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

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1
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3answers
43 views

How Do You solve a differential equation of the form: $y'=yx+x$

How do you solve a differential equation of the form: $y'=yx+x$ In this case you cannot separate the equation indeed: $y'=yx+x \iff \dfrac{dy}{dx}=yx+x$ And I can't separate it? How does one solve ...
1
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1answer
34 views

Solve $\frac{dx(t)}{dt}=-r(x(t)-x_e)+d(t)$

Given is the following ODE: $$\dfrac{dx}{dt}=-r(x(t)-x_e)+d(t).$$ where (d,t) are time functions. Any ideas how to find $x(t)$? What confuses me is that both $x$ and $d$ are two dependent variables ...
1
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2answers
42 views

Unique solution for ode $y' = {\sqrt{1-y^2}} $

I was given to solve the next ode: $y' = {\sqrt{1-y^2}} $ I found its solution: $y=sin(x+c)$ Now, I'm given that $y(0)=0$ and asked to show the only solution is $y=sin(x)$ in the region ...
-1
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0answers
8 views

Understanding the solution of this DE

I was giving an online test for practice where I could not solve the following question. Even after seeing the solution after the test , I am unable to understand it. I've learnt about linear, ...
1
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0answers
20 views

Converse of this theorem about existence of Green's function

I've been solving some problem which asks us to find the Green's functions for some problems when it exists. Now, there's a theorem which allows us to guarantee that it exists. The theorem is as ...
0
votes
2answers
41 views

What is y'' if $\sin y = y + 5x$?

I got $ 5\sin y / (\cos y - 1)^2$ as my answer, but the correct answer was given as $25\sin y / (\cos y - 1)^3$. My thought process: Derive the original equation to get $y'\cos y = y' +5$ $$y'(\cos ...
0
votes
4answers
47 views

Power series solution (Why the constant of the recurrence relation can be chosen arbitrarily?)

Please help me understand this: Solve $(x+1)y''-(2-x)y'+y =0$ First, since $x_0=0$ is an ordinary point, it can be guaranteed that we can find two independent power series solutions centered at ...
2
votes
2answers
22 views

Is there a solution to this differential equation?

I am trying to find a function $y(x)$ that is a solution to $$ \left(a_3 x^3+a_1 x\right) y''(x)-\left(3 a_3 x^2+2 a_1\right) y'(x)+3 a_3\, x \,y(x)=a_0 x^4+a_2 $$ I tried using mathematica but it ...
0
votes
0answers
8 views

Existence of Green's function for this problem

Consider the problem: $$\begin{cases}x^2y''+2xy'=f(x), & 0<x\leq 1, \\ \lim_{x\to 0}|y(x)|<\infty, \\ y(1)=\alpha y'(1), \alpha\in \mathbb{R}\end{cases}$$ I want to determine whether the ...
0
votes
1answer
12 views

Second order ODE- general solution

Is there any familiar formula or method to solve an ODE (for $y(x), \ x\in\mathbb{R}$) of the form $$ (\mu(x)\cdot y'(x))'-\mu(x)\cdot y(x)+r(x)=0 $$ where $\mu(x)$ is real smooth non vanishing ...
0
votes
2answers
46 views

solve the Following Differential equation: $y''=2yy'$

Solve the following equation: $y''=2yy'$ My attempt: $y''=2yy'$ integrate on both sides: $y'=\int(2yy')$ Let y'=s We have $\int 2y \dfrac{ds}{dy}dy = 2 \int yds = y^2$ Is that a way to ...
2
votes
0answers
22 views

Analytical Solution to an ODE

I was wondering if anyone could help me in finding a general analytical solution to this particular ODE $$f''' + c \bigg[ 3 \text{sech}^{2} \bigg( \frac{\sqrt{c}(x-a)}{2} \bigg) - 1 \bigg] \cdot f' + ...
0
votes
2answers
9 views

Deduce that: $\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}[mv\frac{du}{dx}+nu\frac{dv}{dx}]$

Deduce that: $$\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}(mv\frac{du}{dx}+nu\frac{dv}{dx})$$ When I differentiate $\frac{d}{dx}(u^{m}v^{n})$ I get: $$\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}(mv+nu)$$ Is ...
0
votes
1answer
15 views

Existence of Solution to Integral Equation

How do I show that the integral equation \begin{equation*} x(t) = \ln(1+t) + 1/2\int_0^1e^{-t}\sin^2(ts)x(s)ds \end{equation*} has a solution $C[0,1]$?
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0answers
10 views

Convergence of Iterative First Order ODE Method

Let us suppose we have a function $f(x,t)$ which is Lipschitz wrt $x$. Then we have the following iterative method $$ x_{m+1}(t) = c+\int_{t_0}^tf(x_m(v),v)dv$$ for solving the ODE $x'(t) = ...
1
vote
2answers
536 views

How can i convert nonhomogeneous ode to homogeneous ?

