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

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2k views

Exponential of the differential operator

I am not sure whether this question is even well-posed. But today I learnt that $e^Df(x) = f(x+1)$ where $D$ is differential operator and $$e^D \triangleq \sum_{i=0}^{\infty} \frac{D^i}{i!}.$$ (ref. ...
7
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2answers
39k views

difference between implicit and explicit solutions?

What is difference between implicit and explicit solution of an initial value problem? Please explain with example both solutions(implicit and explicit)of same initial value problem? Or without ...
7
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1answer
230 views

Energy estimate of the differential equation $\dot{x}=Ax$

Conside the differential equation $$\dot{x}=Ax,\qquad x(t):{\bf R}\to{\mathcal H}$$ where $\mathcal{H}$ is a Hilbert space and $A$ is a bounded linear operator. With the initial condition, one can ...
7
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2answers
94 views

Is there differentiable $f: {\mathbb R}_+ \to {\mathbb R}_+$ such that $f'(x) > f(x)^2$ for all $x$?

Is there positive differentiable $f: {\mathbb R}_+ \to {\mathbb R}_+$ such that $f'(x) > f(x)^2$ for all $x$? It seems like the answer is no because such a function should have a vertical asymptote ...
7
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2answers
432 views

Sturm Liouville with periodic boundary conditions

Background and motivation: I'm given the boundary value problem: $$y''(x)+2y(x)=-f(x)$$ subject $y(0)=y(2\pi)$ and $y \, '(0)=y \, '(2\pi)$. EDIT: These were not given to be zero !! Maybe this ...
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2answers
3k views

Question regarding usage of absolute value within natural log in solution of differential equation

The problem from the book. $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6 -y$ I understand the solution till this part. $\ln \vert 6 - y \vert = x + C$ The solution in the book is $6 - Ce^{-x}$ ...
7
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1answer
458 views

Sloppy notation for differential equations

Why does one often use the following notation for differential equation: $$ y'=f(t)y$$ (this is just a particular example) ? What bothers me with this notation, which I have encountered in ...
7
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3answers
574 views

A simple question about sine and cosine

I have been thinking about all of the different ways that I have encountered sine and cosine in my studies. There are no courses on trigonometry at my school, so perhaps that's why I feel like ...
7
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1answer
192 views

Reduction to standard form.

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
7
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2answers
130 views

Integrating Factor - Not getting the answer given?

$y' - 4y = t$ My integrating factor is $e^{-4t}$ $\int e^{-4t}y'$ - $\int 4e^{-4t}y$ = $\int te^{-4t}$ $\int (e^{-4t}y)'$ = $\int te^{-4t}$ $e^{-4t}y$ = $-4te^{-4t}$ - $e^{-4t}$ I end up with ...
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3answers
733 views

Solving separable differential equation

Seems straight-forward but I've been unable to get it right. Here are my steps: $$y'(x) = \sqrt{-2y(x) + 28},\hspace{20 pt} y(-4)=-4$$ $$\int {1 \over \sqrt{28-2y} }\hspace{2 pt}\text{d}y = \int ...
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1answer
150 views

My first partial differential equation attempt

have I solved this correctly? My textbook is asking for the relation between $ \alpha $ and $ \beta $: $$ \frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{x^2}} $$ Textbook's proposed ...
7
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2answers
186 views

Can't seem to solve this differential equation

Disclaimer: This IS homework. So I will outline the steps I've taken an where I'm stuck. I have the following DE: $$ xy' = y + x\cos^2\left(\frac{y}{x}\right) $$ I then rule out the possible ...
7
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1answer
422 views

Is there a basis-independent proof of Abel's identity?

Abel's identity states that if $X(t)$ and $A(t)$ are $n\times n$ matrix-valued functions such that $X'(t)=A(t)X(t)$, then $\frac{d}{dt}(\det X(t)) = \mathrm{tr}\,A(t) \cdot \det X(t)$. The ...
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1answer
84 views

Using Poincaré-Bendixson to prove that there is a periodic solution

I want to use the Poincaré-Bendixson theorem to show that there exists a nontrivial (and periodic) solution to $$z'' + [\log (z^2 +4(z')^2)]z' + z = 0.$$ Therefore I substituted $u = z'$ to get $$u' ...
7
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1answer
190 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 ...
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1answer
514 views

How do you solve $f'(x) = f(f(x))$?

