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

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13
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5answers
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Functions that are their Own nth Derivatives for Real n

Consider (non-trivial) functions that are their own nth derivatives. For instance $\frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x$ $\frac{\mathrm{d}^2}{\mathrm{d}x^2} e^{-x} = e^{-x}$ ...
7
votes
2answers
16k views

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order ...
3
votes
1answer
316 views

Nonlinear Differential Equation question

I have a nonlinear Diffeq: $$\frac{d^2x}{dt^2}+\beta \frac{dx}{dt}+\epsilon \times e^{- \lambda x} = f(t) $$ where $f(t)$ is a function that is known, and $\beta$ and $\lambda$ are constants that ...
7
votes
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 ...
5
votes
1answer
2k views

Looking for a logically coherent book for the self-study of differential equations

I'm looking for a logically coherent book for the self-study of differential equations. Let me clarify. By logically coherent, I don't mean proofs of the limit laws, uniqueness theorems etc. By ...
5
votes
5answers
696 views

Does this ODE question have closed form solution?

These days, I am struggling with following ODE problem when I build up my research model: $1/2f''(x)+a(b - x) f'(x) -(c+ e^{A+Bx})f(x)=0$ where f(x) is a smooth function, and $a,b,c, A,B$ ...
3
votes
1answer
6k 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 ...
3
votes
1answer
2k views

system of ode with non-constant coefficient matrix

I am sorry but I haven't learn any method to solve this kind of problem if the given matrix is non-constant. $$\begin{pmatrix}x\\y\end{pmatrix}^\prime=\begin{pmatrix} 1&-\cos t \\ \cos t & ...
14
votes
2answers
1k views

Sum of derivatives of a polynomial

Let $p(x)$ be a polynomial of degree $n$ satisfying $p(x)\geq 0$ for all $x$. That is, for all $x$, $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \geq 0$, $a_n\neq 0$. Show that ...
8
votes
1answer
208 views

Solving the differential equation $\frac{dy}{dx}=\frac{3x+4y+7}{x-2y-11}$

How do we solve the differential equation $$\frac{dy}{dx}=\frac{3x+4y+7}{x-2y-11}$$? I tried substituting $v=yx$ but I do not seem to be getting anywhere.Putting $u=x-2y$ yielded nothing better. ...
6
votes
2answers
4k 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 ) ...
4
votes
1answer
240 views

How to solve this recurrence Relation - Varying Coefficient

Sir,I have two questions related to this recurrence relation. It has been messing with me for long. Because of this I couldn't proceed my work for some time .This contains a polynomial term n+2 in ...
4
votes
3answers
189 views

Solve $x=y\frac{dy}{dx}-\left(\frac{dy}{dx}\right)^{2}$

I've recently been learning about differential equations, and my teacher has been giving some particularly difficult examples out for those of us who finish early. He gave us the following ...
3
votes
2answers
3k views

second derivative of the inverse function

I know that the derivative of the inverse function of $f(x)$ is $g'(y) = \frac{1}{f'(x)}$ But how to derive the formula for the second derivative of g(y) knowing that $\left[\frac{1}{f(x)}\right]' = ...
1
vote
1answer
3k views

The number of solutions to an $n^{th}$ order differential equation.

For an $n$th order differential equation, why are there always $n$ solutions? Why exactly $n$, not $n - 1, n+1$ or infinite many? Addendum by LePressentiment : This is motivated by P176 on ...
12
votes
3answers
338 views

Differential equations that are also functional

I was toying with equations of the type $f(x+\alpha)=f'(x)$ where $f$ is a real function. For example if $\alpha=\frac{\pi}{2}$ then the solutions include the function $f_{\lambda,\mu}(x)=\lambda ...
9
votes
4answers
360 views

Differential Equation Math Puzzle

Dog race: Edit 2: I posted a possible answer below. However, I am unsure how the authors arrived at the solution. Maybe someone can offer an explanation. Four dogs are positioned at the corners of ...
8
votes
3answers
716 views

How can I solve the differential equation $y'+y^{2}=f(x)$?

$$y'+y^{2}=f(x)$$ I know how to find endless series solution via endless integral or endless derivatives and power series solution if we know $f(x)$. I also know how to find general solution if we ...
7
votes
1answer
562 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
votes
2answers
452 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
2answers
331 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 $f(x) = ax^b$ ...
5
votes
1answer
98 views

Ordinary Differential Equations used in Cosmology

I'm just reading over some Cosmology notes and there is a little ODE solve that I am not quite understanding. I have an equation of the form: $$ \ddot{R}=-\frac{GM}{R^{2}} $$ Integrating gives: $$ ...
4
votes
1answer
373 views

Using the Lambert W to express a solution of a differential equation.

I solved a differential equation some time ago and I need to solve for $y$. How can we solve for $y$ using the Lambert W function? $$C_1+x = e^y+Cy$$
3
votes
3answers
728 views

Integrating factor in linear differential equations

I'm watching various videos on differential equations and they all say that linear differential equations are on the form: $y' + P(x)y = Q(x)$ where $P(x)$ is the integrating factor and equals ...
3
votes
4answers
1k views

4 Bugs chasing each other differential equation

This is from a problem seminar and I need help figuring out the solution. Four bugs, $A,B,C,D$ are initially placed at the corners of a unit square. From a given initial moment, all four crawl ...
2
votes
1answer
206 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
2
votes
1answer
1k views

How to reduce higher order linear ODE to a system of first order ODE?

