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

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1answer
8 views

Linear-Homogeneous vs Homogeneous ODEs?

Currently in my third week of my first ODEs class and I've already encountered something I'm struggling with. My second homework assignment requires me to classify and solve some ODEs. He gave us four ...
0
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0answers
19 views

Maximum Solution to Differential Equation

For this initial value problem, decide if the solution exists for all $t \le 0$, or only on a finite time interval $0 \le t \lt T$ $$\dot x = x^2(1-x^5), x(0) = \frac12$$ I've done a problem where ...
0
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1answer
14 views

Bifurcation Diagram question for Population harvesting model $P' = rP (1-\frac{P}{K}) - hP$

A deer population grows logistically and is harvested at a rate proportional to its population size. The dynamics of population growth is modeled by $P' = rP (1-\frac{P}{K}) - hP$ where $r$ (the ...
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2answers
11 views

Effect of wronskian on the solution of a differential equation

As far as my understanding goes, the Wronskian $W(t)$ for a second order homogenous differential equation with continuous coefficients can help us govern whether the solutions will be linearly ...
0
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2answers
16 views

How might I go about forming a general solution for the following differential equation?

$\frac{df}{dt}+t^kf=t^k$ where $k\in \mathbb{Z}$ I've solved for the equation previously where $k=2$ to get $f=\frac{-1}{t-t\ln t}+c$ but am not sure how I should go about solving this generally.
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0answers
14 views

Boundary value ODE with unknown functions?

Problem Let $f(x)$ be given such that $F'(x) = f(x)$. Also let $a(x) > 0$ in the interval $[0, L]$. $$ \left\{ \begin{array}{c l} -(a(x)u'(x))' = f(x),\ x \in (0,L) \\ u(0) = ...
0
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1answer
12 views

Maximal Solution to Differential Equation

For the differential equation $$\dot x = x(1-x), x(0)= \frac 12$$ Decide if the solution exists for all $t \ge 0$ or only on a finite time interval $0 \le t \lt T$. By the theorem, for the maximal ...
0
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2answers
32 views

Differential Equations $ v \frac{dv}{dx} = -g \frac{a^2}{x^2}$

Question: A particle is projected vertically upwards from the Earth's surface. Its distance $x$ from the centre of the Earth is connected with its upwards speed $v$ by the differential ...
1
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0answers
28 views

Differentiating a matrix product

In one of the books I found that given that for a linear system $x'=Ax$, there exists a matrix $Q:=\int\limits_0^\infty B(t)dt$, where $B(t)=e^{tA^T}e^{tA}$, and $V(x) = x^T Q x$, ...
0
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0answers
13 views

Singular solutions of a system of nonlinear 2nd order ODEs

I'm faced with the following nonlinear 2nd order system of ODEs: $$ \phi''(r)+\frac{4r^3-1}{r^4-r}\phi'(r)+\frac{r^2 h(r)^2+2r(r^3-1)}{(r^3-1)^2}\phi(r)=0, \\ ...
2
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0answers
11 views

A kind of Sturm-Picone theorem?

My question is very simple: Suppose $u,v:(a,b)\subset \mathbb{R} \to \mathbb{R}^+$ solve \begin{equation} (p(x)u'(x))'=-f(x,u(x)) \end{equation} \begin{equation} (p(x)v'(x))'=-g(x,v(x)) \end{equation} ...
0
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1answer
16 views

An “extra” solution to an initial value problem

So I came up with this example when I was teaching: consider the IVP $$ y'(x) = xy-x-5y+5, y(0)=1. $$ The standard approach is to separate variables: $y'(x) = (x-5)(y-1)$, which allows me to ...
1
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2answers
22 views

no of solutions of the initial value problem?

$x \dfrac{dy}{dx} = y , y (0) = 0, x \geq 0 .$ My Approach : $\dfrac{dy}{y} = \dfrac{dx}{x},$ by variable separable method, we get $lny = ln x +c $ and then raising e to both sides will get $ ...
0
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2answers
21 views

Solve analytically a nonlinear first order ODE

How can one possibly find the general solution to the following nonlinear ODE? $\frac{dy(x)}{dx}=e^{y(x)/2}$ I tried Mathematica, which gives the solution $y(x)=-2 ln[1/2 (-x - c)]$ However I ...
1
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1answer
23 views

What is the difference between “exclusively depends” and “only depends”?

What is the difference when someone says that an expression exclusively depends on $x$ and an expression only depends $x$?
1
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1answer
38 views

solve $\frac{\partial u^2}{\partial x\partial y}=0$

I need to solve $$\frac{\partial u^2}{\partial x\partial y}=0$$ with the boundary conditions: $u(x,y=x^3)=\sin(x^6)$ and $\frac{\partial u}{\partial x}(x,y=x^3)=0$. I got a particular solution, I ...
1
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3answers
44 views

Why are these equations equal to a constant?

