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

learn more… | top users | synonyms (1)

0
votes
1answer
18 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
vote
2answers
31 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
votes
2answers
23 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
vote
1answer
28 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
vote
1answer
44 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
vote
3answers
46 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
votes
3answers
33 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
votes
2answers
46 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 ...
-2
votes
0answers
31 views

How to find the required differential equation [closed]

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
64 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
votes
1answer
19 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
votes
1answer
31 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
vote
1answer
24 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
votes
1answer
43 views

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

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
vote
1answer
52 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
33 views

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

can i get help with this ED? $$2xy-\sin(x)+(x^2+e^x)y'=0$$ Thanks in advance
0
votes
4answers
56 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
votes
0answers
48 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
vote
2answers
31 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
vote
2answers
31 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
vote
0answers
41 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
votes
0answers
17 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
vote
2answers
64 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 ...
1
vote
2answers
52 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
votes
3answers
68 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 ...
35
votes
2answers
927 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
votes
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
votes
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
votes
3answers
45 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 ...
1
vote
0answers
37 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
votes
1answer
31 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
votes
1answer
14 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
43 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
votes
1answer
28 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
vote
1answer
62 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
vote
3answers
35 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 ...
0
votes
1answer
11 views

Expressing $x$ and $z$ as functions of $y$ (non-generate matrix)

Consider the system $$ \dot{x}=x-z+y^2,\quad\dot{y}=x-2y+z+y^2+2x^2z,\quad\dot{z}=-2x+2y+z^2-y^2. $$ and the equilibrium $(0,0,0)$. Now, there is used some statement that I did not know yet: ...
2
votes
1answer
35 views

Prove that if $A\neq B$ then $\exp(A/n) \neq \exp(B/n)$ for some $n\in \mathbb N$

Let $A \neq B \in M_{n\times n}$ be linear maps. I'd like to prove that there exists $n\in \mathbb N$ such that $e^{A/n} \neq e^{B/n}$. I tried assuming that $e^{\frac{A}{n}} = e^{\frac{B}{n}}$ for ...
1
vote
1answer
25 views

Why is this system reversible? What does this mean?

Consider the system $$ \dot{x}=y,\qquad\dot{y}=-x+y^2. $$ Then, it is said that the system is reversible $(t\to -t, y\to -y)$. What does this mean? If I put this into the equations, I get $$ ...
1
vote
1answer
38 views

How to prove $\frac{dh}{dt}= \frac{5 }{h^2} - \frac{1}{20}$ and a couple other related questions (complete information inside)?

This is a differential equation question. Since I might not be able to explain is well, I will attach a link to the question as well as a screenshot of the mark scheme. Question: Mark Scheme: ...
1
vote
1answer
53 views

Reachability from non-zero initial state?

I have the following system: $$ \dot x(t)=\begin{bmatrix} -2 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}x(t)+\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}u(t) $$ The ...
0
votes
0answers
13 views

capillary surface problem [closed]

Consider the capillary surface problem (⋆) )   Du  div 1 + |Du|2 = κu in Ω on∂Ω,  Dηu  1+|Du|2 =β where κ > 0, η is the outward pointing unit normal to ∂Ω and β ∈ C1(Ω) satisfies |β| ≤ 1 ...
1
vote
0answers
29 views

In a recurrence relation, how do we know which order to terminate?

By employing Frobinious or Power Series approach, we my come up with a recurrence relation that is only solvable if we set any constant lower than $a_0$ or higher than $a_n$ vanish. For example, in ...
1
vote
1answer
18 views

checking that an initial condition holds for the heat equation

I'm trying to follow a video lecture on solving the heat equation. $I) \space u_t = ku_{xx}, x \in \mathbb{R}, t > 0$ $II) \space u(x,0)=\phi (x), $ $k$ is const, $\phi (x) $ is a ...
0
votes
1answer
26 views

Wolfram Alpha Step By Step For Systems of differential equation

Does anyone know if wolfram alpha has step by step solutions for systems of differential equations? When I input them, it comes up with an answer but it does not give me the step by step solution. I ...
1
vote
1answer
38 views

Raising e to the power of both sides of an equation

I have a simple question: in differential equations, it has been common in several of my homework problems to raise a base $e$ to the power of both sides of an equation to get variables out of natural ...
0
votes
1answer
17 views

Differential equation with shifted argument.

What are the methods for solving the following class of problems: $$ \frac{df(x)}{dx}=a f(x-\xi), $$ or $$\begin{cases} \frac{\partial F(x_1,x_2)}{\partial x_1}=a_1F(x_1-\xi_{11},x_2-\xi_{12})\\ ...
0
votes
1answer
31 views

Help with this differential equation, nonlinear

How would I solve the following Differential Equation $\frac{dy}{dx}= \sqrt{x+y} $ Clearly, it is nonlinear and non homogeneous, I could not find the way to solve it with Bernoulli or to make it an ...
1
vote
0answers
26 views

Repeated Eigenvector/Eigenvalue matrix method

So I am having trouble with finding the generalized solution and I am not sure why my answer is interpreted as incorrect and I wanted to double check. $$ \overrightarrow{y'} = \begin{pmatrix} -6 ...