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
318 views

Undamped spring mass system

I have this study guide for an upcoming test for DE class I'm trying to figure out. A mass of 400 grams stretches a spring by 5 centimeters. (a) Find the spring constant k, the angular frequency ω, ...
0
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2answers
87 views

Second order ODE - why the extra X for the solution?

Assuming I have the following homogeneous ODE equation: $$a\cdot y'' + b\cdot y' + c \cdot y = 0$$ Why for $(b^2 - 4\cdot a\cdot c=0) \quad $,(meaning, when $m_1=m_2$) then the solution is: $$y = C_1\...
0
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1answer
183 views

How to go about solving this question on differentials?

A ring of a planet has an inner radius of approximately 52,000 km (measured from the center of the planet) and a radial width of 19 km. Use differentials to estimate the area of the ring. (Round your ...
0
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1answer
36 views

Division of differential equations

$$\frac{dx(t)}{dy(t)}=\frac{\alpha x(t) - \beta x(t) y(t)}{-\gamma y(t) + \delta x(t)y(t)}$$ How would one simplify this fraction? Maybe the chain rule could be of any use, but I don't see how.
0
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1answer
71 views

How to solve implicit differential equation?

Suppose we go from the equation and go backwards: $$y=c\,e^x+e^{2x+c}$$where $c$ is any arbitrary constant. Now, $$y'=c\times(e^x)+(2e^c)\times(e^{2x}).$$ Solving for $c$: we get $$c=\ln\left(\frac{y'-...
0
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1answer
338 views

ODE $y''+ 9y = 6 \cos 3x$

I have this equation: $y''+ 9y = 6 \cos 3x$ $$ m^2 + 9m = 0\\ m(m + 9) = 0\\ m_1 = 0;\\ m_2 = -9;\\ y_h = c_1 + c_2 e^{-9x}\\ r(x) = 6\cos3x\\ y_p = K\cos3x + M\sin3x\\ y'_p ...
0
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2answers
53 views

How to solve this first-order differential equation?

I have been trying to solve a differential equation as a practice question for my test, but I am just unable to get the correct answer. Please have a look at the D.E: $dy/dx = 1/(3x+\sin(3y))$ My ...
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6answers
155 views

Solution to a second order ordinary differential equation

Let $\beta >0$, $\gamma > 0$, $\omega > 0 $ and $\xi >\xi_0 $. The question is to show that the solution to the following inhomogenous ordinary differential equation: \begin{equation} \...
0
votes
3answers
42 views

Converting a differential equation

Consider an ODE $\frac{dy}{dx}=h(x,y)$ such that $h(rx,ry)=h(x,y)$, which implies $h(x,y)=k\left(\frac{x}{y}\right)$. (Why?) Show that this ODE can be changed to a separable ODE for $u=u(x)$, if ...
0
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1answer
50 views

Can someone give me an example of how to work out an exact linear second order differential equation?

I have a theorem that states: If an equation $P(x)y''+Q(x)y'+R(x)y=0$ can be written in the form: $$[P(x)y']'+[f(x)y]'=0$$ then the equation is said to be exact. Now I need to expand and equate ...
0
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1answer
247 views

Solving a system of two second order ODEs using Runge-Kutta method (ode45) in MATLAB

I'm trying to solve the system of differential equations outlined in Von Karman's rotating disk flow. I got them into a system of ordinary differential equations: F(n), G(n), H(n) $$F'' = -G^2 + F^2 ...
0
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3answers
696 views

Integrating absolute value function

I'm working on a problem drawing phase plane diagrams in my applied mathematics course. I'm supposed to draw the phase line diagram of $x''+\vert x\vert=0.$ In the process, I get to the differential ...
-1
votes
1answer
90 views

Initial values are lost (diff eq to Transfer function)?

I read eternal Julius O. Smith III and he says that $$x_{n-m} = z^{-m}X(z)$$ Particularly, difference relation $$y_{n} = y_{n-1} + x_{n}$$ is solved by by $$Y = z^{-1}Y + X = {X \over (1-z^{-1})...
-2
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1answer
52 views

Differential Equations: solve the system

Solve the following system: $$dx/dt=-.2(y-2)$$ $$dy/dt=.8(x-2)$$ This is what I have so far, but I got stuck.. $$\begin{eqnarray} dx/dt&=&-.2y-.4\\ x'&=&-.2y-.4\\ x'+.4&=&-....
-6
votes
1answer
302 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 ...
73
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0answers
844 views

Solving Special Function Equations Using Lie Symmetries

The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's ...
28
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1answer
3k views

Why isn't the 3 body problem solvable?

I'm new to this "integrable system" stuff, but from what I've read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of ...
12
votes
3answers
9k views

Links between difference and differential equations?

