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
1k views

Is there a strategy for solving a non-autonomous differential equation?

I'm curious about techniques for solving a nonautonomous* system in the case of a non-linear differential equation. There's a simple example in my textbook (Hirsch, Smale, Devaney) where we obtain ...
2
votes
1answer
30 views

Two point boundary value problem $x''+x =0$

Side conditions x(0)=0 and x(1)=1 I know that I need to find the roots first but don't know how to continue. Using $x=e^{\lambda t}$, the roots are found by $\lambda^2 + 1 = 0$, which gives us $i$ ...
0
votes
1answer
9 views

Integrating Factors Found by Inspection

Can anybody help me solve this differential equation? $y(x^2y^2-m)dx+x(x^2y^2+n)dy=0$, when I solve this by inspection, I got stuck with this equation: $(xy)^2d(xy)+nxdy-mydx=0$. Am I on the right ...
2
votes
1answer
23 views

how to solve strum liouville problem of second order

$(1+x^2)y^{"}+2xy^{'}+\lambda x^2 y=0$ with $y'(1)=0$ and $y'(10)=0$. How do we solve this type of Sturm-Liouville problem?
0
votes
1answer
21 views

Two similar first order differential equations

Solve$$tx'^2-2xx'-t=0$$ and $$t^2-2txx'-x^2=0$$ These would be easy Riccati's equations if their middle terms werent multiplied by 2. In this case I have no idea how to solve that.
20
votes
2answers
455 views

How does one parameterize the surface formed by a *real paper* Möbius strip?

Here is a picture of a Möbius strip, made out of some thick green paper: I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
3
votes
0answers
28 views

A second order differential equation

How does one solve the following differential equation $y^{"}+xy^{'}+(1-x^2)y=y\sin x$? I don't know how to proceed?
0
votes
1answer
7 views

Writing nonautonomous systems as autonomous systems

Apparently any mth order nonautonomous system is equivalent to a first order autonomous system in higher dimensional space. How does this work in practice? I would you write $\displaystyle ...
0
votes
0answers
4 views

Application of Liouville normal form

The Bessel's equation $r^2 u''+ru' +(\lambda r^2 -m^2)u=0$ has general solutin $u=A J_m(\sqrt{\lambda}r)+B Y_m(\sqrt{\lambda}r)$ Require, first, find the self-adjoint form and Louisville Normal ...
1
vote
2answers
45 views

Looking for a matrix A(t)

I need your help, I'm looking for a contraexample, I need to give a matrix A(t), such that $$e^{\int_0^tA(s)ds}$$ is not a matrix solution for $x'=A(t)x$. I really don't have any clue what can it be. ...
0
votes
2answers
30 views

Particular solution of 1st ODE

for the first order liner ODE $$u'+\left(\frac{1}{x} - \frac{\cos x}{\sin x}\right)u=\frac{e^x (1-\frac{\cos x}{\sin x})}{2x}$$ That the IF is $$e^{\int{\left(\frac{1}{x}-\frac{\cos x}{\sin ...
6
votes
3answers
89 views

Use Taylor Series method to solve $y''-2xy+y=0$

I am doing some practice problems for solving second order ODEs, and I am a bit stuck on this one. Here is what I have: $y''-2xy+y=0$ Let $y = \sum_{n=0}^{\infty} C_nx^n \implies y' = ...
1
vote
2answers
343 views

The trace-determinant plane, classification of equilibria of differential equations

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ...
0
votes
0answers
8 views

What are the upper bound and stability conditions for the following simple linear system

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
5
votes
1answer
457 views

Damped Harmonic Oscillator and Response Function

This is another one of those questions that I feel like I am almost there, but not quite, and it's the math that gets me. But here goes: For a driven damped harmonic oscillator, show that the full ...
0
votes
0answers
15 views

The Stiff Vibrating String Partial Differential Equation

I am trying to solve the plucked stiff string PDE. I am following Philip Morse in "Vibration and Sound" and I. Testa et. al. in "Physically Inspired Models for the Synthesis of Stiff Strings with ...
0
votes
1answer
16 views

Sign of energy and solving the Schrodinger equation.