I am reading this part of a research paper where the author states that the left hand side of equations (12) and (13) must be equal to a constant. However I could not understand the explanation he ...
0
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3answers
28 views

Prove that the solution for $y'=y^3(1-\tan^2(\arcsin(y)))$ , $y(0)= {\pi \over 8}$ , is bounded.

I got this problem to prove, and I assume I need to use the existence and uniqueness theorem for non-linear ODE's, so I set $y' = f(x,y)$ and differentiating in respect to $y$ gives: $f_y(x,y)$. And ...
2
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2answers
39 views

Why is Laplace Transform used for ODEs

This part is taken from differential equations with applications and historical George simmons. According to the given information , there are another integral transformation.I wonder why is the ...
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0answers
26 views

How to find the required differential equation [on hold]

How to find the differential equation of tangent lines to the parabola y=x^2? How to find the differential equation of all conics whose axes coincide with axes of co ordinates? I think the equation ...
2
votes
1answer
51 views

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$?

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$? I calculated the integrating factor to be $e^x$. Then $e^x y'+ e^x y=e^x |x|$ hence $\frac {d(e^x y)}{dx}=e^x |x|$ hence $d(e^x y)=e^x|x|dx $ ...
0
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1answer
17 views

How much can I gauge about the domain of a differential equation without actually solving it?

Say I have the differential equation $$y' = \frac{3t^2 - 2ty}{4 - t^2} \text{, where }y(1)=-3$$ Clearly the equation is undefined at $t = \pm2$, and a solution exists at $t = 1$. Can I conclude from ...
3
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1answer
27 views

Algebra of Linear differential operators, question on Commutativity and Association

The following is a discussion on the following second differential equation $$ \frac{dy^2}{dx} - y = 0 $$ So, let us introduce the following, convention and definition, represent the derivative ...
1
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1answer
15 views

Solving inexact DE with multivariable integrating factor provided.

I just spent more time than I care to admit pushing through the algebra of this question and, although I arrived at the answer, I'm curious to know if there is some simplification that could be done ...
0
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1answer
33 views

Path to Self Study Calculus 1-4 and Linear Algebra [on hold]

For the past year I've taken up self studying mathematics. My initial intent was to study so that when I entered college (currently a junior) I would have most of the basic mathematics for studying ...
1
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1answer
39 views

Question about the weak solution of $u''-u=f$

I have a question about the ODE (weak formulation) given by $$u''-u=f$$ where $u\in H^1(\mathbb{R})$ and $f\in L^2(\mathbb{R})$. I want to see if there is an explicit formula for the weak solution. ...
-1
votes
0answers
31 views

Differential equation $2xy-\sin(x)+(x^2+e^x)y'=0$ [on hold]

can i get help with this ED? $$2xy-\sin(x)+(x^2+e^x)y'=0$$ Thanks in advance
-2
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0answers
21 views

Exponential differential equation

I need help with this equation: $$ g( x ) \exp({f' ( x ) }) = 1-\exp\left(f\left(\frac{x}{2}\right)\right)$$ I can't see an applicable method for that one.
0
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4answers
51 views

Show the given series is a solution of $y''-xy'-y=0$

My problem is this: "Show that the function represented by the power series, $$y=\sum_{n=0}^{\infty} \frac{x^{2n}}{2^nn!}=1+ \frac{x^2}{2}+ \frac{x^4}{8}+ \frac{x^6}{48}+...$$ is a solution of the ...
3
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0answers
45 views

Resolving ODE-1 $(x^2 + y^2 +x)\,dx + xy\,dy=0$ am I wrong or my teacher is?

This is how I've resolve this ODE-1 : $$(x^2 +y^2 +x) \, dx + xy \, dy=0$$ Check if the eq is exact: $${\partial M \over \partial y}={\partial \over \partial y}(x^2 +y^2 x)=2y$$ $${\partial N \over ...
1
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2answers
26 views

Another Differential Equation

Having trouble (again) with this DE can someone help me find the general solution for it? I feel like my biggest problem is doing the algebraic manipulations to identify what kind of DE it is. $$y' + ...
1
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2answers
30 views

Differential Equation $y' = 2y/x - 1$

Can I get help solving this DE? $$ y' = \frac 2xy - 1$$ Doesn't look too hard but i just can't get to the correct result. Thank you in advance
1
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0answers
35 views

Acceleration of an air bubble under the sea

An air bubble arises from the bottom of the sea. Find its acceleration if the resistance force is proportional to $\rho$*A*$v$ where $\rho$ is density of water, A is cross section area and $v$ is ...
0
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0answers
15 views