I have an equation system $$y'(t) = M(t)y(t)+h(t)$$ where $[M(t)]_{2\times2}$ square matrix and $[h(t)]_{2 \times1}$ is the nonhomogeneous part of the system. I can solve numerically homogeneous ...
2
votes
1answer
36 views

Solution to second order PDE $u_{xy}-xu_x+u=0$

Given second order partial differential equation $u_{xy}-xu_x+u=0$, where $u=u(x,y)$ find the general solution. I tried to use $u(x,y)=v(ξ(x,y),η(x,y))$ substitution to get ...
7
votes
2answers
16k views

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order ...
0
votes
1answer
21 views

Green’s Function for Regular Sturm-Liouville Problems

$''$ We are interested in solving problems like: $Ly := (py')'− qy = f$ with boundary conditions $β_1y(a) + γ_1 y'(a)=0$ , $β_2y(b) + γ_2 y'(b)=0.$ To this end we define the operator $Ly = ...
44
votes
7answers
984 views

What is the optimal path between $2$ fixed points around an invisible obstructing wall?

Every day you walk from point A to point B, which are $3$ miles apart. There is a $50$% chance each walk that there is an invisible wall somewhere strictly between the two points (never at A or B). ...
0
votes
1answer
38 views

Stuck at successive differentiation problem, suspecting error in question.

This is my very first submission so I wish Hi! to everyone. This is indeed a homework problem that involves using Leibniz's formula for n-th derivative. Although this is uncharted territory to me, I ...
1
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0answers
25 views

How to solve the following system $\frac{\text{d}x}{\text{d} t} = -Ax + \frac{B}{y} - C$, $ \frac{\text{d}y}{\text{d} t} = -Dx + \frac{E}{y} - F$

Is there a way to analytically solve the following ODE system? $$ \frac{\text{d}x}{\text{d} t} = -Ax + \frac{B}{y} - C\\ \frac{\text{d}y}{\text{d} t} = -Dx + \frac{E}{y} - F $$ Where $A,B,C,D>0$ ...
0
votes
1answer
32 views

How do solve this pde problem?

EDIT: I know somehow, we end up with an equation relating the derivative of some coefficients to the rest of the stuff. I'm not sure where this equation, or even the constant that we use to get it, ...
1
vote
3answers
57 views

2nd degree differential equation

Can someone please tell me how to solve this differential equation? $${d^2y\over dx^2} +y=\tan(x)$$ I am a beginner in ODE and have absolutely no idea how to proceed. Can you also site a reference ...
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votes
0answers
30 views

characteristic equation and particular solution [on hold]

Could you please tell me how to solve $2x^2 y''+4xy'+2y=10x^2 -6x$ Have to find characteristic equation and particular solution. Kindly help.
0
votes
0answers
15 views

Positiveness of energy of differential equation

Let $x(t) : [0,T] \rightarrow \mathbb{R}^n$ be a solution of a differential equation $$ \dot x(t) = f(x(t),t). $$ In addition we have functions $E :\mathbb{R}^n \rightarrow \mathbb{R}$ and ...
2
votes
0answers
17 views

$\omega$-limit set of a point $x \in X$

I would like to verify whether the following definition of the $\omega$-limit set of a point $x \in X$ is correct: $$\omega(x) = \{ y \in M : \exists \text{ sequence }\{t_j\}, \text{ where } t_j ...
0
votes
1answer
25 views

Determination of ordinary differential equations using Wronskian

I am a bit stuck in this question that I found in my textbook - "Show that the Wronskian of the functions $x $, $x^2 $, and $x^2\log x$ is non zero. Can these be independent solutions of an ordinary ...
0
votes
1answer
32 views

finding general solutions of second order diffrential equation

find the general solution of $$\frac {d^2y}{dx^2} +9y =18$$ I am not sure how to write it in its complementary form because of the roots one being positive and ...
1
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0answers
12 views

PDEs on higher genus Riemann surfaces, e.g. Klein Curve

I'm trying to solve a PDE on compact Riemann surfaces of genus g > 1. Since these can be obtained as quotients of the upper half plane $\mathbb{H}_2$ by some Fuchsian group $\Gamma$, I suppose it's ...
1
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2answers
49 views

Solve the first order ordinary differential equation $y'(x)=2x \cos^2 y(x)$

Solve $$y'(x) =2x \cos^2 y(x) .$$ \begin{align} \frac{dy}{dx} &= \ 2x \cos^2 y(x) \\ \frac{dy}{\cos^2 y(x)} &=2x \, dx \\ \tan y(x) &=x^2+k, \qquad\qquad k \in \mathbb{R} \\ y(x) ...
2
votes
0answers
24 views

Sufficient Boundary Condition to a General PDE on a General Domain

We know that for an ODE of $n^{th}$ order we need $n$ different boundary conditions. In PDEs, for example, for Laplace equation $\nabla^2 U=0$ (which is a second order PDE) we need only one B.C. (e.g ...
0
votes
2answers
26 views

Fundamental set of solutions to a differential equation

Say I have a linear 2nd homogeneous ODE of the form $$y''(x)+p(x)y'(x)+q(x)=0$$ Now I know that the general solution to this will be of the form $$y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)$$ where $\lbrace ...
5
votes
1answer
279 views

How to compute the values of this function ? ( Fabius function )

How to compute the values of this function ? ( Fabius function ) It is said not to be analytic but $C^\infty$ everywhere. But I do not even know how to compute its values. Im confused. Here is the ...
14
votes
5answers
144 views

Can we abuse traffic patterns to get home earlier?