The particular problem that triggered my question is as follows: A particle of mass m is confined within the box $0 < x < a$, $0 < y < a$ and $0 < z < c$. The potential vanishes ...
1
vote
3answers
27 views

Book for ODEs and numerical solution

I would like to ask you information for a book. I want to (self) study ordinary differential equation and their numerical solution (with MATLAB). I am not a math student (life science) so I want a ...
2
votes
1answer
53 views

Help with Differential Equation

Our differential equation is: $$ y' + 2y/3 = 1-t/2 $$ Consider $y_0$ and find the value for which the solution of our differential equation touches, but does not cross, the $t$-axis. EDIT ...
0
votes
0answers
23 views

Invariant in geodesic

What in general is invariant in geodesic in terms of parameters $u$ and $v$ ( or functions on which they depend) and their derivatives in integrated form? For a surface of revolution, Clairaut's ...
0
votes
1answer
22 views

Question on a derivation regarding the non-linear ODE $x'' = -U'(x)$, $U$ potential

Let $U$ be a potential function, and consider the IVP $$ (*) \quad x'' = -U'(x), \qquad x(t_0) = x_0, \quad x'(t_0) = v_0. $$ We suppose the following: (V) Let $x_0, v_0$ be initial values and let ...
1
vote
0answers
20 views

A problem with Riccati's equation

Solve $$ x'=-\frac {4}{t^2}-\frac 1 t x+x^2$$ knowing that $\gamma (t)=\frac 2 t $ is a particular solution. So I make a substitution $x=\gamma (t)+\frac 1 u$ $$x=\frac 2 t +\frac 1 u$$ ...
2
votes
2answers
42 views

Prove second derivative of $g$ is proportional to $g^2$

From Apostol's Calculus Vol. 1, chapter 6.26, exercise 30: Let $f(x) = \int_0^x (1+t^3)^{-1/2} dt$. $a)$ Prove $f$ is strictly monotonic. $b)$ Let $g$ be the inverse of $f$. Show that the ...
-5
votes
1answer
49 views

How to show that a given function is a solution of differential equation?

I have been trying to prove this for awhile but in any way that I try it doesn't give me the same required answer that I must show, any ideas? If ${y =\sqrt{x} + \dfrac{1}{\sqrt{x}}}$, ...
4
votes
2answers
54 views

Solve $x^2+tx'+x=0$

Solve $x^2+tx'+x=0$ this is clearly a Bernoulli's equation so I make a substitution $z=\frac 1 x$ $$x=\frac 1z$$ $$x'=\frac {-z'}{z^2}$$ $$\frac {1}{z^2}-\frac {tz'}{z^2}+\frac 1 z=0$$ ...
0
votes
0answers
26 views
+50

Calculating the constants in the general solution of second order homogeneous ODE reduced from Riccati equation

I am writing a code to simulate a kind of volumetric flow, and I have encountered the non-linear Riccati equation in its general form near the end of my calculations. I am having trouble finding the ...
0
votes
1answer
25 views

Eigenvalues for the Sturm-Liouville boundary value problem

Please show me how to calculate the eigenvalues for the following boundary value problem: $$x''+\lambda x=0\\x(0)=0\\x(\pi)=0\\x'(\pi)=0$$ This is what I did: let $\lambda=\mu^2$ $$X(x)=A\cos\mu ...
3
votes
3answers
188 views

Solve second order ODE knowing one of its solutions

Solve $t^2x''-4tx'+6x=0$ knowing that $x_1(t)=t^2+t^3$ is a particular solution So I assumed the general solution will be in form of $x(t)=C_1 x_1(t)+C_2 x_2(t)$ and $x_2 = v(t)x_1(t)$ So now I ...
-1
votes
0answers
13 views

differential equtaion-how to find its regular,critical points

for the given differential equation how do we calculate at a given point the following regular point ordinary point critical point a singular point what does all this mean? eg: consider the ...
-1
votes
0answers
55 views

Solving two differential equations [on hold]

$\frac{\partial ^2 B}{\partial t ^2} = - \mu_0^2 * \epsilon_0^2 * \frac{\partial B}{\partial t }$ $\frac{\partial ^2 E}{\partial t ^2} = \mu_0^2 * \epsilon_0^2 * \frac{\partial B}{\partial t }$
0
votes
0answers
11 views

Solving implicit theta-method function numerically faster than using fixed point iteration

When we are using the theta-method to solve an IVP. We have the equation: $$y(x_{n+1}) \approx y(x_n) + h[(1-\theta) f(x_n, y(x_n)) + \theta f(x_{n+1}, y(x_{n+1})]$$ where $f(x,y)=y'(x)$ The only ...
1
vote
3answers
60 views

$2^{nd}$ order ODE $4x(1-x)y''-y=0$ with $y'(0)=1$ at $x=0$

it has two singular point $x_0 =1$ and $x_0=1$ I rearrange the equation to $$y'' - \frac{1}{4x(1-x)}y =0$$ and by $v(x)=\frac{(x-x_0)^2}{4x(1-x)}$ are analytic at $x_0$ for both $0$ and $1$; that ...
-2
votes
0answers
46 views

integrating factor of the function- and exact equation [on hold]

How do we calculate the integrating factor for the following: $(\cos y \sin 2x)dx+(\csc^2 y-\cos^2 x) dy=0$ $\sec^2 y+\sec y \tan y$ $\tan^2 y + \sec y \tan y$ $\dfrac{1}{\sec^2y + \sec y \tan y}$ ...
0
votes
1answer
23 views

Initial value problems with known solutions?

