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

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6
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1answer
139 views

Confluent Heun equation. 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? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
6
votes
3answers
164 views

What went wrong?

Intrigued by this question, one-dimensional inverse square laws, I started to try to find an answer and came up with what follows. However, I calculated the derivatives to double check myself, and ...
6
votes
1answer
263 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 & ...
6
votes
1answer
1k views

Condition of ordinary differential equation to have a periodic solution

Recently I am reading a book about ODE, and I find a question that asks to prove that ODE has a periodic solution under some conditions. Consider the ODE $x'=f(t,x)$, where $x$ is a scalar and ...
6
votes
2answers
2k views

Finding integrating factor when IF will be a function of x and y

I'm not finding any resource or description or systemic methodology to find integrating factors when the integrating factor will be a function of both x and y. I'm on this problem, $$ ( y - xy^2 ) ...
6
votes
1answer
85 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
6
votes
1answer
97 views

Can we reconstruct a DE from its general solution?

If we think of a differential equation as being a 'problem' and the set of all solutions to the DE as being the 'general solution,' then my question is this: Question: Under what circumstances can ...
6
votes
1answer
181 views

Is there an integral form of Newton's method?

Warning : This seems like a silly sort of question, not the kind I'd ask out loud. The contraction mapping theorem is a basic tool for proving existence of, and finding solutions to, equations. Given ...
6
votes
2answers
82 views

I know this DE is solvable…

I need help with a seemingly simple looking diff equ $$ x\frac {d^{2}y} {dx^{2}}+2y=0 $$ $$ \rightarrow \frac {d^{2}y} {dx^{2}}+2\frac {y} {x}=0 $$ $v= (\frac {y} {x})$ substitution isn't working ...
6
votes
2answers
235 views

Differential Equation $y'=x e^y + \cos x$

I am new to differential equations. I tried to find a series solution for this equation, but I don't know how to solve it. \begin{equation} y'=x e^y + \cos x \\y(0)=1 \end{equation} Actually, the ...
6
votes
2answers
163 views

If $f(x)+f'(x)-\frac{1}{x+1}\int_{0}^{x}f(t)dt=0$ and $f(0)=0$, then what is $f'(x)$?

$f\in C^{1}[0,\infty)$, $f(0)=0$ and $$ f(x)+f'(x)-\frac{1}{x+1}\int_{0}^{x}f(t)dt=0 $$ then $f'(x)=$ ? I'v tried in the following ways. First, let $F(x)=\int_{0}^{x}f(t)dt$, then we are left to ...
6
votes
1answer
170 views

Solve $y''+(1-2x \cos x \cos 2x)y=0$

Solve the differential equation $$y''+(1-2x \cos x \cos 2x)y=0 \space $$
6
votes
2answers
225 views

Does $f'(x) = f(\ln(x))$ have an easily expressed solution?

This is motivated by this question, where we can consider a bound of the form $f((n+1)!) - f(n!) \le 2f(n)$. To find a function with a similar growth rate, I wondered if there's a technique for ...
6
votes
3answers
53 views

First order ODE: $tx'(x'+2)=x$

$$tx'(x'+2)=x$$ First I multiplied it: $$t(x')^2+2tx'=x$$ Then differentiated both sides: $$(x')^2+2tx'x''+2tx''+x'=0$$ substituted $p=x'$ and rewrote it as a multiplication $$(2p't+p)(p+1)=0$$ So ...
6
votes
1answer
67 views

Tricky Integral equation - where to start?

