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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1answer
30 views

Can I apply Hartman-Grobman when only one eigenvalue is zero?

Consider $\begin{bmatrix} \dot x\\ \dot y \end{bmatrix} =\begin{bmatrix} xy^2 -xy\\ x-y \end{bmatrix}$. If we take the Jacobian and evaluate it at $(0,0)$, one of the eigenvalues is $-1$ and the other ...
-1
votes
1answer
25 views

Which of the following equations must be true?

$$C =\frac{5}{9} (F - 32)$$ The equations above shows how temprature $F$, measured in degrees fahrenheit, relates to a temprature $C$, measured in degrees Celcius. Based on the equation, which of the ...
0
votes
2answers
36 views

Solving the 1-D diffusion equation

For the equation $$u_t = Du_x$$ where $D$ is a diffusion constant, we can define the system $$u_x=v$$$$u_t=Dv_x$$ However, how does one solve for $v$? $\frac{\partial u}{\partial x}=v \iff {\partial ...
1
vote
0answers
36 views

EDO - Power Series Solution

Well, here I have my equation: $$ y''-x^2y = 0 $$ Assuming $y=\sum_{n=0}^\infty C_n x^n$ is a solution, then we could write (jumping some few steps) $$2C_2+6C_3 x+\sum_{k=2}^\infty[(k+2)(k+1)C_{k+2}x^...
0
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1answer
33 views

Sturm–Liouville equation and the Eigenvalue general Problem (PDE)

As I am studying for my Partial Differential Equations exam, I came across Sturm–Liouville equation where it says that it's solutions $y(x)$ are the eigenfunctions of the general problem $Ly=λy$. I do ...
3
votes
3answers
95 views

Derivation of $1 = x^2+y^2$ with respect to time [duplicate]

I am studying differential algebraic equations. Given the following equation: $1 = x^2+y^2$ Differentiate this equation with respect to time. The correct solution is: $0=2x \dot x + 2y \dot y $ ...
1
vote
3answers
58 views

Linear 1st order differential equation

I tried to solve this equation but need help from seniors what to do next... $$\frac{dq}{dt} + q = 4\cos2t ; q(0) =1 $$ Multiplying both sides by I(t) i.e. $$I(t)= e^t$$ $$e^t\frac{dq}{dt} + e^t q =...
0
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0answers
42 views

Solving differential equation with numerical method

How do I solve this differential equation numerically for $1<\beta<\inf$ knowing $\sigma$ ? $\frac{\gamma(u)}{du}=\beta\cdot\gamma(u)\cdot\gamma(\beta\cdot u)$ $\gamma(0)=\sigma$ Thank you ...
0
votes
2answers
37 views

a general solution of an ODE system of equations

What is the general solution of the following ODE system of equations? $$ \vec{x}'(t)= \left(\begin{array}{ccc} 1 & 0 & 0\\ 2 & 0 & 1\\ 0 & -1 & 0\\ \end{array}\right)\vec{x}(t)...
1
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0answers
29 views

variational method for Laplacian

For $f \in L^2(U)$ (U open and limited) consider the problem $-\Delta u = f$ em $U$ and $u=0 $ em $\partial U$. Let $J:H^1_0(U) \longrightarrow \mathbb{R}$ given by $$ J(u) = \frac{1}{2} \int_U |Du|^...
2
votes
3answers
98 views

Finding entire functions such that $g'(z)-g(z)=2\,z-z^2$

I am stuck on the following problem: If $g(0)=-1$ and $g(z)\neq z^2, \forall{z}\in \mathbb{C}$, find all entire functions $g$ such that $$g'(z)-g(z)=2\,z-z^2$$ I can see that, since $g(z)\...
0
votes
3answers
94 views

Solution to $\frac{d^2f}{dr^2}+\frac{1}{r}\frac{df}{d r}=0$

I know that $f(r)=aln(r)+b$ where $a$ and $b$ are constants is a solution of $$\frac{d^2f}{dr^2}+\frac{1}{r}\frac{d f}{dr}=0$$ are there any other solutions to this, would appreciate it if someone ...
2
votes
1answer
58 views

How to solve this system of PDEs?