Show that the function is identically Zero in certain subset

We are given a open ball D (radius = 1) in $\mathbb R^2$. and let $\{x_n\}$ be the dense sequence in the set D. Around each point $x_n$ we make a hole of radius $r_n$. The sequence $r_n$ satisfy the ...
1
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2answers
62 views

Unable to solve $y''+\lambda y =0$

I wish to find the eigenvalues and eigenfunctions of the following, but am unable to and further don't know what I am doing wrong at all $y''+\lambda y =0$ where $y(0)=0$, $y'(1)+y(1)=0$ My ...
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2answers
47 views

homogeneous differential equations $y' = f(y/x)$

There is a weird Theorem that comes about when considering whether a function is homogeneous (in the sense of the title definition). I was unable to prove it, or to find a proof to it. Can any one ...
3
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3answers
63 views

solution of differential equation $\left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$

The solution of differential equation $\displaystyle \left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$ $\bf{My\; Try::}$ Let $\displaystyle \frac{dy}{dx} = t\;,$ Then Diferential equation convert ...
0
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0answers
19 views

Identifying the ODE systems that form a stable solution and how to state transition in Matlab

does anyone know what the name of this ODE system is called? \begin{bmatrix} \epsilon & -\alpha \\ 1 & \epsilon \\ \end{bmatrix} \begin{bmatrix} \epsilon & -1 \\ \alpha ...
20
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2answers
673 views

Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
0
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1answer
16 views

How to solve a boundary value problem of a Laplace equation?

Suppose $x,y$ are in the range $0 \leqslant x \leqslant 2,0 \leqslant y \leqslant 1$, I can use separation of variables to get $\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ...
0
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0answers
22 views

Does a Liapunov function h to have all the variables explicitly?

If I have for example, a system like this $$\begin{matrix} \dot{x}=f(x,y) & \\ \dot{y}=g(x,y) & \end{matrix}$$ in which i have to prove stability using a Lyapunov function. Now, if i have ...
3
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3answers
41 views

A simple problem on first order differential equations

An ODE (Ordinary Differential Equation) of order $n$ becomes a relation: $$F(x,y,y^{(1)},...,y^{(n)})=0$$ Then $F(x,y,y^{(1)})=0$ defines an ODE of order one. In "basic standard texts", for purposes ...
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0answers
29 views

perturbation of exponentiolly stable system

consider the following system on $\Bbb{R}^n$ $\dot{x} = f(x,t)+g(x,t) $$ $$ $$ $ $ (*) $ assume that f(0,t)=g(0,t) = 0 and 1. 0 is an exponentiolly stable equilibrium of $\dot{x}=f(x,t)$ ...
0
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1answer
30 views

First Order Differential Equation for a Harmonic Oscillator

A box with mass $m$ is attached to a spring with spring coefficient $k$. This system is then placed into a glass case filled with a liquid with drag coefficient $\alpha$. Now I have the following ...
2
votes
1answer
28 views

Differential equation$ (x^2-x)y' = (y^2+y)$

Can i get help solving the differential equation $$y' = \frac{y^2+y }{x^2 -x}$$ I tried searching but could not find anything similar. Thank you!
0
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1answer
12 views

Maximally extended solution of this ODE.

So I am asked to find the positive, maximally extended solutions to this ODE. $$u'(x) = \frac{x}{u(x)}$$ Now a solution is given by $$u(x) = (\int_{y_0}^y t dt )^{-1}\circ \int_{x_0}^x s ds = ...
1
vote
1answer
39 views

Solve $A \partial_t w + B \partial_t\partial_x^4 w + C \partial_x^4 w + \partial_t^2 w = 0$

a non-mathematician wants me to solve a PDE. The problem is that I don't know a lot of theory to solve PDE's except the fouriertransform. This is the PDE $$A \partial_t w + B \partial_t\partial_x^4 w ...
0
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1answer
27 views

Approaches to stability of newtonian systems

I am having some difficulties figuring out how to approach "Test stability problems". I usually test the linearization of the system (since it is very straightforward and easy), and if that doesn't ...
1
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1answer
50 views

When does the same trajectory appear in two dynamic systems from the same point?

Imagine you have two dynamical systems, given by the statespace equations: $\frac{dx}{dt}=F_1(x)$ and $\frac{dx}{dt}=F_2(x)$, and you are concerned with trajectories form a point in phase space $x_0$. ...
1
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3answers
34 views

Differential equation $2f'(t)+tf(t)=0$ with $f(0)=\sqrt{\pi}$.

How to solve the following differential equation: $$2f'(t)+tf(t)=0$$ with $f(0)=\sqrt{\pi}$. I tried to write $2f'(t)+tf(t)=0$ something like $(f(t)g(t))'=0$ for some function $g$ but it was ...