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
50 views

Hamiltonian differential equation involving complex logarithm

Consider the differential equation $$\begin{pmatrix}\dot p \\ \dot q \end{pmatrix} = \frac{1}{p^2+q^2}\begin{pmatrix} p \\ q \end{pmatrix}$$ where $(p,q)^T\in \mathbb R ^2 - \{0\}$. I want to show ...
2
votes
0answers
16 views

Neglecting the coefficient time dependence in differential equation [on hold]

Suppose equation $$ \frac{d^{2}B(t)}{dt^{2}}+f(t)\frac{dB(t)}{dt} + \omega^{2}B(t) = 0 $$ What is the condition of neglecting the time dependence of $f(t)$, i.e., when the solution of this equation ...
0
votes
0answers
19 views

boundary value problems: eigenvalue and eigenfunction

I'm having trouble in understanding eigenvalues and eigenfunctions in BvP the problem is: $y''$ + $\lambda$$y$ = $0$ $y(0)=0$ $y(2\pi)$ = $0$ make characteristic polynomial $r^2 + \...
-1
votes
2answers
50 views

how to solve $y'(x) +{1\over y(x)}(\sqrt {x^3}+{7\over4}\sqrt {x^5}+{1\over2}\sqrt {x^7})-{1\over2xy(x)}=0$ [on hold]

$$ y'(x) +{1\over y(x)}(\sqrt {x^3}+{7\over4}\sqrt {x^5}+{1\over2}\sqrt {x^7})-{1\over2xy(x)}=0 $$ How to solve this equation?? Does it have simillar form?? What should I do for solving this ...
0
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0answers
14 views

Seeking references on solving a system of differential inequalities

Abstracting from the boundary value conditions for a moment, would somebody please direct me to some references on solving differential inequalities of the following form? Find $u(x)$ that satisfies:...
0
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0answers
15 views

Show that $f$ doesn't has peryodic orbits.

Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ a vector field for which exist a Liapunov function $V:\mathbb{R}^{n}\to\mathbb{R}$ define over all phase space. Show that $f$ doesn't has periodic orbits. I ...
3
votes
2answers
66 views

Temperature/heat equation

I solved this problem $$\left\{\begin{array}{ll} u_{t}=ku_{xx}, & x\in(0,1), t>0 \\ u(0,t)=2, u(1,t)=3, & t>0 \\ u(x,0)=x^{2}+x+2, & x\in(0,1) \end{array}\right.$$ and I got this $$u(...
0
votes
1answer
28 views

Nonhomogeneous heat equation [on hold]

I really don't know how to start to solve it: $$\left\{\begin{array}{ll} u_{t}=ku_{xx}-\lambda^{2}u, & x\in(0,\ell), t>0 \\ u(0,t)=u(\ell,t)=0, & t>0 \\ u(x,0)=h(x), & x\in(0,\ell) \...
0
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0answers
27 views

How to resolve this system of equations with Taylor Method (2nd order)?

I am trying to resolve this system of equations with Taylor Method 2nd order. Formula(with 2 variables): $y_{i+1}=y_i+hf(t_i,y_i)+\frac{h^2}{2}f'(t_i,y_i)$ $\left\{\begin{matrix} \frac{\mathrm{d} y}{...
1
vote
2answers
53 views

Rewriting ODE in terms of a different variable ($z=e^x$)

Given the ODE $$x^2M''+xM'+\lambda M = 0$$ where $1<x<L$, with boundary conditions $M(1) = 0$, $M(L)=0$, we can rewrite it in the Sturm-Liouville form and get $$\left[M'\exp\left(\int\limits_0^L{...
0
votes
1answer
67 views

What is the benefit of representing a complex number as e^i(theta) versus e^(a+bi), what is the process of finding a solution to this example?

What is the benefit of representing a complex number as $ e^{i\theta} $ versus $ e^{a+bi} $? Am I correct in saying that these give the same information but offer convenience in different situations? ...
0
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2answers
34 views

Get the general solution for this differential equation

As I am not very familiarized whit differential equations (I'm more from algebra), I don't know how to solve this problem, but I need to understand how it's done because I have to explain it to a ...
0
votes
4answers
64 views

differential equation particular solution

I need help with this calculus problem: Find the particular solution of the differential equation $e^y\frac{dy}{dx}=e^{−9x}$, such that $y=7$ when $x=0$ I got $-\ln(\frac{e^{-9x}}{9}-\frac{1}{9}-e^...
2
votes
1answer
66 views

Analytical Solution to Coupled Nonlinear ODEs

I am looking to solve several coupled nonlinear ODEs like this one: $\hspace{20mm} \frac{d x(t)}{dt} = C_1 \cdot x(t) + C_2 \cdot y(t) + C_3\cdot (x(t)^2 + y(t)^2) x(t),$ $\hspace{20mm} \frac{d y(t)...
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0answers
57 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$

