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

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0
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3answers
66 views

Homogenous ordinary equation - Homogeneous

The question is: $(x-y)dx + xdy = 0$ Trying to solve: $ \\M(x,y) = (x-y) \\N(x,y) = x $ $ \\Kx - Ky = K(x-y) \Rightarrow \text{ Homogeneous} \\Kx = K(x) \Rightarrow \text{Homogeneous}$ $ \\y = vx ...
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 ...
0
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2answers
22 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 $z=y^{\frac{1}{1-a}}$. ...
2
votes
1answer
56 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 ...
1
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1answer
51 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 ...
4
votes
1answer
27 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 ...
1
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0answers
37 views

Solving higher order pdes

So here's my problem, while you solve the Euler Bernoulli beam Equation by separation of variables, how do I have to prove the separated function of space are orthogonal? If so, are hyperbolic sines ...
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0answers
43 views

How to motivate those expansions?

I've been reading a paper where the author needs to solve the biharmonic equation on the plane. In truth, the function being saught is a function $v$ such that $v = \nabla \times U$ and $\nabla^4 U = ...
4
votes
1answer
605 views

How to use the Fredholm alternative in an ODE

I have the following ordinary differential equation $$ \frac{d^2u}{dx^2} + u = \cos x$$ A particular solution to this problem is $x\sin x$, so we can say that $$ u(x) = c_1 \cos x + c_2 \sin x + ...
5
votes
1answer
113 views

Galerkin methods for odes

Could you give me some information about the multi-adaptive Galerkin methods for odes?? What does the term "multi-adaptive" mean?? Are there real-world problems at which we could apply these ...
0
votes
1answer
46 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 ...
1
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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 ...
1
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1answer
41 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
42 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
68 views

Help me understand this solution

$$2x^4yy'+y^4=4x^6$$ The way my teacher did it is: First, he made a substitution: $y=z^m$ $y'=mz^{m-1}z'$ $$2x^4 z^m mz^{m-1} z'+z^{4m}=4x^6$$ ...
1
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1answer
28 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 ...
1
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0answers
23 views

Solve one dimensional wave equation using fourier transform

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

How do I solve first order non-linear system of PDE: $\partial f^i(x,y)/\partial z = F(f^1,f^2,…,f^n)$?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} \frac{\partial f^i(x,y)}{\partial z} = F(f^1,f^2,...,f^n), \qquad i = 1,..,n \end{eqnarray} Where $\partial/\partial z = ...
3
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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 ...
7
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1answer
115 views

Solution to differential equation $f^{(n)}-(n+1)f^{(n-1)}-(n+1)nf^{(n-2)}-\dotsc-(n+1)!f=g$

Let $n$ be a given positive integer and $g$ be a continuous function. We are looking for a function $f \in C^n(\mathbb{R})$ such that $$f^{(n)}-(n+1)f^{(n-1)}-(n+1)nf^{(n-2)}-\dotsc-(n+1)!f=g.$$ It ...
1
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2answers
52 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
2
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1answer
43 views

Solving wave equation by fourier method

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
1
vote
2answers
37 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 ...
0
votes
2answers
33 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. $$ y''-4y'+4y = 2e^{2x}+4x-12 $$ $$ ...
0
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1answer
15 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
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
25 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: ...
0
votes
1answer
40 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?
0
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0answers
36 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
17 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) + ...
4
votes
1answer
856 views

Solution verification: Find the orthogonal trajectories of the family of curves for $x^2 + 2y^2 = k^2$

I need help with the following question: Find the orthogonal trajectories of the family of curves for $x^2 + 2y^2 = k^2$ I have taken the following steps, are they correct? From what I ...
1
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1answer
23 views

differential inclusions vs differential equations

Can someone please clarify what the difference (no pun intended) between the two is? I am reading this tutorial and at the very start they state that a differential inclusion is a solution to $ ...
7
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0answers
96 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 associated autonomous differential system $$x'=-y+f(x)\qquad y'=x+g(y)$$ Is every ...
6
votes
1answer
227 views

Non-linear second order DE, with no x term in it

Okay, I have a second order non linear de, which has no term containing the variable x. assuming $$ y = f(x) $$ , the equation is $$ y'' - Ay' = \cos{y} - B\sin{y} $$ I tried solving it by ...
4
votes
1answer
481 views

Software for numerical solution of a non-linear ODE system?