I'm trying to find a list of IVPs with known solutions to test my implementation of some numerical techniques. The only one I know of is: $$f(x,y)= y' =-\lambda y\;,\;\;\; y(0)=1$$ with the ...
0
votes
2answers
29 views

Comparison theorem for ODE

Here is something I'm trying to prove: Conjecture: Suppose $f'(x) \leq \phi(f(x), x)$ and $f(a)=\alpha$. Suppose $g'(x)=\phi(g(x),x)$ and $g(a)\geq \alpha$. Then $f(x)\leq g(x)\,\,\forall x$. ...
0
votes
0answers
7 views

Zero-stability for numerical methods, only applies to LMM's?

I'm trying to get a better grasp of the notion of zero-stability. Mainly I'm using a book by Leveque (Finite Difference Methods for Ordinary and Partial Differential Equations). Anywho, Leveque ...
2
votes
0answers
36 views

What are the different solution concepts for Matrix-Ordinary Differential Equation [Theory Question]

I was recently given a ODE to solve from a boss at work, with the knowledge that I haven't done them before and this will help me learn. I've spent 10 hours so far learning the basics of ODEs. The ...
1
vote
0answers
25 views
1
vote
1answer
588 views

Differential equation word problem - Malthus's law

Another problem I'm trying to solve that I have no idea what to do with. Here it goes: In 1798, Rev. Robert Malthus proposed that the rate of change of a population is proportional to the actual ...
-1
votes
3answers
36 views

Time period of ODE

Is it possible to find time period of the following non-linear ODE? $$\frac{1}{\cos{y}}\frac{\mathrm{d}^2 y}{\mathrm{d} x^2 } = a \sin{y} + b, \quad y =y(x). $$ If so, how to obtain it? Is there a ...
1
vote
2answers
30 views

Inverse Trigonometric functions - Boyce & Diprima 2.2.19

The problem asks for a solution to the initial value problem: \begin{align} &\sin(2x)dx+\cos(3y)dy=0\\ &y\left(\frac{\pi}{2}\right)=\frac{\pi}{3} \end{align} The problem is separable and I ...
1
vote
1answer
32 views

How to write a non-homogeneous equation in self-adjoint form

How can I write a non-homogeneous equation in self-dajoint form? such as, for equation with $-1\le x \le1$ $$(1-x^2)u''-xu'+2u=x^4+x$$ What is its self-dajoint form? Also, for a ...
2
votes
1answer
53 views

How can I solve this differential equation, what type is it?

How can I solve this differential equation, what type is it? $$(x^2+2x-2y)dx=dy$$ How can I find the integrating factor?
0
votes
0answers
17 views

$\Phi(\cdot ,x):I_x \rightarrow M$ injective?

I don't understand we the map $\Phi(\cdot ,x):I_x \rightarrow M$ from the excerpt from below of the lecture notes of my professor has to be injectiv. (Here $M$ denotes the domain of the function on ...
0
votes
1answer
24 views

A reference for a simple lemma on positive solutions of ODE

Where one may find any reference to lemmas the following kind: If $x(t)$ is $C_1$ in $[0,T], x(0)\gt0, \frac{dx}{dt} + c(t)x(t)\gt 0$ in $[0,T]$ then $x(t)\gt 0$ in $[0,T]$. There is a version with ...
3
votes
3answers
103 views

Partial derivative function definition paradox

I've pondered this question over quite alot and haven't been able to find an answer anywhere. I'm going to ask this question from the standpoint of basic thermodynamics. Let's say I define ...
2
votes
1answer
1k views

Solve a second order DEQ using Euler's method in MATLAB

I need to solve the equation below with Euler's method: $$y''+ \pi ye^{x/3}(2y' \sin(\pi x)+\pi y\cos (\pi x)) = \frac{y}{9}$$ for the initial conditions $y(0)=1$, $y'(0)=-1/3$ So I know I ...
2
votes
1answer
42 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 A ...
6
votes
1answer
175 views

Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
1
vote
2answers
53 views

Simple Derivative paradox

Suppose I define $y(x)=x^3$ $${dy(x) \over dx} = 3x^2$$ $${dy(x) \over dy} = 1 = 3x^2 \frac{dx}{dy} = 0\text{ since }x \neq f(y)$$ $1 \neq 0$ If you take the differential $d()$ where $dy(x)$ then ...