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

learn more… | top users | synonyms (1)

2
votes
1answer
38 views

Can I find a function of $y$ that satisfies the relation $\dfrac{df(y)}{dx} = y^2(3y'+1)$

Suppose we have an unknown function $y=y(x)$ , is it possible to find a function $f(y)$ such that: $$\dfrac{df}{dx}= y^2\left(3\dfrac{dy}{dx}+1 \right)$$? EDIT: of course if there is no $1$ in the ...
0
votes
1answer
14 views

Advice about formula for exact differential equation

When I realize that I have exact differential equation I know that is good to use specific formula. But this formula has two forms. Can you tell me when I must use first and when second form? ...
3
votes
0answers
48 views

Solve the non-linear differential equation

I have been trying to solve the following differential equation: $$ \dot{y} = \frac{3x^2}{y-x^2+1}$$ Substituting $u=y-x^2+1$ we get $\dot{u}=\dot{y}-2x$ we get $\dot{u}=\frac{3x^2}{y}-2x$. But I can'...
1
vote
1answer
20 views

Solving Ordinary Differential Equations Using Derivative with Respect to Time

I am trying to solve this ODE. But I only have the following parameter values: ...
2
votes
0answers
13 views

What is the necessary condition for ODE to have unique solution?

For the ODE: \begin{align} \dot{x}(t)&=f(x,t) \\ x(t_{0})&=x_{0} \end{align} If $\;\;f:\mathbb{R}^{n}\rightarrow{}\mathbb{R}^{n}$ is Lipschitz continuous on $\mathbb{R}^{n}$, then there exists ...
0
votes
4answers
41 views

$x^2\frac{d^2y}{dx^2}-4x\frac{dy}{dx}+6y=3+20\sin ({\ln x})$

How to show that the substitution $x=e^t$ transforms the differential equation $$x^2\frac{d^2y}{dx^2}-4x\frac{dy}{dx}+6y=3+20\sin ({\ln x})$$ into $$\frac{d^2y}{dt^2}-5\frac{dy}{dt}+6y=3+20\sin t$$
0
votes
1answer
28 views

2nd order differential equation with limits

Solve the differential equation $\frac{d^2y}{dx^2}-2\frac{dy}{dx}-3y=2e^{-x}$ given that $y\rightarrow0$ as $x\rightarrow \infty$ and that $\frac{dy}{dx}=-3$ when $x=0$ My attempt, I've already ...
0
votes
0answers
11 views

dominated theorem

If $\phi(t)$ and $\psi(t)$ are fundamental matrices of differential equations $ dX(t)=A(t)X(t)dt$ and $ dX(t)=B(t)X(t)dt $ If $ g(t)$ is a bounded and measurable function then is it correct to ...
0
votes
1answer
69 views

Why can this differential equation be written in $3$ different ways?

Suppose we have the following differential equation using operator notation: $$(D-x)(D+x)y=0\tag{1}$$ where $$D=\frac{d}{dx}$$ Now I could rewrite $(1)$ as $$\begin{align}\require{enclose}(D-x)(D+x)y&...
0
votes
1answer
26 views

solving a differential equation using substitution

ok so I have to use the substitution x=e^t to change the following DE $$x^2y''-3xy'+y = 1+ x^8 \ln^3 x +xe^{5x}$$ into a linear DE, I also have prove the needed chain rule, any help?? all I know in ...
0
votes
1answer
14 views

Lyapunov function guarantees local exponential stability

can someone give me a proof of http://nptel.ac.in/courses/101108047/module13/Lecture%2031.pdf page 15? Suppose all conditions for asymptotic stability are satisfied. In addition to it, suppose $\...
0
votes
1answer
38 views

Exponential matrix definition

The exponential matrix is $e^{tA} : = X(t)$ where $X$ is the unique global solution of the differential equation $x'=Ax$ which satisfies $X(0)=I$. I want to prove, using this definition, that $$ e^{(...
0
votes
1answer
45 views

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

Analytical solution for a non-linear differential equation: $\frac{d^2y}{dt^2} = A \left(\frac{dy}{dt}\right)+ B \sin(2Cy)$ A,B are non-zero constants and y (position) is a scalar-value parameter ...
0
votes
1answer
43 views