How would you go about solving this? $$p(x,t)=C\exp\left[-x+\int_0^t\int_0^\infty y\,p(y,\tau)\,\mathrm{d}y\,\mathrm{d}\tau\right]$$ Here $p(x,t)$ is the time-dependent probability distribution of a ...
6
votes
1answer
78 views

Iterative method for matrix differential equation

Let $A$ and $X(t)$ be $n\times n$ matrices. I want to solve the matrix differential equation $$\dfrac{dX}{dt}(t)=AX(t)$$ with $X(0)=I$ (the $n\times n$ identity matrix) using the Picard iterative ...
6
votes
1answer
77 views

Solution of the IVP $\dot{y}=x^2+y^2,y(0)=0$

The solution of the IVP $$\dot{y}=x^2+y^2,y(0)=0$$ I know this IVP has a unique solution but I can't find the interval in which it has a unique solution can someone help me in finding the interval.
6
votes
2answers
214 views

How to interpret the meaning of “$y$ solves the DE” to have nice properties.

Assume that $I$ is an open interval $0 \in I$ $x$ varies in $I$ $y$ is a differentiable function of $x$. Now in the context of these assumptions, consider the following problem. ...
6
votes
1answer
99 views

Uniqueness of Ordinary Differential Equations in $D^{'}$, the space of Schwartz distribuitions

Let $m \in \mathbb{N}$. For $k=1,...,m$ let $a_k : \mathbb{R} \rightarrow \mathbb{C}$ be a $C^{\infty}$ function. And suppose that: $a_m(x) \neq 0 \; \forall x \in [x_0, \infty[$ And let P be the ...
6
votes
1answer
232 views

Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
6
votes
2answers
260 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 ...
6
votes
1answer
208 views

Completing the differential equation from exercise 10.23 in Tom Apostol's “Mathematical Analysis”

I found this answer, outlining the exercise, to be interesting. However, I have trouble solving the differential equation. The question starts by attempting to solve the following integral without ...
6
votes
2answers
253 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 ...
6
votes
1answer
591 views

Solving differential equation $y^{(5)} + 2y^{(3)} + y' = 2x + \sin(x) + \cos(x)$

I'm trying to solve the differential equation $$L[y] = y^{(5)} + 2y^{(3)} + y' = 2x + \sin(x) + \cos(x)$$ using the method of undetermined coefficients. I'm having a problem in that my solution ...
6
votes
1answer
169 views

Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
6
votes
2answers
145 views

Function whose inverse is also its derivative?

What are some good examples of a function $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ ...
6
votes
1answer
66 views

Assistance Solving A Second Order Nonlinear ODE (Converted into a First Order)

I am trying to find the solution to $y''=y+y^2$ I noticed that if I multiplied by $y'$ on both sides and integrated, the result would be $\frac{1}{2}(y')^2=\frac{1}{2}y^2+\frac{1}{3}y^3+c$ I have ...
6
votes
1answer
82 views

Differential equation of a path

Differential equation of a path of a particle is $$\frac{d^2 u}{d \theta^2}-\frac{5}{4}u=\frac{9}{4}\frac{\alpha}{\beta^2}$$ where $u=\frac{1}{r}$ and $r$ is distance from origin to a particle, ...
6
votes
1answer
99 views

Showing the 3D Ricci flow ODE preserves the order of the curvature tensor eigenvalues

The following system of ODEs arises when studying Ricci flow on 3-manifolds: $$ \frac{dm_1}{dt} = m_1^2+m_2m_3 \\ \frac{dm_2}{dt} = m_2^2+m_1m_3 \\ \frac{dm_3}{dt} = m_3^2+m_1m_2 \\ $$ Going back ...
6
votes
1answer
297 views

Nonlinear first-order differential equation with a simple parametric solution.

I have to solve the nonlinear first-order differential equation $$\frac{a-y'}{\sqrt{1+y'^2}}e^{-a \arctan y'}=bx+c,$$ where $a,b,c$ are constants, and $y$ is a function of $x$. Obviously, there is ...
6
votes
1answer
139 views

Solving differential equation from Cauchy problem

I am getting acquainted with the Cauchy equations and I am trying to solve an exercise, taking the examples from my class notes. The exercise is: $$\begin{cases} y'=xy+x\\y(1)=2 \end{cases}$$ I have ...
6
votes
3answers
271 views