Is there any analytical solution for the following differential equation? $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$ A,B,C,D are non-zero constants and ...
1
vote
3answers
35 views

finding $k$ and $y(t)$

I am looking for help with this homework problem I am really stuck on. A function $y(t)$ is a solution of $$y′+ky=0.$$ Suppose that $y(0)=100$ and $y(2)=4$. Find $k$ and find $y(t)$. I worked it ...
0
votes
1answer
50 views

particular solution of the given differential equation

I need help with this calculus problem I am very confused about how to go through with this problem! Find the particular solution of the given differential equation $$\frac{\text{d}y}{\text{d}x}=−6xe^...
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0answers
24 views

Stability via Lyapunov Functions

Let $\sigma \gt 0$, $\tau \in \{-\infty\} \cup\mathbb R$ and $I = (\tau,+ \infty)$. Then define : $B_\sigma = \{x : |x|\lt \sigma\}$ $\bar B_\sigma = \{x : |x|\le \sigma\}$ $\color {blue} ...
0
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1answer
20 views

Could in-homogeneous ODE has more than one particular solution $x_p$?

I study ODE course at MIT open course, and the professor said several times "any particular solution would be fine". So, Could in-homogeneous ODE has more than one particular solution $x_p$ ? If so I ...
1
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0answers
56 views

In $x''+3x'+2x=e^{at}$, would $x_p$ at $a=-0.99$ looks like $x_p$ at $a=-1$? [closed]

In linear second order DE with constant coefficients: $$x''+3x'+2x=e^{at}$$ The characteristic equation of the DE is "$m^2+3m+2=0$" which roots are {$-1$,$-2$} The polynomial operator $p(D)=D^2+3D+2$ ...
1
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0answers
49 views

general solution of ODE, not exact

In http://math.jhu.edu/~szrebiec/images/exam1.pdf I found the exercises: 9) Solve the general solution to $(1+ty)e^{ty}+(1+t^2ye^{ty})\dfrac{dy}{dt}=1$ 10) Solve the general solution to $\left(\...
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0answers
26 views

How can isoclines show a solution to a differential equation?

Suppose we have this differential equation: $$y'= 2x + y$$ Now, the isoclines are: $$2x + y = m \implies y = m - 2x$$ How can I deduce the solution to this differential equation based on ...
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0answers
30 views

Complex fixed points on the bifurcation diagrams

I'm working with bifurcation diagrams, an extesion that is being made of them is the determination of complex fixed points in addition to the real fixed points, my question is: what information ...
0
votes
2answers
46 views

find the particular solution for the equation

I need help with this calculus problem. I dont know how to go about starting or finishing this problem Find the particular solution of the given differential equation $$\frac{dy}{dx}=−3\bigg(\frac{x^...
0
votes
4answers
73 views

solve $y$ as a function of $t$ in equation $y′=−8\frac{t}{y}$

I need help with this problem i have for homework i got an answer but it isn't right so i need help getting the right answer with some work Solve for y as a function of $t$ when $y′=−8\frac{t}{y}$ ...
1
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2answers
42 views

half life radioactive substance

I need help solving this calculus problem and i am really confused about how to work it Let $y(t)$ denote the mass of a radioactive substance at time $t$. Suppose this substance obeys the ...
0
votes
2answers
58 views

change in a coyote population

I am having a problem with this calculus problem: The rate of change of the number of coyotes $N(t)$ in a population is directly proportional to $650−N(t)$, where $t$ is the time in years. That ...
0
votes
1answer
23 views

a question about Initial value problem

I am trying to solve this Initial value problem below $$ x'_1(t)= 3x_1(t)+ x_2(t) - 7x_3(t) - 3x_4(t) \\ x'_2(t)= 3x_1(t)+ x_2(t) - 7x_3(t) - 3x_4(t) \\ x'_3(t)= 3x_1(t)+ x_2(t) - 7x_3(t) - 3x_4(t) \\...
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0answers
20 views

uniqueness of weak solutions for parabolic pde Evans

Hi I am trying to understand Evan's proof on uniqueness of weak solution in chapter 7. For the proof of theorem 4 below, I can see (35) and (36) make sense. But I have difficulty to see how Gronwall's ...
3
votes
2answers
64 views

Solution to the differential equation $xy''+y'+xy=0$

Show that the differential equation $$xy''+y'+xy=0$$ admits a solution of the form $$\varphi(x)=\int_0^1f(t)\cos(xt)dt$$ for some function $f(t)$. Since $$\varphi'(x)=\frac{d}{dx}\int_0^1f(...
0
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0answers
33 views

How to solve $\frac{d\lambda}{dt} = \frac{2}{3T_v}(f(t)^2 - \frac{\lambda^3}{f(t)})$ where $f(t) = a+b \sin(wt)$

How can I solve $$\frac{d\lambda}{dt} = \frac{2}{3T_v}(f(t)^2 - \frac{\lambda^3}{f(t)})$$ where $f(t) = a+b \sin(wt)$ and $a, b ,w$ and $T_v$ are constants? I have to find $\lambda$ as a function of ...
0
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0answers
41 views