Does there exist any correspondence between difference equations and differential equations? In particular, can one cast some classes of ODEs into difference equations or vice versa?
40
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2answers
3k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
22
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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 ...
21
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1answer
1k views

When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
10
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1answer
11k views

General Solution of a Differential Equation using Green's Function

My father recently lent me an old textbook of his, called Mathematical Methods of Physics by Mathews and Walker. I am working on the following exercise. Consider the differential equation $$y'...
17
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1answer
462 views

Dual Bases: Finite VS Infinite Dimensional Spaces

Motivation I know that in finite dimensional spaces like a $n$ dimensional Euclidean space $\mathbb{E}^n$, for every basis $G=\{g_1,g_2,...,g_n\}$ we can define a dual basis $G'=\{g^1,g^2,...,g^n\}$ ...
13
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4answers
489 views

Why solutions of $y''+(w^2+b(t))y=0$ behave like solutions of $y''+w^2y=0$ at infinity

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^\infty |b(t)| dt <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{t\to\infty} [(\phi(t)-\sin(wt))...
11
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1answer
542 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 ...
5
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1answer
4k views

Locally or Globally Lipschitz-functions

Determine if the following function satisfies a local or a a uniform Lipschitz condition. The definition of locally Lipschitz and globally lipschitz are as follows: (i) We say that f is (uniformly) ...
39
votes
4answers
2k views

How do we know that we found all solutions of a differential equation?

I hope that's not an extremely stupid question, but it' been in my mind since I was taught how to solve differential equations in secondary school, and I've never been able to find an answer. For ...
14
votes
1answer
599 views

Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
14
votes
3answers
2k 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?
10
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3answers
1k views

Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?

Is Lipschitz's condition necessary condition or sufficient condition for existence of unique solution of an Initial Value Problem ? I saw in a book that it is sufficient condition. But I want an ...
9
votes
3answers
5k views

Differential Equations without Analytical Solutions

In many talks, I have heard people say that the differential equation they are interested in has no analytical solution. Do they really mean that? That is: Can you prove a differential equation ...
3
votes
2answers
116 views

Closed form for $f(z)^2 + f ' (z)^2 + f ' ' (z) ^2 = 1 $?

Can we give a closed form for $f(z)$ in $$ f(z)^2 + f ' (z)^2 + f ' ' (z) ^2 = 1 $$ Apart from $f(x)= 1$ or $f(x)= -1$. Where " closed form " means in terms of standard functions , integrals and ...
13
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4answers
2k views

Examples of nonlinear ordinary differential equations with elementary solutions.

I am looking for nice examples of nonlinear ordinary differential equations that have simple solutions in terms of elementary functions. (But are not trivial to find, like, for example, with ...
11
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1answer
637 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 ...
10
votes
1answer
712 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: $$d_s\...
8
votes
4answers
1k views

Methods to solve differential equations

We are given the equation $$\frac{1}{f(x)} \cdot \frac{d\left(f(x)\right)}{dx} = x^3.$$ To solve it, "multiply by $dx$" and integrate: $\frac{x^4}{4} + C = \ln \left( f(x) \right)$ But $dx$ is not a ...
3
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3answers
1k 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 ...
7
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1answer
195 views

Properties of the solutions to $x'=t-x^2$

Let $f_c$ be the solution to $$ \left\{ \begin{array}{c} x'=t-x^2 \\ x(0) =c \end{array} \right. $$ I'm trying to prove: If $c \geq 0$ then $f_c(t)$ is defined for all $t>0$ There is a ...
7
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1answer
145 views

dropping a particle into a vector field, part 2

Okay, so earlier I posted this question "dropping a particle into a vector field " as sort of a feeler question as i study line integrals in order to go into surface integrals and eventually ...
5
votes
2answers
386 views

The Green’s function of the boundary value problem

What is the Green’s function of the boundary value problem $$ \frac{\mathrm d^2 y}{\mathrm d x^2}-\frac{1}{x}\frac{\mathrm dy}{\mathrm dx}=1,\quad y(0)=y(1)=0, $$ this boundary problem is not self ...
4
votes
3answers
194 views

Solve differential equation $f'(z) = e^{-2} (f(z/e))^2$

I'm curious if there's a simple closed solution to the following DE and, if so, what it is. $$\begin{align} f'(z) &= e^{-2} (f(z/e))^2 \\ f(0) &= 1. \end{align}$$
4
votes
2answers
585 views

Advection Diffusion Equation on Semi-Infinite Domain

Regarding the BVP $$u_t(x,t) - v u_x(x,t)=k u_{xx}(x,t),\qquad x\geq0$$ with BC $u_x(0, t)=0$ for $t\geq 0$, and parameters $v,k>0$, I have some questions. Does an expression for the Green's ...
3
votes
2answers
132 views

Using the Jordan form Complex

Let $C$ be a complex $n \times n$ matrix with $\det C \neq 0$. Prove that there is a complex matrix $B$ such that $C = e^B$ Hint: use the Jordan form matrices for comlexas
2
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1answer
723 views

Cancelling differentials

I'll start with an example. In physics, $x(t)$ represents the $x$-position of a particle, and $v(t)$ its ($x$-)velocity. To determine the total displacement of a particle on the interval $[a, b]$, we ...
14
votes
1answer
592 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2 = \pm \sqrt{-A \cos(x) - B \cos^2(x)+y'(x)-C},$$ where $C, A$ and $B $ are parameters. (The case that either $...
10
votes
6answers
694 views

When to write “$dx$” in differentiation

I'm taking differential equations right now, and the lack of fundamental knowledge in calculus is kicking my butt. In class, my professor has done several implicit differentiations. I realize that ...
8
votes
1answer
944 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 ...
8
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1answer
2k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
7
votes
1answer
199 views

Finding a Lyapunov function for a given system

I need to find a Lyapunov function for $(0,0)$ in the system: \begin{cases} x' = -2x^4 + y \\ y' = -2x - 2y^6 \end{cases} Graph built using this tool showed that there should be stability but not ...
7
votes
2answers
267 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 ...