I have been given a nonlinear system of ODEs which has arisen out of a colleague's engineering research: $$\begin{array}{rcl} \dot{x}_0&=&x_1\\ ...
0
votes
1answer
24 views

Steady states of a system

How can I find the steady states? I am aware that the condition is to equal 0 but I am not able to say how many steady states there are... $$\begin{cases} \dot x=x-y^2 \\ \dot y= -x+2y-z^2 \\ \dot z= ...
0
votes
2answers
39 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
vote
1answer
53 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: ...
0
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1answer
31 views

Help with a Differential Equation - I am getting the wrong Answer

Please consider the following problem. I feel I am going about it the right way but the answer I am getting is not right. Problem: Given that $y = x$ is a solution of \begin{equation*} (x^2 - 2x + ...
0
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1answer
30 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 ...
0
votes
2answers
17 views

Solving ODE numerically, using derivative steps

Assume I have the ODE $\dot{p}(t) = f(t,p)$, with $p(0) = p_0$ and assume $f(t,p)$ (for simplicity) is only a function of $p$. I want to solve the ODE numerically, using derivative steps, kind of like ...
1
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2answers
49 views

Differential Equations and Eigenvectors/Eigenvalues

I am trying to solve $\dfrac{d\mathbf{x}}{dt} = \left[\begin{array}{cc} -4 &1\cr -6 &1 \end{array}\right] \mathbf{x}$ and I need to find the general solution of the system in the form ...
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votes
0answers
61 views

What are the solutions for $xy^2 (dy/dx)=y^3-x^3$? [closed]

Would someone please help me with solving the following differential question: $xy^2 (dy/dx)=y^3-x^3$
0
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1answer
103 views

What is the difference between an integral curve and the solution of a differential equation?

Can you please explain what the difference between an integral curve and the solution of a differential equation is? My book gives an example that $$\frac {dy}{dx}=\frac {y}{x}$$ defines a direction ...
0
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0answers
40 views

Solve x' = -2x + 2y, y' = 2x - 5y

Solve the equations $$x' = -2x + 2y$$ $$y' = 2x - 5y$$ I got $$x=2(c_1) e^{-t}+(c_2) e^{-6t}$$ $$y=(c_1) e^{-t}-2(c_2) e^{-6t}$$
2
votes
1answer
552 views

Comparison of Adams-Bashforth and Runge-Kutta methods of order 4

I have a system of ODE, that must to solve with numerical methods. I solve it by Adams_Bashforth with order4 and Runge-Kutta with order4 methods. Do you know with same length step which methods ...
0
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0answers
20 views

how to get this specific matrix differentiation?

Currently, I am reading this article with title "neighborhood components analysis", http://papers.nips.cc/paper/2566-neighbourhood-components-analysis.pdf. Everything went well until I encounter the ...
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0answers
27 views

Differential equation (species competing for their food supply) [closed]

i'm a financial engineering 1st year student.this subject is new to me so i couldn't understand how to attempt these questions properly can anyone help me out please? A population model of species ...
2
votes
1answer
31 views

Not contradiction of Picard-Lindelof theorem

I have this homework problem: Consider the following initial value problem: $$\frac{dy}{dt}=6t\sqrt[3]{y^2}$$ $$y(0)=0$$ Demonstrate that this I.V.P has a different solution of $y(t)=0$, ...
1
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0answers
18 views

Differential Equations: Confocal Ellipse and Hyperbola

I am currently brushing up on Conic Sections, and I am having some problems on solving a first order quadratic differential equation. I would appreciate any help on the topic! I know that confocal ...