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

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955 views

solution of first order differential equation and maximal interval

Find the solution of $x' = x^2t$ with initial value $x(0) = x_{0}$. Determine the maximal interval where it exists, depending on $x_{0}$ Please help me find the maximal interval!
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1answer
2k views

The number of solutions to an $n^{th}$ order differential equation.

For an $n$th order differential equation, why are there always $n$ solutions? Why exactly $n$, not $n - 1, n+1$ or infinite many? Addendum by LePressentiment : This is motivated by P176 on ...
0
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1answer
101 views

Basic Reference material about ODEs such as saparability with calculations and examples?

I am trying to show this kind of non-linear $y''''=y'y''/(1+x)$ in normal form. For example here if $y=e^{x}\rightarrow y^{(n)}=e^{x}\rightarrow x=-1$, where $y^{(n)}$ ...
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4answers
429 views

Find a continuous function $f$ that satisfies…

Find a continuous function $f$ that satisfies $$ f(x) = 1 + \frac{1}{x}\int_1^x f(t) \ dt $$ Note: I tried differentiating with respect to $x$ to get an ODE but you get one that contains integrals - ...
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2answers
201 views

Function whose inverse is also its derivative?

What are some good examples of a function $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ ...
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2answers
218 views

How to prove $(x^2-1) \frac{d}{dx}(x \frac{dE(x)}{dx})=xE(x)$

$$E(x)=\int_0^{\frac{\pi}{2}} \sqrt{1-x^2 \sin^2 t}\, dt$$ Where $E(x)$ is complete elliptic integral of the second kind. $u=\sin t$ $$E(x)=\int_0^{1} \frac{\sqrt{1-x^2 u^2}}{\sqrt{1-u^2}}\, du$$ ...
4
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1answer
193 views

Solution of $ f \circ f=f'$

Let $f:\mathbb R \to \mathbb R $ be a function such that $f \circ f=f'$ and $f(0)=0$ , I proved that $f$ is the null function. Can we prove that the same result holds if we change $f \circ f=f'$ by ...
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1answer
155 views

How do you solve this differential equation using variation of parameters?

$\color{green}{question}$: How do you solve this differential equation using variation of parameters? $$y"-\frac{2x}{x^2+1}y'+\frac{2}{x^2+1}y=6(x^2+1)$$ $\color{green}{I~tried}$ . . . ...
4
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1answer
325 views

Sturm-Liouville Questions

In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on. The first issue I have is that my book began with the statement that given $$L[y] = a(x)y'' + b(x)y' + ...
3
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1answer
93 views

Bessel Equations Addition Formula

So, I'm considering yet another tricky proof involving Bessel Functions. Basically, I'm trying to figure out how the following is true: $$J_n(\alpha + \beta) = \sum_{m = -\infty}^\infty ...
3
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1answer
1k views

Possible ways to do stability analysis of non-linear, three-dimensional Differential Equations

For example Lorenz system, $$ \frac{d}{dt}\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} -\sigma & \sigma & 0\\ \rho & -1 & -x\\ y & 0 & -\beta ...
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3answers
295 views

General solution of second-order linear ODE

I am trying to look a bit deeper into the mathematics the equation of motion used in physics and engineering. I have some specific questions at the end, but please correct me if I make a mistake in ...
3
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1answer
356 views

Exercise from Stein with partial differential operator

I have again something from Stein-Shakarchi I would really appreciate some help with. Any references are also welcome! Suppose $L$ is a linear partial differential operator with constant ...
2
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1answer
131 views

Classification of operators

I have a collection of questions about the limit point/circle concept and self-adjointness that are kind of connected, so I would like to ask them in a row. Apparently, an operator that is limit ...
2
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1answer
97 views

:How to find the general solution of $(y+ux)u_x+(x+yu)u_y=u^2-1$?

question : find the general solution of $(y+ux)u_x+(x+yu)u_y=u^2-1$ $\frac{dx}{dt}=y+ux,\frac{dy}{dt}=x+yu,\frac{du}{dt}=u^2-1$ I dont know how to start. is this quasilinear ? edit 1: tried ...
2
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2answers
281 views

Expressing an oscillator as a series of ODEs

Consider an oscillator satisfying the initial value problem $u''+w^2u=0$, where $u(0)=u_0$, $u'(0)=v_0$. Let $x_1 = u$, $x_2=u'$, and transform the equations given into the form $x' = Ax, x(0)$. Then ...
2
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1answer
178 views

dropping a particle into a vector field

I'm independently studying Colley's Vector Calculus and am on the section on line integrals. I understand that the line integral gives the amount of work done on a vector field for a predetermined ...
2
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1answer
151 views