The 1-D Euler’s equation for constant pressure can be written in terms of the two equations $$u_t + uu_x = 0, x\in\mathbb{R}, t> 0, u(x,0)=f(x)$$ $$\rho_t+\rho u_x+u\rho_x=0, x\in\mathbb{R}, t>0,...
0
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0answers
28 views

Second-order differential equations methods

I'm looking for a method called 'Inexact Method' Idk if it goes by another name, here's what I do know: It's one of the two 2nd order differential equations methods. The other method is called '...
0
votes
1answer
65 views

How does one get following solution for $ay''+(x^3y)'=0$?

I'm little confused how one can conclude the following: if $$ ay''+(x^3y)'=0 $$ Then $$ y=C\exp \left(-\frac{x^4}{4a}\right) $$ where $C$ is a constant. The way I'd go about it, it's clear ...
0
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1answer
27 views

Finite difference formula to approximate second derivative

I have one question which asks to derive a finite difference formula to approximate $f''(x)$ in the form of $$f''(x)\approx Af(x+2h)+Bf(x+h)+Cf(x)$$ with the method of undetermined coefficients. ...
2
votes
1answer
62 views

Inconsistency between analytical solution and numerical solution

I have a differential equation as below: $$u^{'}=y$$ $$y^{'}=-k_1u-k_3u^3$$ Analytical solution of this equation would be: $$(u_0,y_0)=(\pm\chi_esech\sqrt{\kappa}\tau,\mp\chi_e\sqrt{\kappa}sech\...
1
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0answers
27 views

Initial Value Problem using Laplace transformation: What is the ${\cal L}$ transform of $u(t-5)$?

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses a single difficulty. If it's any indication of difficulty, the exercise ...
0
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0answers
40 views

Substitution in differential equation

I have a differential equation in the following form, where a, b and R are constants and $\delta (x,t)$ Dirac function. I know that substitution holds $z=x-c y$, where $c$ is a constant also $$a \...
2
votes
0answers
47 views

Perturbation ODE with boundary layer problem

I encountered the following ODE and tried to solve using perturbation theory: $$y'=(1+\frac{1}{100x^2})y^2-2y+1$$ $$y(1)=1,\ x\in[0,1]$$ I am asked to find an approximation correct to $O(\epsilon)$. ...
0
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0answers
21 views

How to programatically solve the optimal control problem?

I have to programatically (write a program) find a control function $u(\cdot)$ to minimize the following functional: $$ J(u,x) = \int_0^T { f_0(x(t), u(t), t)}dt + \Phi(x(0)) \rightarrow \min$$ ...
0
votes
1answer
35 views

Fourier Series of cos(ax)

I would like to ask some help on this problem.. 01) Expand the following function in fourier series. $f(x)=cosax,−π<x<π$ where 'a' is not an integer. Hence, Show that $\frac{1}{sint} = \frac{...
0
votes
1answer
39 views

Green Function problem PDE

Someone can give me tips for solve this PDE, please! $\frac{d^4G}{dx^4}=\delta(x-x_0)$ with $G(0;x_0)=G(L;x_0)=\frac{dG}{dx}(0;x_0)=\frac{d^2G}{dx^2}(0;x_0)=0$ I do not know how to start this, ...
0
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0answers
16 views

Is $w(x,y)=\frac{x^2+3y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}} \ dx - \frac{3x^2+y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}} \ dy$ an exact differential form?

$$a_1=\frac{x^2+3y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}}$$ $$a_2=- \frac{3x^2+y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}}$$ I verify if $w$ is a Closed differential form: $$\frac{d \ a_1}{d \ y}=\frac{d \ a_2}{d \ ...
1
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0answers
30 views

Help with solving ODE

This is actually a part of an SDE problem, which I am having problem to solve. However i think it requires only ODE knowledge to solve. I have $$\mu(u) = x^3 + 3 \int_t^u a \mu(s) ds + 3 \sigma^2 \...
2
votes
2answers
28 views