General solution of linear ODE

What is the most general solution of $\frac{\mathrm{d}y}{\mathrm{d}x}+Py=Q$ where $P,Q$ are constants and $u,v$ are two particular solutions? How do I proceed from here? $ye^{Px}=Q\int (e^{Px})\,dx$...
1
vote
1answer
73 views

Prove that $\exp(A+B)=\exp(A)\exp(B)$ iff $[A,B] = 0$

I have searched throughout the forum and online as well, and I got that with condition of $[A,B]=0$, $e^{(A+B)t}=e^{At}e^{Bt}$. Now the question is, to show for any matrices $A$ and $B$, it is true ...
1
vote
2answers
32 views

Inital value problem, differential equations

I'm asked to solve the problem: $7\frac{dy}{dt}+y= 28t$ with $y(0)=2$ When I integrated I ended up getting $y= (14t^2+C)/(7+t)$, then I plugged in 0 for $t$ and 2 for $y$ to find that $C$ is 14. ...
1
vote
0answers
30 views

A vector field in a star shaped set

I'm having problems trying to proof Poincaré's lemma for Star-Shaped sets Let $F:U\to \mathbb R^{2}$ be $C^{1}$ functions,where $U\subset \mathbb R^{2}$ is a Star-shaped set. If $F_x:U\to \mathbb ...
2
votes
0answers
32 views

Does any Riccati equation admit local solutions?

Is it true that any differential equation of the form: $$w'(t)=a(t)w^2(t)+b(t)w(t)+c(t),$$ where $a,b,c:[\alpha,\beta]\to\mathbb{R}$ are continuous functions, admit a solution in the neighbourhood ...
0
votes
3answers
54 views

economics using differential equations

I need help with this calculus problem The producer of a certain commodity determines that to protect profits, the price p should decrease at a rate equal to half the inventory surplus $S−D$, where $...
2
votes
2answers
49 views

Method of Annihilators Tedium…

One of the exam preparation questions for MIT's online Honors Differential Equations course asks for a general solution of \begin{align} (D^2 - 1)^4(D^3 + 1)^5y = 3e^t \end{align} The fact that the ...
0
votes
0answers
21 views

Functional differential equation separatrix

I've been spinning my wheels with the following differential equation, and would greatly appreciate any guidance on ways to attack it. I have $u(x) \geq 0$ for all $x$. Further, $x \geq 0$. The ...
0
votes
0answers
42 views

Calculus: Derivative of a summation and dot product

I'm trying to implement a speed boost to an eye-tracking algorithm (found here: http://www.inb.uni-luebeck.de/publikationen/pdfs/TiBa11b.pdf). I need to take the derivative of the eye-tracking ...
2
votes
1answer
40 views

Stability of a line of equilibria

I'm working with a nonlinear autonomous system $x'=f(x)$. This system stays in $\mathbb{R}^n_+$ whenever it begins there, and it has a ray of equilibria, i.e. there is a positive vector $x_0$ so that ...
0
votes
0answers
22 views

Obtaining error between exact and finite element solution of a PDE when exact solution is not available

How does one obtain the error between the finite-element (FE) solution and the exact/analytical solution when the latter in not available? After all, isn't the purpose of a numerical method to find ...
0
votes
1answer
43 views

Differential equation without analytic solution - comparative statics

I am facing a differential equation - with boundary condition $v(T)$ given - without an analytic solution but still need to understand how the solution is affected by a change of the function's value. ...
1
vote
2answers
48 views

A simple second order ODE

This might be a very naive question, but is there a solution for a second order ODE of the form $$y''(x) = f(x)y(x)$$ where $f(x)$ is a general function? Any information is appreciated. Thanks.
1
vote
1answer
25 views

Dampened mathematical pendulum

I have the system of ODEs $$\begin{align*}\dot y &= v \\ \dot v &= -\lambda v - q(y) \end{align*}$$ for an increasing function $q$ such that $q(0)=0$ and the energy function $E= \frac 1 2 v^2 +...
0
votes
1answer
49 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
49 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
votes
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
votes
0answers
14 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
65 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
votes
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
52 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
64 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
votes
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 ...
1
vote
4answers
60 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
65 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)...
-1
votes
0answers
56 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
48 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^...
0
votes
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
votes
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
vote
0answers
56 views

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

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
vote
0answers
45 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(\...
0
votes
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 ...