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

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3
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1answer
127 views

Inverse Laplace transform of $\operatorname{arccot}(s)$, $\arctan(s)$

How would one find inverse Laplace transforms of $\operatorname{arccot}(s)$ or of $\arctan(s)$ without knowing in advance that this is related to $\dfrac{\sin x}{x}$?
2
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0answers
47 views

Non-Conservative System

I'm having a bit of trouble understanding the concept of a conservative system mathematically. A problem in a textbook (Arnold's Mathematical Methods for Classical Mechanics) is asking me to give an ...
1
vote
0answers
43 views

Trajectories of predator prey equation

I am studying the predator prey equation recently, and here is an example: Let $x'=x(1-0.5y)$ and $y'=y(-0.75+0.25x)$. This is a predator prey equations. Then $$\frac{dy}{dx}=\frac{y'}{x'}=\frac{y(-0....
1
vote
1answer
71 views

Using Multiple Scale Analysis to solve a non-linear differential equation

I would like to know if there are other methods to solve equations such as this one below. I don't really understand the theory behind the multiple scale analysis and why it works I understand some of ...
1
vote
1answer
68 views

2 to 1 dimension in linear PDE with non-constant coefficients

I have a question that can majorly help in my physics. Problem Say, we have a linear PDE \begin{equation} \hat{D}~F(x,y)=0, \end{equation} with $\hat{D}$ being a (second order) differential ...
1
vote
1answer
75 views

What is $\int\sinh(x)^pdx$?

What is $$\int\sinh(x)^pdx$$, where $0<p<1$?. I tried using Mathematica, but it came up with some Hypergeometric2F1 function. Is there a simpler answer in this integral?
0
votes
0answers
45 views

Can I have variables extreme of integration?

Suppose you have a function $v(t)$ that you want to find. The condition is that it's integral is some fixed quantity. The integral is done between $0$ and $u(t)$, where $u(t)$ is an increasing ...
1
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0answers
34 views

Stieltjes differential equation

So I have the following differential equation that I want to solve: $$ dy(t) = -d[\alpha(t)\cdot t]\,\,\,\,,y(0) = 50$$ where $[\cdot]$ is the greatest integer function. My guess is that $$y(t) = ...
0
votes
2answers
75 views

Finding second derivative for $x=\sin t$ and $y= \sin 2t$.

If $x=\sin t$ and $y= \sin 2t$, how to find second derivative of $y$ w.r.t $x$ ? Or rather how to prove $(1-x^{2})\frac{d^{2}y}{dx^{2}}-x\frac {dy}{dx}+4y=0$? Is there any shortcuts to find these ...
2
votes
1answer
66 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the ...
0
votes
1answer
30 views

Inverse Laplace Transform and the Unit Step Function

I need to find the inverse Laplace transform of the following function: $$ F(s) = \frac{(s-2)e^{-s}}{s^2-4s+3} $$ I completed the square on the bottom and got the following: $$ F(s) = (e^{-s}) \...
0
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0answers
30 views

Simple differential equation question [duplicate]

I think this is pretty easy but it's been forever since I've done this. Can someone help me start or give me some tips?
2
votes
2answers
65 views

Prove that a classical solution of $-\langle\nabla,A\nabla u\rangle=f$ is also a weak one

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable and $A(x)$ be symmetric, for all $x\in\Omega$ $u\in C^2(\Omega)$ with $A\nabla u\...
2
votes
2answers
50 views

Differential equation Physical Example.

I am Learning Differential equation with ordinary differential equation. How to tell students the actual geometric meaning of differential equation? What is first order differential equation actually ...
2
votes
1answer
42 views

Laplace transform and IVP at $\infty$

Solving the following differential equation $$ty^{''}\left ( t \right )+\left ( t-1 \right )y^{'}\left ( t \right )-y\left ( t \right )=0$$ with initial values $$y\left ( 0 \right )=5, y\left ( \...
1
vote
0answers
39 views

Differential equation with $\sqrt{1-\cos(f)}$

I'm currently trying to solve the differential equation $$ \sqrt{2} a \cdot \sqrt{1 - \cos(f)} = f' $$ where $a$ is a constant and I can freely choose $f(0)$ to simplify the solution and calculation. ...
1
vote
2answers
65 views

Solving recurrence using analogy with continuous $x_{n+1} = \frac{r^2}{2d - x_n}$

What's up lovely friends, I'm facing a physics problem and felt on a recurrence that one does not see everyday. This one: $x_{n+1} = \frac{r^2}{2d - x_n}$ or $f(n+1) = \frac{a}{b-f(n)}$ if you will ...
2
votes
3answers
126 views

Find a second solution of the given differential equation.

$$ xy''+y'=0; y_1=ln(x) $$ I solved this all the way to the end and found my second solution to be $y_2=-1$, but the book says it is $y_2=1$. I am checking my algebra and the method I used was to get ...
2
votes
0answers
24 views

Regarding continuity and the value of the function at the point of discontinuity.

