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

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4
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3answers
184 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 ...
4
votes
5answers
667 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 $a,b,c, A,B$ ...
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 ...
8
votes
1answer
201 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
6answers
249 views

Are the any **non-trivial** functions where $f(x)=f'(x)$ not of the form $Ae^x$

This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is ...
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 ) ...
5
votes
1answer
2k views

Finding Weak Solutions to ODEs

I'm wondering if anyone has a reference to a good set of notes on finding weak (distributional) solutions to ODEs, or has any tips or tricks. For example, $$ xy^\prime=0 $$ has a classical solution ...
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
1answer
357 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
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]' = ...
2
votes
2answers
9k 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
218 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
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.
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 ...
13
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 ...
12
votes
3answers
333 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 ...
10
votes
3answers
317 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" ...
7
votes
1answer
542 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
408 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 ...
5
votes
1answer
89 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: $$ ...
3
votes
3answers
680 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
1answer
136 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
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 & ...
2
votes
1answer
201 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 ...
1
vote
3answers
10k 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.
3
votes
3answers
483 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
456 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
171 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 ...
1
vote
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
743 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
846 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
150 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
236 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
331 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 ...
19
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 ...
11
votes
2answers
13k views

Explanation and Proof of the fourth order Runge-Kutta method

Runge-Kutte 4th order method is a numerical technique used to solve ordinary differential equation of the form $dy/dx=f(x,y), y(0)=y_0$ It gives $y_{i+1}$ in the form $y_{i+1} = ...
21
votes
2answers
355 views

Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$

Fix some even functions $f$ and $g$, differentiable, such that $f(0)=g(0)=0$ and $f'(0)=g'(0)=0$, and consider the autonomous differential system $$\left\{\ \begin{array}{lcr}x'&=&-y+f(x)\\ ...
13
votes
3answers
514 views

Fourth Order Nonlinear ODE

I was looking at an ode $w^{(4)} + w^3 = 0$ with initial conditions $[w'''(0),w''(0),w'(0),w(0)]=[1,0,0,0]$. I can see via maple that there is a blowup around 3.7. I was wondering if there was a way ...
13
votes
4answers
4k views

Can this gravitational field differential equation be solved, or does it not show what I intended?

This is the equation I'm having trouble with: $$G \frac{M m}{r^2} = m \frac{d^2 r}{dt^2}$$ That's the non-vector form of the universal law of gravitation on the left and Newton's second law of ...
10
votes
2answers
2k views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ...
10
votes
4answers
9k views

Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
9
votes
2answers
1k 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 ...
7
votes
3answers
300 views

Why it is absolutely mistaken to cancel out differentials?

In many of my physics courses, (don't worry, this is a mathematics question!) My teachers cancel out differentials, and every time, they say: "If a mathematician saw me canceling out this ...
8
votes
1answer
9k 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 ...
12
votes
1answer
817 views

Recursive solutions to linear ODE.

When finding the solutions to the simple ODE $$ y'- mxy= x^n \text{ ; } y(0) = 0$$ I found the following: Let $P_n$ be the particular solution for each integer exponent $n$. Then if we define ...
11
votes
3answers
17k views

Definition of a Differential Equation?

Here is one definition of a differential equation: "An equation containing the derivatives of one or more dependent variables, with respect to one of more independent variables, is said to be a ...