Dimensions analysis in Differential equation

Differential equation of solitary wave oscillons is defined by, $$ \Delta S -S +S^3=0 $$ How can we write this equation as, \begin{equation} \langle(\vec{\nabla}S)^2\rangle+\langle S^2\rangle-\langle ...
2
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1answer
208 views

Help with Initial value problem : $y'= x^2+ xy^2, y(0) = 0$; Picard–Lindelöf Approximation.

i need solve this: $$y'=x^2+xy^2 , y(0)= y(t_0)= 0$$ a) Compute, starting from the constant function $u_0=0$ the successive approximations $u_1,u_2,u_3$ (in the sense of the theorem of ...
2
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1answer
198 views

Explain Dot product with Partial derivatives in Polar-coordinates

Related to page 819 prob 4 in this book. I am incorrectly calculating the left-hand-side (def. LHS), some newbie error with commutativity probably. Ideas? Errors? I propose ...
2
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1answer
938 views

Boundary conditions of an elastic bar

I was following some online lecture relating to an elastic bar with length $L$ that obey the differential equation $\displaystyle \frac{d^{2}u}{dx^{2}} = f(x)$, where $f(x)$ is its own weight or some ...
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2answers
274 views

Using laplace transforms to solve a piecewise defined function initial value problem

I want to use laplace transforms to solve the following: $$\frac{d^2 y}{dt^2}+16 y = f(t) = \left\{\begin{array} 1 1&t\lt\pi\\0&t\geq \pi\end{array}\right.\text{ with } y(0)=0 \text{ and } ...
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0answers
56 views

How to solve a system of two differential equations describing the concentration in a leaky tank?

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
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3answers
255 views

Differential equation $\sin \theta \frac{dr}{d \theta}+r\cos \theta =\tan \theta,0<\theta<\pi/2$ [closed]

This problem has been stumping me for over an hour how can I set it up, I think I have done it wrong over and over. Solving for $r$.
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3answers
4k views

How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system. I don't know how to use Mathematica/Matlab ... :( If anyone can make this portrait and post a print screen here, I would thank you ...
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1answer
101 views

Questions concerning the differential operator

Consider the differential equation:- $a \phi + (bD^3 - cD)w =0$, where $a, b$ and $c$ are constants, $D$ denotes the differential operator $\dfrac{d}{dx}$, and $w$ is a function of $x$. I'm ...
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3answers
15k views

What exactly is steady-state solution?

In solving differential equation, one encounters with steady-state solution. My textbook says that steady-state solution is the limit of solutions of (ordinary) differential equations when $t ...
1
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1answer
324 views

How to show that the geodesics of a metric are the solutions to a second-order differential equation?

On $\mathbb R^n$, let $\rho: \mathbb R^n\to\mathbb R$ be a smooth function, and $g$ be the metric given by scaling the usual flat metric by $e^{2\rho}$. I want to know how to show that the geodesics ...
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0answers
56 views

A different variation of parameters technique

I discovered a variation on the variation of parameters technique (I'll call it "VOP2") after a student asked me yesterday why we can make the assumption $u_1'(x)y_1(x)+u_2'(x)y_2(x)=0$. I didn't know ...
0
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1answer
133 views

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the green's function. I have to solve the given differential equation using Green's function method $\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta ...
0
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1answer
79 views

van der pol equation

Consider the van der Pol equation below: $(x'')+a(x^2-1)(x')+(x)=0$ I need to : Find an equilibrium point and linearize this equation near it Find solutions of the linearized equation depending on ...
0
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0answers
76 views

Proof the first-order equation

$$ \begin{align} \tau\dot y + y &= KF(t) \tag{1}\\ y(t) &= C_0 + C_1 e^{-t/\tau} \tag{2} \end{align} $$ How these two equations can form $$y(t)=KA+(y_0-KA)e^{-t/\tau}?$$ Note ...
0
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2answers
678 views

differential equations in SIR epidemic model and obtain Ro

I need to know why the differential equation system that expresses epidemic's model SIR in some texts appears: $$\frac{dS}{dt} =-\beta\frac{S}{N}I$$ $$\frac{dI}{dt}= \beta \frac{S}{N}I - \gamma I$$ ...
0
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2answers
448 views

Substitution $x=\sinh(\theta)$ and $y=\cosh(\theta)$ to $(1+x^{2})y'-2xy=(1+x^{2})^{2}$?

After this substitution I got to the point $$\cosh^6 (\theta)y'-\sinh(2\theta)-\cosh^4 (\theta)=0$$ and now let $$z=\cosh^2 (\theta)$$ so $$z^3 y'-z^2-\sinh(2\theta)=0$$ but then I ...
0
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1answer
3k views

System of differential equations in Maple

I have problems entering a system of differential equations to Maple 13. Equations are: $x' = -4x + 2y$ $y' = 5x - 4y$ Solve for $x = 0, y = 0$ Thank you in advance
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3answers
433 views

Why is it legitimate to solve the differential equation $\frac{dy}{dx}=\frac{y}{x}$ by taking $\int \frac{1}{y}\ dy=\int \frac{1}{x}\ dx$?

