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

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32 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 ...
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0answers
34 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} + ...
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1answer
15 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 ...
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1answer
29 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
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1answer
52 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, $y_h=c_1+c_2x+c_3x^2+c_4e^{-x}$ ...
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0answers
54 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
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2answers
55 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: $y'=ky$ ...
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1answer
25 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
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1answer
106 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
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1answer
28 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 ...
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1answer
46 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 ...
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2answers
285 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:$$\frac{dy}{dx} = y^2(1-2x)$$ ...
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1answer
17 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 ...
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0answers
75 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
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1answer
202 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\\ ...
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1answer
40 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 $e^{\lambda_1 t}$, ...
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2answers
49 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} ...
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0answers
38 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, ...
7
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1answer
166 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
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1answer
58 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 ...
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1answer
46 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 ...
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0answers
27 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 $x(0)=x_0$. ...
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0answers
69 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 ...
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2answers
38 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) ...
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0answers
205 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
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1answer
35 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
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1answer
42 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
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2answers
71 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: $$z=y/x$$ $$x\cdot(dz)/(dy)=dy/dx-z$$ $$x\cdot(dz)/(dy)=-((x^4+x+y)/(-x))-z=-1-x^3-z+z$$ $$x(dz)/(dx)=-1-x^3$$ $$dz=((-1-x^3)/(x))dx$$ ...
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3answers
41 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$. ...
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0answers
23 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 ...
2
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1answer
134 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
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2answers
90 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 ...
2
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1answer
47 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 $$\frac\partial{\partial q}f(s,q)= s f(s+1,q)$$ ...
1
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1answer
86 views

Series Solutions Near an Ordinary Point

I am attempting to solve this problem for practice: $y"-(x-3)y' - y = 0$ at $x_{0} = 3$. But it appears as though I don't have an idea of the best approach to employ to go about solving it. Can ...
0
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1answer
29 views

How many independent parameters in $e^{c_1x}+e^{c_2x}$

While studying differential equations my friend had this doubt. We might say two but exchange of non equal values of these gives the same curve so infact they are not independent, so how many ...
3
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3answers
355 views

Solving inhomogenous ODE

I have an inhomogenous ODE. The main issue here is variables are matrices. It is bit of matrix calculus. A solution would be highly appreciated interms of x . I guess we can use same methods for ...
4
votes
2answers
53 views

Is there a unique solution to this simple differential equation?

I am trying to establish uniqueness for a solution to a bigger problem, and it boils down to whether or not the following differential equation has a unique solution: $$f'(t)⋅(f(t)-t)=K$$ Clearly, ...
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1answer
84 views

show that bessel function satisfies the differential equation

so this is a question I have been given and I have no idea where to even start, my bessel functions is not all that great but if someone could just help me get a handle on this thing, i.e. just help ...
3
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0answers
23 views

Differential Equations: Say whether the equation has bounded solutions at $x = 0$

Its been forever since I've done Diff EQ and I can't remember how to go about solving this problem: Say whether the equation has bounded solutions at $x = 0$ and whether all solutions are bounded: $$ ...
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2answers
34 views

Two very similar solutions to a differential equation through two different methods

In our differential equation class, we learned of two methods to solve elementary differential equations: integration factors and seperation. We had to solve the differential equation (k is a ...
3
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2answers
50 views

how to solve this differential equation which cannot be reduced to homogeneous using standard methods

I came across this interesting question. $$(y^2 + xy + 1)\,dx + (x^2 + xy + 1)\,dy =0.$$ I tried to make it homogeneous by using an integrating factor but could not proceed through. NOTE: I am ...
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0answers
83 views

Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...
1
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1answer
38 views

Sloution to first order non linear system of ordinary differential equations

Can some one help me how to solve the following system of differential equations. \begin{eqnarray*} \frac{dP_0}{dt}& = & -C_1\lambda P_0 P_1+ \frac{C_1}{2}P_1^2 \\ \frac{dP_1}{dt} &= ...
0
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2answers
77 views

Solve differential equation!

Hi im trying to solve this differential equation: $$\frac{\mathrm dx}{\mathrm dt} = k_1(a-(x-c))(b-(x-c)) - k_2(x)$$ The equations is from a chemical problem but I dont now how to solve it. Should ...
2
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1answer
37 views

nonlinear, nonhomogeneous ODE 1. order

Solve $x'(t)-\dfrac{a}{t}x(t)=b(t),~a=const,~x(0)=0$. Homogeneous Solution: $\dfrac{x'(t)}{x(t)}=\dfrac{a}{t}\quad|\int\\ \ln(x(t))=a\ln(t)+c,~c=const\quad| e\\ x(t)=t^ae^c$ Is that correct? No ...
0
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3answers
61 views

How to go about solving a second order differential equation

How do I go about solving: $$ y'' = -e^{-y}$$ Am I supposed to do reduction of order?
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0answers
31 views

How to solve ordinary differential inequations with vector variables?

Given $a\in\mathcal{R}_+^d$ and $s\in\mathcal{R}^d$,we wanna a function f(.) which maps s to a vector $f=\begin{bmatrix}f(s_1),\cdots,f(s_d)\end{bmatrix}^T$ and satisify the following inequation. ...
0
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0answers
52 views

What is the Riemann surface of the exponential integral?

I have recently encountered a differential equation whose solution has a term \begin{equation} \frac{1}{2}e^{-\frac{1}{2 \varepsilon} e^{i \tau}} \int_{\tau_0}^\tau e^{\frac{1}{2 \varepsilon} e^{i ...
0
votes
1answer
23 views

chebshev series expansion

How to use the Chebshev series to expansed this function $$f(x)=\frac{4}{5-3x}=\sum_{n=0}^{\infty}a_n T_n(x)$$ ...
0
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2answers
37 views

Why write the solution of the harmonic oscillator form 1 is equal to writing form 2?

Why write the solution of the harmonic oscillator form $$\psi=A\cos\omega_0 t+B\sin \omega_0t$$ is equal to writing form $$\psi=C_1e^{i\omega_0t}+C_2e^{-i\omega_0t}$$? I would like to see how one ...