I had a heated discussion with my co-worker today, and was wondering if someone here could shed some light on this situation. The post is a bit lengthy, but I wanted to put all my intuition down in ...
0
votes
2answers
19 views

Specific solution for ODE

Can somebody explain step-by-step, as I can't understand, how to find the particular solution of the ODE? 1) $y' + y = 1$ 2) $y' + 2y = 2 + 3x$
0
votes
0answers
16 views

Largest interval for existence of solution

Consider the first order IVP: $$\sin(t)y' + y=\frac{\sin(t)}{\ln(t-1)}, \qquad y\left({2\pi\over3}\right)=17$$ How to prove that the largest interval where the solution to the above problem is certain ...
1
vote
1answer
41 views

Solving a differential equation of the form $y'' = f(x,y)$ [on hold]

The question is $$y''=\frac{c}{x} +\frac{d}{y^2}+r$$ $y=y(x)$ $c,d$ and $r$ are constants.
0
votes
1answer
9 views

Decoupling coupled differential equations with time dependent coefficients

Consider the following system of coupled differential equation. $$\left[ \begin{array}{c} \frac{dc_1}{dt} \\ \frac{dc_2}{dt} \end{array} \right] = \begin{bmatrix} -B & -V(t) \\ -V(t) & B ...
2
votes
2answers
65 views

Periodic solutions of $x'=x^2-1-\cos t$

Consider $x'=x^2-1-\cos t$. What can be said about the existence of periodic solutions for this equation? I'm not sure if periodic solutions exist, but if they do, they must have period equal to $ ...
-1
votes
3answers
86 views

How do you integrate $e^{-st}t\cos(t)$?

I'm doing differential equations and specifically studying Laplace Transformations, where of course the Kernel is: $K(s,t) = e^{-st}$ And the Laplace Transformation $\mathcal{L}$ of a function ...
0
votes
1answer
21 views

solution of second order ODE with non-constant coefficients

What is the general solution of $\frac{d^2y}{dx^2}+P[Q-R\cosh(Sx)]y=0$ where $P,Q,R,S$ are real and positive? I tried transforms but cannot get a solution.
1
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1answer
25 views

Solving constrained Euler-Lagrange equations with Lagrange Multipliers (Geodesics)

I'm trying to solve a calculus of variations geodesics problem using Lagrange Multipliers, showing that the geodesics of a sphere are the so-called great circles. I am using a constrained Lagrangian ...
2
votes
0answers
52 views

Relation between a linear second order differential equation and Riccati special differential equation

Consider the following differential equation \begin{equation} \frac{d}{dx}\left[N(x)\frac{dw}{dx}\right]+\sigma^2\rho(x)w=f(x,\sigma),~~ 0<x<l, \end{equation} $0<N\in C^1(0,l)$, $0<\rho\in ...
0
votes
0answers
22 views

how do I resolve equations that are both dependant on each other

I'm working on a project concerning the ideal power equation of aerodynamic bodies seen here: $$P = \frac{1}{2}C A D v^3 + \frac{W^2}{Db^2v}$$ where $P$ = power, $C$ = coefficient of drag, $A$ = ...
0
votes
0answers
15 views

O.D.E. in Homogeneity Lemma

Let $\psi: \mathbb{R}^{n} \to \mathbb{R}$ smooth such that $\psi(x) > 0$ for $x \in B(0,1)$ and $\psi(x) = 0$ for $x \notin B(0,1)$. Let $c \in S^{n-1}$ fix and arbitrary and consider the O.D.E. ...
0
votes
0answers
33 views

Convergence of $\dot{x}(t) = -\alpha(t)x(t) + b\mathrm{e}^{-\lambda t}$

Let $x(t)\ge 0$ obey the following differential equation: $$ \dot{x}(t) = -\alpha(t)x(t) + b\mathrm{e}^{-\lambda t}, $$ where $b>0$, $\lambda>0$, $\alpha(t)\in\mathbb{R}$ is both lower- and ...
0
votes
1answer
21 views

Show that $\frac{dx}{dt}=\frac{1}{14}(15-x)$ given $x=15-12e^{\frac{-t}{14}}$

A biologist is researching the growth of a certain species of hamster. She proposes that the length, $x$cm, of a hamster $t$ days after its birth is given by $x=15-12e^{\frac{-t}{14}}$ Show that ...
1
vote
1answer
24 views

General solution to differential equation, given a polynomial general solution

I am solving one DE and I have to consider the following: $$(y+ax)^n(y+bx)$$ to come up with a general solution to the following differential equation: $$\frac{dy}{dx} = \frac{10x-4y}{3x-y}$$ I ...
2
votes
1answer
76 views

Solution to a particular Wave Equation

Consider the partial differential equation \begin{align} \frac{1}{c^{2}} \, \frac{ \partial^{2} U}{\partial t^{2}} &= \frac{\partial^{2} U}{\partial x^{2}} + x \, \frac{\partial U}{\partial x} + ...