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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3answers
50 views

Solving differential equation with $y^{'}y^{''}$

How can this differential equation be solved? $$y'y'' - t = 0$$ I can't figure this out and the $y'y''$ is giving me trouble since they are attached together.
1
vote
0answers
9 views

Matrix linearization of the Lagrangian points.

I have to solve a long problem, and I´m in trouble in a step. The step is to linearize the next differential equation, by writtin its correspondient Jacobian, and then, calculate the eigenvalues of ...
1
vote
0answers
28 views

Ordinary Differential Equation with a trigonometric function: radius of convergence?

For the equation $$x^2y'' + y' + \tan(x)\,y = 0$$ establish lower bounds for the radius of convergence about the point $$x_0 = 1.$$
4
votes
3answers
57 views

Show that the solution of an initial value problem is always less than a given constant

My try is that $$\frac{dy}{dt} =(y-3)e^{\cos ty}$$ $$\frac{dy}{y-3}= e^{\cos ty}dt$$ $$\ln (y-3)=-\frac{e^{\cos ty}}{\sin ty} +c$$ my steps is correct or I made mistakes ? please help to solve ...
0
votes
2answers
32 views

Differential Equations, first order

I am taking an Engineering Math class. When it comes of differential equations with $y^2$ included in the question I am lost on the procedure used to solve it. Bernoulli method does work because its ...
1
vote
0answers
14 views

Example of cyclic vectors in linear differential equations

Suppose $f(x)$ obeys the first-order differential equation $f'(x) = P(x) f(x)$, and $g(x)$ obeys the first order differential equation $g'(x) = Q(x)g(x)$. Is there a second-order differential ...
2
votes
0answers
21 views

Find a series solution to $(x^2-2)y''+6xy'+4y=0$.

Find a series solution to $(x^2-2)y''+6xy'+4y=0$. A. Find the recurrence relation to $a_n$: My answer is $a_{n+2}=a_n\cdot \frac{n+4}{2(n+2)}$ which is correct. B. Using A, write two independent ...
1
vote
0answers
21 views

Second order equation.

(i)Show that the ODE $$y''+[b'(x)/b(x)]y'-[a^2/b^2(x)] y=0$$ has a pair of linearly independant solutions that are reciprocals, where $a$ is a constant and $b(x)$ is a function of x. Find them in ...
0
votes
0answers
24 views

Linearly independant functions.

Show that any two linearly independent functions ${y_1(x),y_2(x)}$ with sufficient differentiability satisfy a unique second order homogeneous linear DE of the form $$Ly=y''+p_1(x)y'+p_2(x)y=0$$ ...
2
votes
1answer
252 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
2answers
26 views

Find solutions for an differential equation system

I have a differential equation system $x_1'(t) = -x_2(t)$ $x_2'(t) = -x_1(t)$ I see that I can write $\dot{x} = Ax$ where $A = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$ The complete ...
-2
votes
0answers
39 views

Proving unique maximal interval of existence

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
0
votes
0answers
19 views

Find real solution for an inhomogene system

I have an inhomogene differential equation system $\begin{pmatrix}\dot{x}_1 \\ \dot{x}_2\end{pmatrix} = \begin{pmatrix}-1 & 3 \\ -3 & -1\end{pmatrix} \begin{pmatrix}x_1 \\ x_2\end{pmatrix} + ...
-5
votes
0answers
14 views

Partial Differential Equation : [closed]

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0
votes
1answer
12 views

System of ODE with initial values

I'm very bewildered by the following task in differential equations. We have $$x'' = -x -z +e^{-t} \\ z' = -2z-2x +3e^{-t}$$ and need to find solutions satisfying $x(0)=0,\ x'(0)=0,\ z(0)=1$. But ...
0
votes
1answer
15 views

Help to model time variant system

Let's imagine we have a system comprised of nodes and links. We give each node an initial value. After simulation is started flow is present between nodes through connecting links. Relationship ...
2
votes
1answer
49 views

high order DE :$y''''+y'''=1-x^2\,e^{-x}$

I am doing some exercise and I got to this question: SOLVE: $ y''''+y'''=1-x^2e^{-x}$, the associated homegeneous eqation solution is simple to calculate that is, $y_h=c_1+c_2x+c_3x^2+c_4e^{-x}$ ...
0
votes
0answers
20 views

differential equation with offset, characteristic polynomial equation

I have seen a lot of example problems on differential equations on forming a characteristic polynomial equation with the following diff-eq form: $\ddot{y}^2 + y = 0$ But what do you do when there is ...
3
votes
2answers
15 views

Differential Equation Word Problem involving y=Ce^(xk) (y=y')

"The rate of change of y is proportional to y. Write and solve the differential equation that models the verbal statement." This part of the problem is easy. My work is such: $y'=ky$ ...
1
vote
1answer
18 views

