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

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7
votes
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
votes
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 ...
5
votes
1answer
1k 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
332 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
585 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 ...
2
votes
1answer
159 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
1answer
8k 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 ...
2
votes
2answers
8k views

Find the differential equation of all circles of radius a

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.
1
vote
1answer
79 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 ...
10
votes
3answers
308 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
127 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
421 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
147 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
404 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
1k 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
2answers
689 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
1answer
776 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
147 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
204 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
1answer
208 views

About the Legendre differential equation

Consider the Legendre differential equation $$ (1-x^2) y'' - 2xy' + n(n+1)y = 0 $$ Then its solution is given by $$ y = c_1 P_n (x) + \text{an infinite series} $$ In fact $y = c_1 P_n (x) + c_2 Q_n ...
1
vote
4answers
321 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 ...
18
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 ...
35
votes
2answers
2k views

Is it mathematically valid to separate variables in a differential equation?

I read the following statement in a book on Calculus, as part of my mathematics course: Technically this separation of $\frac{dy}{dx}$ is not mathematically valid. However, the resulting ...
8
votes
2answers
10k 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} = ...
13
votes
3answers
495 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
3k 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 motion ...
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 + ...
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 ...
11
votes
3answers
15k 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 ...
7
votes
4answers
6k 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 ...
8
votes
1answer
216 views

When do Harmonic polynomials constitute the kernel of a differential operator?

Let $f$ be a real polynomial of two variables. Let $\partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$. Let $H$ denote the space of harmonic polynomials, i.e., ...
5
votes
2answers
366 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 ...
7
votes
1answer
167 views

solution of $y' = \exp \left(-\frac yx\right) + \frac yx$

Could you help me to solve equation $$y' = \exp \left(-\frac yx\right) + \frac yx;\quad y(e) = 0$$ I know how to solve 1st order linear de like $y' = \exp \bigl(-\frac 1x\bigr) + \frac yx$ but here ...
5
votes
2answers
304 views

Proving Nonhomogeneous ODE is Bounded

I am trying to prove the following: Let $x(t)$ be a solution of the IVP $$ \dot x=A(t)x+h(t), $$ where $A(t), h(t)$ continuous on $1\le t<\infty$. Further assume that $$ \int_1^\infty \| ...
3
votes
1answer
119 views

Lipschitz continuity and differential equations

Does anyone have any ideas for this one? could use some help.
2
votes
3answers
567 views

How does an integrating factor geometrically “uncurl” a vector field?

We know that certain 1-D forms $m(x,y,z)\,dx + n(x,y,z)\,dy + p(x,y,z)\,dz$ admit integrating factors as we teach in basic differential equations. How does the integrating factor geometrically turn ...
11
votes
8answers
5k views

how do you solve $y''+2y'-3y=0$?

I want to solve this equation: $y''+2y'-3y=0$ I did this: $y' = z$ $y'' = z\dfrac{dz}{dy}$ $z\dfrac{dz}{dy}+2z-3y=0$ $zdz+2zdy-3ydy=0$ $zdz=(3y-2z)dy$ $z=3y-2z$ ...
9
votes
1answer
288 views

Solution of differential equation related to Normal density

Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ Given $0<\sigma\le 1$. I wish to know whether there ...
8
votes
2answers
385 views

Deriving the addition formula of $\sin u$ from a total differential equation

How do we derive the addition formula of $\sin u$ from the following equation? $$\frac{dx}{\sqrt{1 - x^2}} + \frac{dy}{\sqrt{1 - y^2}} = 0$$ Motivation Let $u = \int_{0}^{x}\frac{dt}{\sqrt{1 - ...
8
votes
3answers
1k views

To what extent can you manipulate differentials like dy and dt like actual values?

I have been thinking about the differentials that we use in derivatives and integrals. For example, I have an equation: $${\int{w}{dr}} = \text{other stuff}$$ The context for this strange equation ...
7
votes
1answer
192 views

prove that the following function is: $f(x) = 0$

let $f: [0,1] \to \mathbb R$ , $f$ is differentiable $f(0) = 0$ $|f'(x)|\le|f(x)|$ for $x\in [0,1]$ prove that $f(x) =0$ for $x\in [0,1]$ i believe that i need to somehow use the ...
6
votes
2answers
3k 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
3answers
137 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
4
votes
1answer
114 views

Under which conditions a solution of an ODE is analytic function?

If I'm not wrong there is a theorem that says that if the conditions for Picard's theorem are satisfied, for an ode $\dot x=f(x,t)$, then the solution of the ode is as smooth as $f$. I think I'm not ...
4
votes
3answers
3k views

Fourier Series for $|\cos(x)|$

I'm having trouble figuring out the Fourier series of $|\cos(x)|$ from $-\pi$ to $\pi$. I understand its an even function, so all the $b_n$s are $0$ $$a_0 = \frac 2 \pi \int_0^\pi |\cos(x)|\,dx = ...
4
votes
2answers
663 views

Applied ODEs in trajectory problem

I'm having a hard time solving this problem: Let there be a town $A$ in a shore of a river. Let $x=0$ be the shore. Let $(0,0)$ be the location of the town. Let $B$ be another town, in the ...
4
votes
4answers
407 views

Differential equation with a constant in it

Solve $$y'' + s^2y = b \cos sx$$ where $s$ and $b$ are constants. I have tried undetermined coefficients, but it makes such a mess that I keep getting lost, I also tried variation of ...