Answers to this question Homogeneous differential equation $\frac{dy}{dx} = \frac{y}{x}$ solution? assert that to find a solution to the differential equation $$\dfrac{dy}{dx} = \dfrac{y}{x}$$ we may ...
10
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1answer
235 views

Riccati differential equation $y'=x^2+y^2$

$$y'=x^2+y^2$$ I know, that this is a kind of Riccati equation, but is it possible to solve it with only simple methods? Thank you
7
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1answer
148 views

Nonlinear 1st order ODE involving a rational function

$$y'=\frac{-6x+y-3}{2x-y-1}$$ Is there a foolproof method for tackling equations of the form $y'=\dfrac{ax+by+c}{dx+ey+f}$ ? I've tried a few substitutions (like $y=vx$ and $v=2x-y-1$, neither of ...
7
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3answers
965 views

Why does the absolute value disappear when taking $e^{\ln|x|}$

I have noticed that if you have an equation (after integrating) such as $$\ln|y| = \ln|x| + c,$$ and you further simplify it using the law of exponents, you get $$e^{\ln|y|} = e^{\ln|x|+с},$$ which is ...
4
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2answers
136 views

A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$

Let: $f(x)$ and $g(x)$ be twice differentiable, non-decreasing functions. $f''(x) = g(x)$ and $g''(x) = f(x)$. $f(x) \cdot g(x)$ is a linear function. Then we have to show that $f(x) = g(x) = ...
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2answers
2k views

How can I solve this Initial Value Problem using the Euler method?

My Problem is this given Initial Value Problem: $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ I am looking for a way to solve this problem using the Euler method. I have a given Interval of $[1,2]$ and a ...
4
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1answer
173 views

Nonlinear equation (oscillon) comparison

Lagrangian for a spherically-symmetric, real scalar field in d spatial dimensions, $$L=c_d \int r^{d-1}dr\left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
4
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3answers
328 views

Closed form solutions of $\ddot x(t)-x(t)^n=0$

Given the ODE: $$\ddot x(t)-x(t)=0$$ the solution is: $$x(t)=C_1\exp(-t)+C_2\exp(t)$$ If we square the $x(t)$ we have: $$\ddot x(t)-x(t)^2=0$$ and the solution is given by: $$x(t)=6\wp(t+C_1;0,C_2)$$ ...
4
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2answers
286 views

Formula for integration bounds of recursively defined polynomial sequence

We can recursively define a sequence of polynomials by $$P_0(x) := 1$$ and then with the definite integral $$P_n(x) := \int_{c_n}^x P_{n-1}(t) ~\mathrm dt$$ where the $c_n$ are to be chosen so ...
3
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3answers
54 views

Solve $(x^2 + 1)y'' - 6xy' + 10y =0$ using series method

Use series methods to solve: $(x^2 + 1)y'' - 6xy' + 10y =0$ a) Give the recursion formula b) Give the first two non-zero terms of the solution corresponding to $a_0 = 1$ and $a_1 = 0$ ...
3
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3answers
111 views

Ordinary differential equation $y'(t)=\sin(f(t,y))$

One whose solution never makes me happy is the following: $$y'(t)=\sin(y+t)\text{.}$$ I would start by substituting $z(t)=y(t)+t$ to get an ODE in $z(t)$, but then I'm not sure about how to substitute ...
3
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1answer
3k views

How to Solve the Coupled Differential Equations?

I came across the set of following coupled equations while studying cycloid motion in Griffiths' Intro to ED $\ddot{y}=\omega \dot{z}$ $\ddot{z}=\omega (\frac{E}{B}-\dot{y})$ I am at a loss as to ...
3
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1answer
163 views

Initial value problem $t\frac{dx}{dt}=x+\sqrt{t^2+x^2}$

Another question on ODEs, this time just wondering how I should start with this one. $$t\frac{dx}{dt}=x+\sqrt{t^2+x^2}, \qquad x(1)=0.$$ It looks like a linear ODE and so after playing around with it ...
3
votes
1answer
2k views

Frobenius Method to solve $x(1 - x)y'' - 3xy' - y = 0$

So, Im trying to self-learn method of frobenius, and I would like to ask if someone can explain to me how can we solve the following DE about $ x = 0$ using this method. $$ x(1 - x)y'' - 3xy' - y = 0 ...
3
votes
2answers
211 views

Solving ODE with substitution

I have this as homework: $$(xy^2+y)dx+(x^2y-x)dy=0$$ I tried to solve it by substituting $z=xy+1$, but got the answer like $y=Cxe^{xy}$, which, I guess, is wrong. I tried to solve it couple of ...