First-Order ODE Problem

I'm currently taking an ODE course at my school and one of the problems given follows: Suppose that a trajectory of $$(3x^2 - y)dx + (3y^2 - x)dy = 0$$ contains the point $(1,1)$. Show that it also ...
1
vote
1answer
56 views

solving differential equations with function coefficients using Laplace Transform

Does there exits a method to solve an $n$-th order liner differential equation with "function coefficients" using Laplace transform. It is well known that the identity $$L\left\{ {{t^n}f\left( t ...
4
votes
1answer
20 views

Second order ODE $y''+p(t)y'+q(t)y=0$

Let consider ordinary differential equation of the form $$t^2y''+3ty'+y=0$$ This is equivalent to $$y''+\frac{3}{t}y'+\frac{1}{t^2}y = 0$$ which looks better. But how does one find the solutions ...
0
votes
1answer
31 views

Why aren't exact differential equations considered PDE?

Exact differential equations come from finding the total differential from some multivariable function. In the exact differential equation $M\mathrm{d}x+N\mathrm{d}y=0$ M and N are considered to be ...
0
votes
2answers
109 views

Find the largest $x$ interval containing $0$ on which $y$ is well-defined.

I'm currently taking an intro course on ordinary differential equations and was given this homework problem: Find the solution of the following differential equation:$$\frac{dy}{dx} = y^2(1-2x)$$ ...
0
votes
1answer
13 views

Unique Indicial Equation of DE - Help Figuring Out?

So I'm currently working on the following: $$ x^2y'' + x(1+x+x^2)y' + x(2-x)y = 0 $$ I am a little thrown-off by there being an x-term in front of (2-x)y. Initially my thought was that this had no ...
0
votes
0answers
28 views

Using the Lyapunov-Perron method to find the local stable/unstable manifolds

Hello Stack Exchange community. I am currently having an issue finding the local stable/unstable manifolds of this system. After going at it for a few hours I believe the person who wrote this ...
2
votes
1answer
32 views

System (in a 6x6 matrix) of ordinary differential equations

One must find general solution for $$y' = \left(\begin{matrix} 1&2&-1&-2&1&2\\ -1&-2&1&2&-1&-2\\ 2&4&-2&-4&2&4\\ ...
-1
votes
0answers
11 views

The boundary value problem $y'' + \lambda y = 0, y(0)=0, y(\pi)+ky'(\pi)=0$ is self adjoint [closed]

Which is/ are correct: The boundary value problem $y'' + \lambda y = 0, y(0)=0, y(\pi)+ky'(\pi)=0$ is self adjoint only for $k\in $ {0,1}$ $, only for $k\in $ (-$\infty,\infty)$, only for $k \in $ ...
0
votes
1answer
36 views

ODE of second order

Let assume I have the following ODE of second order $$y''-y'=(y+1)^2 - y^2$$ Normally, using roots of the characteristic polynomial of this equation, I'd say the solutions are $e^{\lambda_1 t}$, ...
1
vote
2answers
36 views

Differentiation - simple case

In the book calculus made easy, page 22 the case of the negative power for $y=x^{-2}$ $$\begin{align} y+dy & =(x+dx)^{-2}\tag{1}\\ \\ & = x^{-2}\left(1+\frac{dx}{x}\right)^{-2}\tag{2} ...
0
votes
0answers
29 views

Global existence of ode system without solving it explicity.asdf

Here is the ode system that I am looking at $x' = -y-z$ $y' = x + ay$ $z' = b + z(x-c)$ where a,b,c are positive constants. By the local existence theorem, I know that there is a local solution, ...
6
votes
1answer
149 views

$\nabla \cdot f + w \cdot f = 0$

Let $w(x,y,z)$ be a fixed vector field on $\mathbb{R}^3$. What are the solutions of the equation $$ \nabla \cdot f + w \cdot f = 0 \, ? $$ Note that if $w = \nabla \phi $, then the above equation is ...
2
votes
1answer
55 views

Matrix with eigenvalues no negatives: What is $\lim_{t\to\infty} e^{tA}$?

Here's a homework question I've been stuck on for a while. My question is what can you tell about $$\lim_{t\rightarrow\infty}e^{tA}$$ if $A$ is $n\times n$ matrix and you know that every eigenvalue of ...
1
vote
1answer
43 views

IVP: $y'=\frac{y}{3x-y^2}$, $y(1)=1$

Solve implicitly the initial value problem: $ \left\{ \begin{array}{l l} y'=\frac{y}{3x-y^2} & \quad x\geq 1\\ y(1)=1 \end{array} \right. $ The equation is not exact and trying to ...
0
votes
0answers
23 views

basic differential equation question

The following statement arises in a proof I am reading, and I do not understand why this is: Suppose $J$ is an open interval containing zero and $x: J \to W$ satisfies $x'(t)=f(x(t))$ and $x(0)=x_0$. ...
1
vote
1answer
51 views

