so special about sine? (Concerning $y'' = -y$)

In an attempt to actually grok sine, I came across the $y''= -y$ definition. This is incredibly cool, but it leads me to a whole new series of questions. Sine seems pretty prevalent ...
13
votes
3answers
872 views

Solving Differential Functional Equation $f(2x)=2f(x)f'(x)$

Find all functions satisfying $f(2x)=2f'(x)f(x)$ Under given condition, can't we find explicit solutions?
11
votes
1answer
462 views

Osgood condition

Let $h$ and $g$ be continuous, non-decreasing and concave functions in the interval $[0,\infty)$ with $h(0)=g(0)=0$ and $h(x)>0$ and $g(x)>0$ for $x>0$ such that both satisfy the Osgood ...
15
votes
1answer
423 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2-y'(x)+C = -A \cos(x) - B \cos^2(x),$$ where $C, A$ and $B $ are parameters. (The case that either $A$ or $B$ ...
10
votes
1answer
594 views

What is, how do you use, and why do you use differentials? What are their practical uses?

What is a differential? And how is it useful? What is its practical use? For example, in Electromagnetic Wave Theory as it pertains to diffraction gratings, we have an equation like this one: ...
8
votes
1answer
599 views

Eigenfunctions of the Laplacian

I am willing to offer a bounty for this one, so I will give you an exact idea of what I need: I am looking for solutions of $$\Delta \Psi(r,\theta)=k^2\Psi(r,\theta)$$ where $k\in \mathbb{R}$. Such ...
4
votes
1answer
304 views

Generalized Legendre differential equation

In an application I encountered the ODE $$ \left( x^2-1 \right) \frac {{\rm d}^{2}}{{\rm d} x^2} f ( x ) +x \left( \frac {\rm d}{{\rm d}x} f (x) \right) ( 8x^2-7 ) -4 (C+1) f( x ) =0. $$ which is ...
3
votes
3answers
277 views

Numerical Analysis References

Could anyone suggest any good (perhaps online ref papers) reference material on numerical analysis focusing on determining accuracy/estimated errors, rates/orders of convergence especially when ...
11
votes
3answers
607 views

When do the Freshman's dream product and quotient rules for differentiation hold?

This is motivated by looking at the calculus exams of some of my undergraduate students. A recurring mistake is assuming that the derivative of the product of functions is a product of derivatives and ...
8
votes
0answers
427 views

Complicated exercise on ODE

I have this exercise extracted from a examination of qualitative theory of ODE (in which we study the existence and uniqueness of solutions, and stability using the function of Lyapunov) I don't know ...
6
votes
2answers
674 views

Particular solution to a Riccati equation $y' = 1 + 2y + xy^2$

The equation is $y' = 1 + 2y + xy^2$. I've tried $mx+n$, $ax^m$, even $\tan x$ as candidates for particular solution where $a,m,n \in \mathbb Q$, but it did not work. Can anyone find one particular ...
5
votes
1answer
166 views

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...
4
votes
3answers
174 views

How $\frac{dx}{dy}=f(x)g(y) \Leftrightarrow \int \frac{dx}{f(x)} = \int g(y)dy$?

In my intro differential equations class we have often used the "equivalence" stated in title. It seems to me that somehow, the intermediate step $$ \frac{dx}{f(x)} = g(y)dy$$ is being used, in which ...