Difference between 'stable node' and 'stable spiral'

Suppose I have a pair of 2 non-linear differential equations of the form: $$\begin{matrix} \frac{dy}{dt}=f(x,y)\\ \frac{dx}{dt}=g(x,y) \end{matrix}$$ Equilibrium points are where the trajectory ends ...
1
vote
2answers
23 views

2nd Order Runge-Kutta Method

Could someone please help me with the next step of this 2nd order Runge-Kutta method. I am solving the ODE \begin{align*} x'=-\frac{x(t)}{2}, \ \ x(0)=2. \end{align*} I wish to use the second order ...
1
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0answers
27 views

solution for the Cauchy problem of an elastic bar forced vibrations

I am looking for the solution of the following problem: $\frac{\partial^2 w}{\partial t^2}+a^2\frac{\partial^4 w}{\partial x^4}=\Phi(x,t),\quad -\infty<x<\infty, t>0\\ w(t=0)=0,\quad \...
1
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0answers
41 views

Numerical Solution of Parametric Differential Equations

I have to numerically solve a differential equation $(dx/dt)^2+(dy/dt)^2=dx/dt$. I can't figure out the way to use standard numerical methods (Euler, RK, etc). Any suggestion will be greatly ...
0
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0answers
30 views

Is this 1st order homogeneous ordinary differential equation analytically solvable?

It is an equation about the quantities $\{ \psi_n | -\infty < n < \infty \}$. $$ i \frac{ \partial \psi_n}{\partial t} = - (\psi_{n-1} + \psi_{n+1} ) + V\sin (\omega t) \delta_{n,0} \psi_0 ....
1
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0answers
37 views

Picard's theorem proof

I have a very simple question. On the proof of Picard's theorem about the existence and uniqueness of a solution to a differential equation, we have to prove that the map T defined by $$(T\varphi)(t)=...
2
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1answer
24 views

Solving a nonhomogeneous linear system using variation of parameters

I have to apply the method of variation of parameters to find a particular solution of the following system: $x' = A\vec x + \vec f(t)$ for $$ A = \begin{bmatrix} 2 & -4 \\ 1 &...
1
vote
2answers
43 views

Equilibrium in a system of nonlinear differential equations

I have two questions about a specific system of differential equations. First, if a complex number can be an equilibrium point. Second, and related with the first question, how can I verify that $(0,0)...
1
vote
1answer
66 views

First integrals for solving system of ODEs

Assume a problem $$ \begin{cases} \frac{\mathrm{dx}}{\mathrm{dt}} = \frac{y}{x-y}, \\[2ex] \frac{\mathrm{dy}}{\mathrm{dt}} = \frac{x}{x-y}. \end{cases}$$ Additionally, $x = x(t)$ and $y=y(t)$. ...
0
votes
2answers
31 views

Differential equation system with both derivatives

How can I rewrite those two differential equations into a matrix and solve them? $$c_1y_1'+r_1y_1=cos(vt)-my_2'$$ $$c_2y_2'+r_2y_2=-my_1'$$ $c=(c_1,c_2),r=(r_1,r_2),m\quad$ are constants. What ...
0
votes
0answers
39 views

Any hope of a closed form solution of this differential equation:

I have a differential equation of the form $$\left(\frac{\partial \log (\alpha (x))}{\partial x}\right)^2+2 f(x) \left(f(x)+\frac{\partial \log (\alpha (x))}{\partial x}\right)+2 \frac{\partial }{\...
1
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0answers
20 views

derivative of convolution integral

I'm confused for a derivation related to the derivative of convolution. Given that $$ C_{im}(x,t)=\omega e^{-\omega t}*C_m(x,t)+C_{im}(x,0)e^{-\omega t} $$ By taking derivative of the above equation ...
1
vote
1answer
71 views

Equation of the form $\mathbf{\Phi}'(t)=\mathbf A(t)\mathbf{\Phi}(t)$.