A friend told me to solve the following differential equation: $$f'(x)=f(f(x))$$ I have no idea how to solve this! This doesn't seem to be an ordinary differential equation and I can't even solve ...
7
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3answers
123 views

Integral formulation for the solution of $xy'' + y' = y$

Let's say that $y$ satisfies the following ODE: $$xy'' + y' = y$$ I want to formulate $y$ as a contour integral. I know that the final result I should get is: $$y(x)=\frac{1}{2i\pi} ...
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1answer
140 views

Reference request: a differential equation arising in geometry

$$ \frac{d\beta}{d\alpha} = \frac {\sin\beta}{\sin\alpha} $$ In what contexts (if any) is this equation known to occur?
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1answer
184 views

What's the name of this chaotic system? (Cool pics included.)

I found this playing with a 2D-ODE-system plotter I'm writing. Surely, since it's so simple, it's been found and extensively studied by someone. What's it called? I'd like to look it up and learn a ...
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2answers
368 views

Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer?

This is a slightly farcical question, for which I apologise. An undergraduate tutee of mine was faced with the following problem: Q. A particle of mass $m$ moving along a line is subject to a force ...
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1answer
288 views

$f(x)=\int_{0}^{+\infty} e^{-(t+\frac{1}{t})x}dt$ how to find $f(x)$?

$$f(x)=\int_0^{+\infty} e^{-(t+\frac{1}{t})x}\;dt$$ if while $ x>0 $ , $ f(x) $ has values I noticed some interesting relations for $f(x)$ as shown below: $$ \begin{align} t & ...
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1answer
252 views

A higher-order differential equation involving absolute values and trigonometry

For a smooth function $f: (-\pi/2,\pi/2) \to \mathbb{R} $, if $\displaystyle\frac{|f''(x)|}{\sqrt{(1+f'(x))^3}} = \cos{x}$, and $f(0) = f'(0) = 0$, $f''(0) = 1$, ...
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2answers
393 views

Prove that a map is continuous

Let $r:[0,1]\to\mathbb R$ be a continuous function and let $u_\lambda$ be the unique solution of the Cauchy Problem: $$\begin{cases}u''(t)+\lambda r(t)u(t)=0,\quad\forall t\in [0,1],\\ ...
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1answer
33 views

Initial value problem

Solve the following initial value problem: $$\frac{d^2y}{dt^2}+2\frac{dy}{dt}+5y=0; y(0)=0 \text{ and } y'(0)=2 $$ I started off with the characteristic equation which is: $$ r^2+2r+5=0 $$ Using ...
7
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3answers
258 views

Distribution theory and differential equations.

How does distribution theory plays role in solving differential equations? This question might seem to be very general. I will try to explain, please bear with me. I understand, distributions make it ...
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2answers
62 views

Show that any solution of second order differential equation has atmost a countable number of zeroes $?$

Question : Considered the second order differential equation $y''(t) + a(t) y'(t) + b(t) y(t) = 0$. then any solution of second order differential equation has atmost a countable number of zeroes ...
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1answer
166 views

$\nabla \cdot f + w \cdot f = 0$

Let $w(x,y,z)$ be a fixed vector field on $\mathbb{R}^3$. What are the solutions of the equation $$ \nabla \cdot f + w \cdot f = 0 \, ? $$ Note that if $w = \nabla \phi $, then the above equation is ...
7
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3answers
244 views

Differential equations solveable independently of coordinate system?

Looking from a physics viewpoint ODEs tend to look very differently when setting up the problem in different coordinate systems. For instance the Laplacian in spherical coordinates involves way more ...
7
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4answers
388 views

A Handwaving Proof of a Specific Existence and Uniqueness Theorem

My problem is as follows: Given the second order homogeneous linear differential equation with constant coefficients $$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+c\,y(x)=0,$$ is there a good heuristic ...
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3answers
538 views

How to find the general solution of $xy''-(2x+1)y'+x^2y=0$ when we know the general solution of $y''+2y'+xy=0$?

Given that the general solution of $y''+2y'+xy=0$ is $y=C_1\int_0^\infty ...
7
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1answer
428 views

ODE Laplace Transforms: what impulse brings an oscillating system to rest?

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
7
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2answers
57 views

Can $\sin(x^2)$ be solution of the diff equation $y''+p(x)y'+q(x)y=0$ in some interval containing $0$

If $p(x)$ and $q(x)$ are continuous functions for any $x$, can $y(x)=\sin(x^2)$ be solution of the diff equation $y''+p(x)y'+q(x)y=0$ in some interval $I=[a,b] $containing $0$? I think it is not as ...
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2answers
197 views

What is the value of $x$ such that $\frac{\text{d}^2y}{\text{d}x^2}=0$ where $\frac{\text{d}y}{\text{d}x}=-ae^{-bx}y-cy+d$?