Suppose while solving a boundary value problem, we have a two piece solution $f_1(x)$ and $f_2(x)$ where $f_1(x)=f(x)$ for $x < x_0$ and $f_2(x) = f(x)$ for $x>x_0$. If there is a matching ...
0
votes
1answer
34 views

Problem in Identifying Homogeneous Differential equation

The following equation is Homogeneous (source:wolfram alpha), and has the answer $\frac{x}{y}+\mathrm{e}^{x^3}=c$ as solved by putting $y=vx$. $$y\mathrm{d}x-x\mathrm{d}y+3x^2y^2\mathrm{e}^{x^3}\...
3
votes
1answer
55 views

Verify my solution of $y'=(1-y)\sqrt{y}$

I have to solve $$y'=(1-y)\sqrt{y},\ \ \ y(0)=y_0$$ My approach: $$\begin{align} \int{\frac{1}{(1-y)\sqrt{y}}dy}=x+c\\ 2\int{\frac{1}{1-w^2}dw}=x+c\\ 2\text{arctanh}\sqrt{y}=x+c\\ y=\tanh^2{\frac{x+...
0
votes
2answers
25 views

Help me with this differential equation

$$xy'-y=x(1+e^{\frac{y}{x}})$$ Please give me a hint on how to solve this. If I'm not mistaken, this is a Bernoulli equation, but I can't seem to solve it using the substitution $z=y^{\frac{1}{1-a}}$. ...
0
votes
1answer
118 views

How to Identify a homogeneous first order first degree ODE

The following equation is homogeneous edit: y dx - x dy + 3x^2y^2e^(x^3) dx = 0 (source: Wolfram alpha) but it is not of the form of $f(zx,zy)= z(f(x,y))$. How do I identify such type of special ...
2
votes
1answer
100 views

How do you read a partial differential equation?

In calculus we can read the "normal derivative", $\frac {df}{dx}$, as the rate of change of our function $f$ with respect to $x$. With partial derivatives of multivariate functions it is very much the ...
0
votes
2answers
48 views

Homogeneous differential equation $F(x, y, y',y'')$

I'm studying an example of a different equation's solution in my maths textbook. The equation is: $$ xy'(yy'' - (y')^2) - y(y')^2 - x^4y^3 = 0$$ The author concludes that it is a homogeneous ...
0
votes
1answer
409 views

Find the differential equation of all circles of radius 1 and centers on $y=x$

Find the differential equation of all circles of radius 1 and centers on $y=x$, I've answered several problems with circles finding its equation but not like $y=x$ can someone please explain this to ...
0
votes
0answers
42 views

If $\partial\Omega\in C^{2+\alpha}$ and $-\Delta\Theta=f\text{ in }\Omega$ with $f\in C_0^\infty(\Omega)$, then $\Theta\in C^{2+\alpha}$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with $\partial\Omega\in C^{2+\alpha}$ for some $\alpha>0$ $f\in C_0^\infty(\Omega)$ $\Theta\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be the ...
2
votes
5answers
138 views

$T: V \to \Bbb{R^2}$ by $T(f)=(f'(0),f(0))$.

Let V be the space of twice differentiable function on $\Bbb{R}$ such that $$f''-2f'+f=0.$$ Define $T: V \to \Bbb{R^2}$ by $$T(f)=(f'(0),f(0)).$$ The I could see that $T$ is one-one, but is $T$ onto?...
1
vote
1answer
49 views

Differential of a tricky function

I have a function that I'm strugling to take the differential of. $$F(t) = F(t-a)G(t).$$ My attempt is the following: $$ dF(t) = F(t-a)dG(t) + G(t) dF(t-a)) $$ but I have a feeling something is not ...
2
votes
1answer
121 views

Finite Difference for Hamilton-Jacobi-Bellman without boundary conditions

Let $t\in\mathbb{R}_+$ denote time, $x \in X$ is the state and $u \in U$ the control. The objective function is $F:X \times U \to\mathbb{R}$ and $f:X \times U \to\mathbb{R}$ is the law of motion for ...
-1
votes
1answer
45 views

ode and area of triangle

Question: find a curve $x$ so that the area bounded between it's tangent at some point $t$ and the time axis on the interval between the point of contact of $x$ and it's tangent ( $t$ ), and the ...
2
votes
1answer
384 views

How can i find the basis solutions of homogeneous linear ODE?

Second order linear differential equation is given below. $y''+\frac{2}{x}y'+k^2y=0,$ where $k$ is constant and $x\neq 0$ I already know that the basis are $y_1=\frac{e^{-ikx}}{x}$ and $y_2=\frac{e^...
2
votes
1answer
242 views

Solve one dimensional wave equation using fourier transform

I'm trying solve this wave equation using Fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
3
votes
2answers
63 views

What is the type of differential equation?