Incorrect answer - Simultaneous Differential Equations

The questions states solve for y such that $$y' = \begin{bmatrix} -4 & 2 & 1 \\ 1 & -3 & 1 \\ 3 & -3 & -2 \\ \end{bmatrix}y , y(0)= c = \begin{bmatrix} 1\\5\\3 \end{...
0
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0answers
24 views

Cauchy equation number of solutions [closed]

What is the number of solutions ? $$\begin{array}{l} x^6 + y^4 = y'\\ y(2)=-1 \end{array}$$
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0answers
18 views

Design with Matlab the equation of the position of a drone

I want to design with Matlab the equation of the position of a drone which is $u=m(\ddot z_{des}+K_pe+K_v\dot e+g)$ where $e$ and $\dot e$ can be calculated from the current and desired states $(z,...
0
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0answers
22 views

How to find the result of a differential equation in Matlab?

Given a variable $x$ defining the position of a drone, $\dot x$, its velocity, how do we find its acceleration $\ddot x$ in mathlab? I know that we can found the solution of a differential equation ...
-3
votes
1answer
61 views

How to solve $\frac{d^2y}{dx^2}+8\frac{dy}{dx}+16y=0$? [closed]

How to solve $\frac{d^2y}{dx^2}+8\frac{dy}{dx}+16y=0$ ? I found this question in a certain exam paper and the solution goes as follows. Auxillary equation is $m^2+8m+16=0$ $⇒(m+4)2=0⇒m=−4$ ∴ ...
0
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0answers
32 views

How to solve the differential equation $\frac{dy}{dx}-\frac{3x^2}{1+x^3}=\frac{\sin^2(x)}{{1+x}}$ and how to find its integrating factor?

How to solve the differential equation $$\frac{dy}{dx}-\frac{3x^2}{1+x^3}=\frac{\sin^2(x)}{{1+x}}$$ and how to find its integrating factor? The integrating factor is given as $$\frac{1}{1+x^3}$$....
1
vote
1answer
47 views

Stationary solution of a Fokker-Planck equation

I have a question that's driving me crazy. I have a Fokker-Planck equation $$\frac{\partial P}{\partial t}=x\frac{\partial P}{\partial x}+D\frac{\partial^2 P}{\partial x^2}$$ I look for the ...
0
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0answers
24 views

Bifurcations in two dimensional systems

Given the following autonomous differential equations, which illustrates an elliptic limit cycle, $r' = \alpha -\left(\frac{r}{a}\right)^2 \cos^2\theta - \left(\frac{r}{b}\right)^2 \sin^2\theta \\ \...
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0answers
16 views

Implicit Euler using Taylor

I was reading script about differencial equatations. More specific about schemes that help calculate them - implicit Euler. That method was analyzed using something similar to Taylor but i am not sure ...
1
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2answers
39 views

Differential equation with one derivative: $y'=y\cos(x)+x\cos(x)-1$

$y'=y\cos(x)+x\cos(x)-1$, I tried to make it in the form $ay''+by'+c=0$, but I can't find the roots.
1
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1answer
53 views

Differential equation, Solution is a Bessel fucntion

this is my first post here. I knocked my head on a differential equation yesterday, this one: $$ \frac{12 \nu}{x^2} \frac{S(x)''}{S(x)} = -\lambda^2 $$ Where $nu$ is a constant. The book says the ...
4
votes
1answer
41 views

First order differential equation solution [closed]

I couldn't solve the following problem, can you please help? $$y' =\tan (x+y) -1$$ $$(x^4 -2xy^2 +y^4 ) dx - (2x^2 y -4xy^3 )dy = 0$$ $$(y \sec^2 x +\sec x \tan x ) dx +(\tan x +2y )dy =0$$
2
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2answers
59 views

Differential equation questions

I am studying differential equations of order $1$ and $2$ and I had these questions on my mind: 1. Is it true that every differential equation has infinitely many solutions if there are no initial ...
1
vote
1answer
43 views

finding the general solution of linear system

Under what conditions does the solution of the following system exist? How can one find the general solution of the linear system $$\frac{dX(t)}{dt}=A(t)X(t)+B(t)$$ where: - $A(t)$ is an $n \times n$...
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0answers
35 views

Differential Equations confused over question

Interpret the statement as a differential equation. On the graph of y = \Phi(x) , the slope of the tangent line at a point P(x, y) is the square of the distance from P(x, y) to the origin. From my ...
1
vote
2answers
27 views

Trajectories of Differential Equation Systems with Complex Eigenvalues

In an autonomous system of 1st order differential equations in order to find the trajectories one must solve $$\frac{dy}{dx}=\frac{Ax+By}{Cx+Dy}$$ In the case of complex eigenvalues my notes say ...
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0answers
18 views

Need help solving this cauchy problem [closed]

$Y'=(x^3)\sqrt{y}$, $y(2)=1$ this is the problem
0
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0answers
37 views

How to approximate the largest eigenvalue of a monodromy matrix [closed]

Would you happen to know of a method to calculate the largest eigenvalue of a monodromy matrix? For my case the fundamental matrix cannot be calculated explicitly but it exists!