How can you find the values of $x$ such that $$\frac{\text{d}^2y(x)}{\text{d}x^2}=0$$ where $$\frac{\text{d}y}{\text{d}x}=-ae^{-bx}y-cy+d$$ with $$y(0)=y_0$$ and $$a,b,c,d>0$$ If it helps I can ...
7
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1answer
98 views

Solve $y' + \frac1y + \frac1x =0$ Differential Equation

Do you have any suggestions for how to sole this differential equation? $y'+\frac1x + \frac1y =0$ ? :) I tried solving this by changing variable in the form of $v=x^\alpha*y^\beta$ but it didn't ...
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2answers
94 views

Why is Existence and Uniqueness for Navier-Stokes Easier in 2-D than in 3-D?

I know that existence and uniqueness for incompressible viscous flow in the 2-D case has already been established$^1$, and that doing the same for the 3-D case has yet to be shown. Not only that, but ...
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1answer
266 views

First-term approximation for singular perturbation of ODE (with two turning points)

I'm reading "Introduction to Perturbation Methods" by Mark Holmes, and I came across an exercise that I don't know how to approach. As I decided to independently read this book, I have no ...
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1answer
177 views

how to solve this differential equation $\frac{dy}{dx} = \frac{1+xy}{x(1-xy)}$ by substitution?

I've tried with this differential equation $\displaystyle \frac{dy}{dx} = \frac{1+xy}{x(1-xy)}$ , put $u=xy$ then $\displaystyle\frac{du}{dx}=x\frac{dy}{dx}+y$ So, It will be after editing ...
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1answer
117 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 ...
7
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1answer
370 views

$\frac{dS}{d\rho}$ Factor arising

To get details see: equations 29,30,31,34,44,50,51 We have known some solitary wave solutions, given by(equations 1 to 5) $$ \phi_1=p_1\cos \tau \tag{1}$$ $$\phi_2=\frac16 ...
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1answer
2k views

A Differential Equation with Trigonometric Coefficients

Suppose we have the following second-order differential equation: $\cos^2(x)y'' -\sin(x)y' + y = 0$ How do we determine its general solution? I couldn't even guess a particular solution; all my ...
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2answers
354 views

Deriving the addition formula for the lemniscate functions from a total differential equation

The lemniscate of Bernoulli $C$ is a plane curve defined as follows. Let $a > 0$ be a real number. Let $F_1 = (a, 0)$ and $F_2 = (-a, 0)$ be two points of $\mathbb{R}^2$. Let $C = \{P \in ...
7
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1answer
206 views

$\int_3^5{\frac{x^2}{1+x^2}dx}$ by differentiation under the integral

I'm trying an easy problem to get my bearings using the method here. The integral is $$\int_3^5{\frac{x^2}{1+x^2}dx}$$. I would like to proceed, if possible to solve by defining: $$F(y) = ...
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3answers
311 views

Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem

A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume". At that time for me to write ...
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1answer
61 views

Semantics of Writing Differential Equations

Let $f : \mathbb R \to \mathbb R$, and consider the differential equation $$ f'(t) = f(t) $$ it is easily seen that it has the solutions $f(t) = a\cdot \exp(t)$ for $a \in \mathbb R$. Now another way ...
7
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1answer
232 views

Checking the stability of an equilibrium point

I have the linearization of a non-linear system about an equilibrium point as follows $$ \dot x = (-A+M)x, $$ where $x\in\mathbb{R}^3$, $A$ is a positive definite matrix and $M$ has its eigenvalues ...
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3answers
334 views

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in polar coordinates $$ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial U} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 U} {\partial \theta^2} = 0$$ ...
7
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1answer
88 views

Solve second order differential equations

There are two differential equations that I could not solve. Can someone please help me solve them? $$ (x^2+y^2)y′′-y(y^{′})^3+xy′-y=0 $$ and $$ xy^2y′′+2y^2y′-4xy(y^{′})^2+2x^2(y^{′})^3=0, $$ ...
7
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1answer
73 views

Stochastic Calculus Question

I'm new here and was hoping someone could help me answer this question. I'm reading a paper and I'm a bit confused on how they go from 1 equation to the next. They say: Let \begin{align} x(t) = {} ...
7
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1answer
103 views

$\cos(x+y)\frac{\partial z}{\partial x}+\sin(x+y)\frac{\partial z}{\partial y}=z+\frac{1}{z}$

Having difficulty in solving the following partial differential equation: $$\cos(x+y)\frac{\partial z}{\partial x}+\sin(x+y)\frac{\partial z}{\partial y}=z+\frac{1}{z}.$$ Will it be easier if we ...