Given the differential equation: $$\left( \frac 1x - \frac{y^2}{(x-y)^2} \right)\, dx = \left( \frac 1y - \frac{x^2}{(x-y)^2} \right)\, dy$$ I can't determine a type of this equation. Perhaps, this ...
1
vote
2answers
294 views

Find the differential equation of all tangent lines of parabola $y^2=4x$

My professor said that it's $x(y')^2-yy'+1=0$ but how? I drew it and I think it open to the right $90^\circ$ but I can find the solution to differentiate
0
votes
2answers
60 views

When do I have to respect the $C$ constant and when can I combine?

Question Verify that the given two-parameter family of functions is the general solution of the non-homogeneous differential equation on the indicated interval. $$ y''-4y'+4y = 2e^{2x}+4x-12 $$ $$ y=...
2
votes
1answer
70 views

Solving wave equation by fourier method

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
0
votes
1answer
61 views

Sketch and find the differential equation of all lines through the origin

We've just finished discussing about elimination of arbitrary constants, so I'm confused on how to solve and sketch this type of problem, I was told that the answer in this problem is $xy'-y=0$ but ...
1
vote
2answers
51 views

two-point concentrated load

I am trying to solve the following problem with two point load: $$ \frac{d^2u}{dx^2} = \delta(x-1/4) - \delta(x-3/4) $$ With boundary conditions $u'(0) = 0$ and where $u'(1) = 0$ From the definition,...
-1
votes
1answer
42 views

ajuda com a solução desta EDO

Alguém poderia me ajudar no desenvolvimento da Questão: $y''-a(x^n)y=0$ ? Ps:. A solução desta questão eu conheço, mas o desenvolvimento eu não consigo manipular de modo a chegar na solução. English ...
3
votes
2answers
41 views

Laplace Transforms of Step Functions

The problem asks to find the Laplace transform of the given function: $$ f(t) = \begin{cases} 0, & t<2 \\ (t-2)^2, & t \ge 2 \end{cases} $$ Here's how I worked out the solution: $$\...
1
vote
0answers
40 views

What is the differential equation, given a certain solution?

I am a little stuck on this problem. The question asks, Write a first order autonomous differential equation such that $y(t)=\cos(t)$ is a solution. I understand that first order means that it ...
2
votes
0answers
20 views

Closed representation of Ladder operators in One Dimensional Second Order Homogeneous Differential Equations

(1) Has anyone published the closed representation of ladder operators for second order differential equations? More specifically the second order differential equation $$ -\partial_x^2\Psi_m(x) + V(...
0
votes
1answer
61 views

Solving differential equation (Numerical & Analytical)? [closed]

I want to solve the following differential equation $y''(x)\ /\ y(x)= \frac{\lambda\ x^{\frac{3}{4}}}{\sqrt{1 - x}}\ ,\ 0\lt x\lt 1$ But do not know how to actually solve it. Any suggestion?
4
votes
1answer
31 views

Differential Equation with Cross Products [without separating into system of equations]

I need to solve the following equation: $$ \frac{d m}{d t}=-m\wedge b-\alpha m\wedge (m\wedge b), $$ where $b$ is constant However, I was instructed specifically not to separate the calculation into ...
0
votes
2answers
54 views

Solve the differential equation. $\frac{dy}{dx} + 2y = f(x),$ where $f(x) = 1,$ if $ 0 \leq x \leq 1;$ $ f(x) = 0, x > 1, y(0) = 0.$

Solve the differential equation. $\frac{dy}{dx} + 2y = f(x),$ where $f(x) = 1,$ if $ 0 \leq x \leq 1;$ $ f(x) = 0, x > 1, y(0) = 0.$ Find $f(\frac{3}{2}).$ I am confused whether to use the ...
-1
votes
2answers
128 views

Fundamental Matrix (Floquet theory)

Let $\begin{pmatrix} \dot{x}_1 \\\dot{x}_2\end{pmatrix}=A(t)\begin{pmatrix}x_1\\x_2 \end{pmatrix}$ where $$A(t)=\begin{pmatrix}\alpha(t)+\cos(t)&\sin(t)\\ -\sin(t)& \alpha(t)+\cos(t)\end{...
1
vote
1answer
55 views

Differential ordinary equation. Can it be solved?

Is the following ODE solvable? $C'(t)=\lambda+\dfrac{1}{C(t)},\ \forall t\in I$-interval. This one arises from a model of the blood alcohol concentration. See here: http://s3.amazonaws.com/...
0
votes
1answer
35 views

Is $ y = \int_0^x \frac{du}{\sqrt{x^4 - u^4}} $ an increasing or decreasing function of $x$? Find an integral expression of $y'$.

Is $$ y = \int_0^x \frac{du}{\sqrt{x^4 - u^4}} $$ an increasing or decreasing function of $x$? Find an integral expression of $y'$. I don't understand what I need to know to solve this question. It ...
2
votes
1answer
105 views

An applied problem in ODE leading to Interval of Existence issue

ODE books have a section on interval of existence of a solution with a standard set of problems, such as what is the interval of existence for $y'+(\tan t) y = t/(t-2), y(3)=5$, or $y'=y^3, y(0)=2$. I ...