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

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2
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1answer
28 views

If $l_i$ is the first weak eigenvalue of $-\Delta$ in a domain $G_i$ and $G_1\subseteq G_2$, then $l_1\ge l_2$ and equality is possible

Let $\Omega_i\subseteq\mathbb{R}^n$ be a domain $\lambda_i$ be the first weak eigenvalue of $-\Delta$ in $\Omega_i$ It's easy to verify that $\Omega_1\subseteq\Omega_2$ implies $\lambda_1\ge ...
1
vote
1answer
31 views

First ODE problem solution different than WolframAlpha solution

$-y'' +2y' - y = x$ , with conditions $y(0) = y(1) = 0$ I am supposed to find a solution for this problem, so I started with finding the result for the homogeneous equation, and i got $y = c_{1}e^x + ...
0
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1answer
35 views

Inverting the differential operator $D^2-3D+2$ [closed]

I am trying to calculate $$(D^2-3D+2)^{-1}(xe^{3x})$$ that is, find a function $f$ such that $(D^2-3D+2)(f)=xe^{3x}$ where $D=\frac{d}{dx}$. Using inverse operator, I am getting an incorrect answer. ...
0
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1answer
27 views

How do I find the Laplace Transform of $ \delta(t-2\pi)\cos(t) $?

How do I find the Laplace Transform of $$ \delta(t-2\pi)\cos(t) $$ where $\delta(t) $ is the Dirac Delta Function. I know that it boils down to the following integral $$ \int_{0}^\infty ...
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1answer
25 views

Conditions of a differential equation

Consider the differential equation \begin{align} 2 x^2 y'' + x(x^2 - 1) y' + (2 x^2 - x +1)y = 0 \hspace{5mm} y(0) = 0, y'(0)=1. \end{align} A solution readily found is \begin{align} y(x) &= B_{0} ...
0
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0answers
21 views

clarity in the solution of the following problem

$$(D^2+D)y=x^2+2x+4$$ I found the solution as $$CF=C_{1}+e^{-x}C_{2}$$ and PI=$$\left(\frac{x^3}{3}\right)+4x$$ but the solution from my teacher is PI = $$\left(\frac{x^3}{3}\right)+4x+C3$$ Where ...
1
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5answers
223 views

I need help with a Finite Series

Problem: Find the sum to $n$ terms of \begin{eqnarray*} \frac{1}{1\cdot 2\cdot 3} + \frac{3}{2\cdot 3\cdot 4} + \frac{5}{3\cdot 4\cdot 5} + \frac{7}{4\cdot 5\cdot 6}+\cdots \\ \end{eqnarray*} ...
2
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0answers
22 views

Phase line and Equilibrium Points

Consider the differential equation $dy/dt=y^8+3y^6-y^2-1$. Sketch the phase line and classify the equilibrium points. Since when $y=0$, the derivative is negative and when $y>1$ the derivative ...
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0answers
8 views

Representing solutions of a second order linear differential equation as the solutions of 2 first order linear differential equations.

Consider $xy''+2y'+xy=0$. Its solutions are $\frac{\cos x}{x},\,\frac{\sin x}{x}$ . Neither of those solutions (as far as I could find) can be the solutions of a first order linear homogeneous ...
3
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1answer
50 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
23 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
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0answers
32 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 ...
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0answers
33 views

Using Multipule 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
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1answer
52 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
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1answer
67 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?
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0answers
36 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
22 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
61 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
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0answers
25 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
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1answer
18 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}) ...
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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
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2answers
50 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 ...
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2answers
18 views

Finding the equation of parabolas with axis parallel to the x-axis

I've seen a post like this but it's on hold and doesn't really help, can someone give me a hint or a step by step solution on how to solve this? I think the other guy is from another section of my ...
0
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0answers
20 views

Develop a concept of weak solvability for $-\langle\nabla,A\nabla u\rangle=f$

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable with $A(x)$ is symmetric, for all $x\in\Omega$ $\exists c_1,c_2>0$ with ...
1
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2answers
31 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
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1answer
26 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
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0answers
35 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
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2answers
56 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
72 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
19 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
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1answer
24 views

Problem in Identifying Homogeneous Differential equation

The following equation is Homogeneous (source:wolfram alpha), and has the answer $(x/y)+e^(x^3)=c$ as solved by putting $y=vx$. $$y dx - x dy + 3*x^2*y^2*e^(x^3) dx = 0$$ or $$(dy/dx) = (y + ...
2
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0answers
15 views

Finding intersections of tori/toruses

I am looking for intersections of three tori. Is this possible? If so, how? To put things in perspective: I am looking for the coordinates of point P in space, and I have a triangle on the 'ground'. ...
3
votes
1answer
44 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\\ ...
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
16 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
57 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
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2answers
42 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
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1answer
58 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 ...
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0answers
21 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
65 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 $T$ ...
1
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1answer
42 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 ...
1
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1answer
56 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 ...
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votes
1answer
43 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 ...
1
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1answer
29 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 ...
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0answers
24 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
57 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 ...
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2answers
69 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
34 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. $$ y''-4y'+4y = 2e^{2x}+4x-12 $$ $$ ...
2
votes
1answer
43 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
16 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 ...