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

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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?
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10 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?
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1answer
15 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.
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1answer
6 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 ...
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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 ...
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2answers
29 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 ...
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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 ...
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1answer
25 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 $i$ ...
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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 ...
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3answers
88 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' = ...
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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 ...
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1answer
21 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 ...
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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 ...
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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 $x=\gamma (t)+\frac 1 u$ $$x=\frac 2 t +\frac 1 u$$ ...
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1answer
48 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 ${y =\sqrt{x} + \dfrac{1}{\sqrt{x}}}$, ...
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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 $z=\frac 1 x$ $$x=\frac 1z$$ $$x'=\frac {-z'}{z^2}$$ $$\frac {1}{z^2}-\frac {tz'}{z^2}+\frac 1 z=0$$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
42 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. ...
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Solving two differential equations [on hold]

$\frac{\partial ^2 B}{\partial t ^2} = - \mu_0^2 * \epsilon_0^2 * \frac{\partial B}{\partial t }$ $\frac{\partial ^2 E}{\partial t ^2} = \mu_0^2 * \epsilon_0^2 * \frac{\partial B}{\partial t }$
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2answers
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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 ...
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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 ...
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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, ...
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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 ...
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2answers
28 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 $f(x)\leq g(x)\,\,\forall x$. ...
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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 ...
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integrating factor of the function- and exact equation [on hold]

How do we calculate the integrating factor for the following: $(\cos y \sin 2x)dx+(\csc^2 y-\cos^2 x) dy=0$ $\sec^2 y+\sec y \tan y$ $\tan^2 y + \sec y \tan y$ $\dfrac{1}{\sec^2y + \sec y \tan y}$ ...
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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 ...
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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 ...
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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 ...
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1answer
41 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 ...
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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 ...
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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 ...
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0answers
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Find the fundamental matrix of a system of ODEs?

To linearize a system, in one of the steps I am required to find the fundamental matrix $\Phi$(t) of a system such that $\Phi$(0)=I. The example system my professor used: $\dot{x} = x - y - x^3 - ...
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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 ...
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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 ...
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1answer
31 views

Qualitative Ordinary Differential Equations (Terminal Velocity)

The velocity $v(t)$ of a skydiver falling to the ground is governed by $m\dot{v} = mg - kv^2$, where $m$ is the mass of the skydiver, $g$ is the acceleration due to gravity, and $k \gt 0$ is a ...
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1answer
33 views

Determine stability of the fixed points of $\dot x=x-x^3$

Find all of the fixed points of the system $\dot x=x-x^3$, and determine whether they are stable or unstable (or neither). Make sketches of at least 2 solutions of this system (corresponding to ...
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1answer
12 views

Qualitative Ordinary Differential Equations Problem

Find some continuously differentiable function f:R→R and some number a such that the unique solution to the initial value problem x(t)=f(x(t)) with the first x having a dot over it x(0)=a is ...
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1answer
16 views

Bigger Interval in which ivp has a differentiable solution?

find the bigger interval in which the ivp has a differentiable solution, there is no need to solve the equation. i know that the the problem has unique solution in the interval (-2,2), but how know ...
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1answer
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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 ...
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1answer
21 views

Integration of nonlinear ODE

In Gelfand Fomin well known Calculus of Variations text, when dealing with the isoperimetric problem in the upper half plane they obtain the differential equation $$ ...
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How t find z (unknown) in Runge-Kutta question

I'm trying to solve the below question solve $\dfrac{dx}{dy}=\dfrac{1}{x+y}$ for $x=0.5$ to $z$ using R-K (order $4$) with $x_0=0$, $y_0=1$ (take $h=0.5$). Please help me and tell me how to ...
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2answers
57 views

Verifying possible solutions of the differential equation $(y')^{2}-1-y^2=0$ [duplicate]

Which of the following functions is/are solutions to the differential equation $$(y')^{2}-1-y^2=0$$ Note: There may be more than one solution. A. $y(x)=\frac{(\cos(x)-\sin(x))}{\sqrt{2}}$ B. ...
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25 views

Solving a system of first order differential equations

So, I have (another) problem with differential equations (from an optimal control problem). I am trying to solve the following system of DEs (is this even a system?): $$ \lambda'(t) = r \lambda(t) + ...
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1answer
61 views

Which of the following functions is/are solutions to the differential equation $(y'')^2-1-y^2=0$? [on hold]

Which of the following functions is/are solutions to the differential equation $$(y')^{2}-1-y^2=0$$ Note: There may be more than one solution. A. $y(x)=\frac{(\cos(x)-\sin(x))}{\sqrt{2}}$ B. ...