Question about continuity

If $f:\mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and we have that $u$ is continuous and satisfy $-(p(t)u'(t))'=f(t,u)$ Why $p(t) u'(t)$ and $u'(t)$ are continuous? where ...
0
votes
0answers
24 views

solving a state transition matrix for linear time variant system

given $x'(t) = [x(t)]^4$ , $to=2$, $x(to)=-2$ obtain the state transition matrix, $I(t,to)$ , required to map the initial state correction, $xo$, to a future time, $t$. so i know by setting $x'(t)=F$ ...
1
vote
0answers
44 views

Numerical analysis- Runge Kutta

I have: $$y'(x)= \sin(y); y(0)=1$$ I need to calculate the function values with Runge-Kutta. The range is [0,1]. My problem is that I need to choose the h (=dx) such that the error will be in order ...
1
vote
2answers
35 views

Computing a messy convolution

Consider the functions $$ x(t) = u(t - \frac{1}{2}) - u(t - \frac{3}{2}) $$ and $$ h(t) = tu(t) $$ where $u(t) = 1$ if $t \geq 0$ and $u(t) = 0$ if $t < 0$. I'm trying to compute $$ (x*h)(t) ...
1
vote
0answers
35 views

Local truncation error of Euler method

Wikipedia and this book say the local truncation error of Euler method is $O(h^2)$. But this book and A friendly Introduction to Numerical Analysis say it's $O(h)$. Which is correct? I have a similar ...
1
vote
1answer
19 views

Uniqueness of solution to linear first-order ODE with singular points

I want to solve a linear first-order ODE for $y(x)$, $x\in[0,1]$, $$ \gamma(x)y'-ay=-a\gamma(x),\quad y(0)=0, $$ where $\gamma(x)$ is a known function with $\gamma(0)=0$, and $a>0$ is a known ...
1
vote
1answer
25 views

II order nonlinear ODE, regularity of the solution

I have the following ODE $$ \frac{d^2}{ dt^2 }x(t) = F(x(t)),\: x(0) = x_0, \quad (t,x) \in [0,T]\times, \mathbf{R}^d $$ where $F$ is a nonlinear term. The question is: what kind of conditions on ...
0
votes
2answers
33 views

differential equation $(x^4+x+y)dx-xdy=0$

I have proplem $$(x^4+x+y)dx-xdy=0$$ I have been doing so: $$z=y/x$$ $$x\cdot(dz)/(dy)=dy/dx-z$$ $$x\cdot(dz)/(dy)=-((x^4+x+y)/(-x))-z=-1-x^3-z+z$$ $$x(dz)/(dx)=-1-x^3$$ $$dz=((-1-x^3)/(x))dx$$ ...
0
votes
3answers
40 views

First order differential equation ${{dy} \over {dx}} = {( - 2x + y)^2} - 7$

I am doing some exercise and I got to this question: Solve ${{dy} \over {dx}} = {( - 2x + y)^2} - 7$ with $y(0) = 0$. My approach has been to first set $u = - 2x + y$ then I got $ - 2 = {u^2} - 7$. ...
0
votes
0answers
20 views

Solving differential equation when $x=0$ for $u$ when $u \,\,{du} = \frac{-k}{mx^2}dx$ where $u =u(t)=dx/dt$, $u(0) = 0$ and $x(0) = x_0$ and $x_0 >0$

The differential equation is following: $$u \,\,du = \frac{-k}{mx^2}dx$$ where $u =u(t)=dx/dt$, $u(0) = 0$ and $x(0) = x_0$ and $x_0 >0$. $k,m,x_0$ are positive constants. How do you solve this ...
-1
votes
1answer
20 views

Question particular solution, differential [closed]

Show that $u(t) = \frac{\sin t}{t} - \cos t$ is a particular solution of the differential equation $t\left( {{{dx} \over {dt}}} \right) + x = t\sin t$ .
2
votes
1answer
23 views

$yy''=y^2y'+(y')^2$ method of reduction (differential equation)

I have a question about using reduction to solve $$yy''=y^2y'+(y')^2$$ This is how I have been thinking: put $y'=p$ and $p''=(dp)/(dy)*p$ $yp*dp/dy-y^2p-p^2=0$ ... $dp/dy-y=p/y$ but now I don't ...
3
votes
2answers
61 views

Volterra equation for a Bessel type IVP that appears in inverse scattering

I have the following i.v.p. (Colton Kress-Inverse acoustic and electromagnetic scattering theory, Springer) $$y''(r)+(k^2n(r)-\frac{l(l+1)}{r^2})y(r)=0$$ $$y(0)=0, y'(0)=1$$ using the Liouville ...
0
votes
0answers
21 views

definition of classical comparison theorom

Does any one know the definition of classical comparison theorem? I'm studying theory of impulsive differential equations. The author used this theorem to prove another theorem but he didn't describe ...
2
votes
1answer
38 views

How to solve the following delay differential equation?

What is the solution for the following equation? $$\frac\partial{\partial q}f(s,q)= \frac s2 f(s+2,q)$$ Note, it is known that the solution for $$\frac\partial{\partial q}f(s,q)= s f(s+1,q)$$ ...