Is there any general and systematic way of reducing the higher order linear ODE to a system of first order ODE? For example, assume we have $a_3x^{(3)}+a_2x^{(2)}+a_1x^{(1)}+a_0x=0$, then how do we ...
2
votes
2answers
10k views

Complementary Solution = Homogenous solution?

I have calculated solutions to homogenous equations but is the complementary solution mentioned here the same as the homogenous solution? Let's take example $y''-3y'+2y=\cos(wx)$ and now ...
1
vote
1answer
225 views

Differential equations - Relation between the number of solutions and the order

The case $\mathbb{C}[z, e^{\lambda z} \mid \lambda \in \mathbb{C}]$: I want to show that in the ring $\mathbb{C}[z, e^{\lambda z} \mid \lambda \in \mathbb{C}]$ each differential equation has a ...
1
vote
2answers
126 views

Easiest way to solve $y''+y=\frac{1}{\cos x}$

I know how to solve it using Lagrange method of variation of constants, but is there easier way?
1
vote
1answer
88 views

About the solution of a difference equation

Let $r>4$ be a positive integer. Let us consider this difference equation: $$u_{n+1}=(1+r²ⁿ⁺¹)u_{n}-r²ⁿ⁻¹u_{n-1}+2$$ I want to find a closed form, bu I am not able to find the good idea.
1
vote
2answers
1k 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
3answers
11k views

Find the differential equation of all circles of radius a [closed]

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.
0
votes
2answers
2k views

`“Variation of Constant”` -method to solve linear DYs?

My school instructs to use some method called "variation of constant" (first page here) to solve linear DY more in my earlier question here. I think I solved the ...
10
votes
3answers
319 views

Nicer expression for the following differential operator

I have the following sequence of differential operators: $$D_n = \underbrace{t \partial_t t \partial_t \dots t \partial_t}_{\text{$n$ times}}.$$ Is there any expression involving a sum of "normal" ...
3
votes
1answer
141 views

Conditions on a $1$-form in $\mathbb{R}^3$ for there to exist a function such that the form is closed.

What are the conditions on a $1$-form in $\mathbb{R}^3$ for there to exist a function such that the form is closed? More precisely, given a point, $p$, what are conditions on the coefficients of a ...
3
votes
3answers
500 views

Differential equation: autonomous system

This isn't homework. I have no idea what theorems I should be looking at to solve this. Guidance, partial and total solutions are all welcomed. Let $f$ be a locally lipschitz function in an open ...
3
votes
2answers
148 views

Find a general control and then show that this could have been achieved at x2

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
3
votes
2answers
475 views

Generating unitary matrices numerically - “close” to the identity element

EDIT: broke this into two parts - for these were two different questions. For numerically obtaining the stabilities of a matricial equation, i need to generate an ensemble of matrices that are ...
2
votes
2answers
2k views

A problem For the boundary value problem, $y''+\lambda y=0$, $y(-π)=y(π)$ , $y’(-π)=y’(π)$

For the boundary value problem, $y''+\lambda y=0$ $y(-π)=y(π)$ , $y’(-π)=y’(π)$ to each eigenvalue $\lambda$, there corresponds Only one eigenfunction Two eigenfunctions Two linearly ...
1
vote
1answer
316 views

exact solution to lotka-volterra equations [closed]

I am looking for exact or perturbative solution realistic lotka-volterra (the one with logistic term in one of the equations) equations in population dynamics. Any reference where they have done it ...
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2answers
1k 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
3answers
115 views

Solving $\frac{dx}{dz}-\frac{2x}{z}=1$

Please can someone solve this? $$\frac{dx}{dz}-\frac{2x}{z}=1$$ Please this is only part of my homework question. I am stuckwith here. Please teach me this solution thank you:)
1
vote
2answers
831 views

Legendre Polynomials: proofs

Does any one know, how to compute any of those two things? The relationship between Legendre polynomials and Shifted Legendre Polynomials. $\displaystyle\int_{-1}^1P_n^2(x)dx=\dfrac{2}{(2n+1)}$ for ...
1
vote
1answer
878 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
152 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
1k views

How to apply reduction of order to find a 2nd linearly independent solution?

I have some questions about writing a general solution, $y$, for $y''-y=0$ when $y_1 = e^x$ is a known solution. I do not understand the logic of the method of reduction of order. How do we apply ...
1
vote
2answers
258 views

Use two solutions to a high order linear homogeneous differential equation with constant coefficients to say something about the order of the DE

OK, this one utterly baffles me. I am given two solutions to an nth-order homogeneous differential equation with constant coefficients. Using the solutions, I am supposed to put a restriction on n ...
1
vote
4answers
332 views

Finding all functions $f$ satisfying $f'(t)=f(t)+\int_a^bf(t)dt$

I am trying to find all functions f satisfying $f'(t)=f(t)+\int_a^bf(t)dt$. This is a problem from Spivak's Calculus and it is the chapter about Logarithms and Exponential functions. I gave up ...
21
votes
3answers
5k 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 ...