Existence of an extremum for the solution of the ODE $\ddot{x}+\frac32x^2=0$

Consider the 2nd order ODE $$ \ddot{x}+\frac32x^2=0. $$ Denote by $u$ the maximal solution of the associated Cauchy problem with initial condition $(x(0),\dot{x}(0))=(0,1)$. The problem is to prove ...
6
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1answer
1k 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 ...
6
votes
1answer
270 views

Does the Wronskian have anything to do with the product rule in calculus

Does the Wronskian have anything to do with the product rule in calculus. I ask this because i noticed the form looking similar to the product rule. $$W=g(x)f'(x)-g'(x)f(x)$$ where as the ...
6
votes
1answer
198 views

Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
6
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0answers
65 views

Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
6
votes
1answer
96 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 ...
6
votes
0answers
119 views

ODE system and Lie symmetries

The ODE system (see below), where $F$ is a given function together the algebraic condition (SYM) imply that $$\boxed{y(t)=k-x(t)} \tag{*}$$ for some $k$ constant. The result that $y$ is a translation ...
6
votes
2answers
139 views

What comes after Differential Equations?

First of all, please do excuse the lack of correct terminology, I've haven't learnt Differential Equations at school (yet) so this question comes from just a bit of research I did for my own ...
6
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0answers
177 views

Non-linear second order DE, with no x term in it

Okay, I have a second order non linear de, which has no term containing the variable x. assuming $$ y = f(x) $$ , the equation is $$ y'' - Ay' = \cos{y} - B\sin{y} $$ I tried solving it by ...
6
votes
1answer
92 views

Existence of solution of ODE $y^{\prime}=f(y,t)$ where $f(y,t)$ is not defined in initial value.

Consider a differential separable equation $$y^{\prime}=f(y,t)$$ with initial solution $y(t_0)=y_0$. Suppose that $f(y_0,t_0)$ is not defined. Is there a theorem which can be used to prove the ...
6
votes
1answer
96 views

Non-linear first order differential equation

I've found this particular equation rather tough, can you give me some hints on how to solve $$\dot{y}+t\cos\frac{\pi y}{2}+1-t=y$$ Thanks a lot.
6
votes
0answers
390 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
6
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0answers
184 views

Hints/Help studying an Abel Differential Equation

I want to know more than qualitative information about the Abel differential equation $\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$ Since I don´t know how to solve this and as far as could see, this ...
6
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1answer
108 views

Special functions and diff eq's …

They're are all these methods of dealing with linear second order diff eq's: generating function; recurrence relation; Rodrigues differential form; Schlafi integral form; associated form; second ...
5
votes
2answers
295 views

Solution to the differential equation $\frac{f}{f^\prime}=\frac{f^\prime}{f^{\prime\prime}}$

I am wondering if anyone could please post the solution to the following differential equation for the function $f(x)$: $$\frac{f}{f^\prime}=\frac{f^\prime}{f^{\prime\prime}}$$ Thanks!
5
votes
3answers
2k 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 ...
5
votes
3answers
682 views

Solving $-u''(x) = \delta(x)$

A question asks us to solve the differential equation $-u''(x) = \delta(x)$ with boundary conditions $u(-2) = 0$ and $u(3) = 0$ where $\delta(x)$ is the Dirac delta function. But inside ...
5
votes
4answers
307 views

How do I solve $yy'+x=\sqrt{x^2+y^2}$?

I tried this: $yy'+x=\sqrt{x^2+y^2}$ $y'=-\frac{x}{y}+\frac{1}{y}\sqrt{x^2+y^2}$ $y'=-\frac{x}{y}+\sqrt{(\frac{x}{y})^2+1}$ Substitution: $v=\frac{y}{x}$ ...
5
votes
2answers
482 views

How to solve differential equations of the form $f'(x) = f(x + a)$

What could one do to find solutions for $f'(x) = f(x + a)$ for various values of $a$? I know that $c_1\sin(x + c_2)$ is solution when $a = \frac{1}{2}\pi$, and of course $c_1e^x$ when $a = 0$. ...