Let $\mathbf{\Phi}(t)$ and $\mathbf A(t)$ be matrices satisfying the differential equation $$ \mathbf{\Phi}'(t)=\mathbf A(t)\mathbf{\Phi}(t)\ . $$ If I am not mistaken, if $\mathbf A$ and its integral ...
0
votes
0answers
29 views

Solving a system of differential equations with a repeated eigenvalue

$\vec{x}' = \begin{bmatrix}4&-2\\8&-4\end{bmatrix}\vec{x},$ I'm getting $\lambda = 0$ as an eigenvalue And the resulting eigenvector $\vec{v} = \begin{bmatrix}1\\2\end{bmatrix}$ I ...
0
votes
1answer
37 views

Solve biharmonic equation with boundary conditions

I was wondering a biharmonic equation $u\left(x\right)$ satisfy $\nabla^{4}u\left(x\right)=0$ $\in V$ subject to the boundary conditions $u=0$, $\frac{\partial}{\partial n}\left(\nabla^{2}u\right)=0$, ...
1
vote
2answers
43 views

how to solve this non linear ode

$$ y y'(x) +y(x)^2(\sqrt {x^3}+{7\over4}\sqrt {x^5}+{1\over2}\sqrt {x^7})-{1\over2x}=0 $$ How to solve this equation?? I searched text book , and I only found bessel, legandre. But they are not same ...
0
votes
1answer
63 views

Prove that orbits of one system are orbits of another

Let $$\left\{\begin{align} \dot x &= x- \frac{xy}{1+\alpha x}\\ \dot y &= -y + \frac{xy}{1+\alpha x}+\delta y^2 \end{align} \right.$$ be a predator-prey model. Prove that the following ...
0
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0answers
18 views

Linear system of advection diffusion equations

I am trying to find the eigenvalues and eigenfunctions of the coupled PDE system $$ \partial_t \vec{u} = - \stackrel{\leftrightarrow}{A} \partial_x \vec{u} + \stackrel{\leftrightarrow}{D} \partial_x^2 ...
0
votes
1answer
33 views

Solve $\left(1+{y'}^2 \right) y''' - 3{y''}^2 y' = 0$ with Legendre Transformation

I have got a question concerning the use of Legendre Transformation for solving ordinary differential equations. We did this stuff in an ordinary differential equations class. The professor stated ...
0
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0answers
44 views

ODE inside ODE question

Given the equations of motion: $$ x'' = \frac{F - .375(\theta'' \cos\theta - (\theta')^2 \sin\theta)}2$$ and $$\theta'' = \frac{2g\sin\theta - \cos\theta (F+.375(\theta')^2 \sin\theta)}{1.5 - .375\...
1
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0answers
16 views

Peano's theorem , initial value problem, Banach Spaces

I'm Taking a Course in Differential Equations and this is one of the exercises those I have to do at home, I can't come up with these short questions: Let X be an infinite-dimensional Banach space. ...
0
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0answers
29 views

A differential equation and a Jacobi field

I have a question about the following answer here: Zero Sectional Curvature implies exp is a local isometry We have $w\in T_v(T_pM)\cong T_pM$, the geodesics $\gamma_s(t)=\exp_p(t(v+sw))$ and the ...
0
votes
2answers
34 views

Limit in simple differential equation

Given the differential equation: $$\frac{dv}{dt}=-g-kv$$ How can one deduce directly from this equation (without solving the differential equation first) that: $$\lim_{t\to \infty}v(t)=-\frac{g}{k}$$
0
votes
2answers
36 views

If $y_1$, $y_2$ are linearly independant solutions of a second order homogeneous ODE then $y_1^2(x) + y_2^2(x) \neq 0 \ \forall x$

Let be an homogeneous second order linear ODE in an open interval $I$ : $$y''(x) + p(x)y'(x) + q(x)y(x) = 0$$ $p, q : I \rightarrow \Bbb R$ continuous and let be $y_1, y_2 : I \rightarrow \Bbb R$ two ...
1
vote
1answer
43 views

Variable change to make differential equation separable

I would like to understand why the following statement is true: Let there be a homogeneous differential equation whose unknown function is $y(x)$, with $x\in]0,\infty[$. If we rewrite the equation ...