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

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1answer
15 views

Is there a close-form solution for the non-linear difference equation?

is there a close-form solution for the difference equation below? $$(x_{n+2}-x_{n+1})-(x_{n+1}-x_n)=(\frac{x_{n+1}}{c})(x_{n+2}-x_n)$$ Any comments are appreciated.
0
votes
0answers
8 views

Dermining stable and unstable manifolds - is my result ok?

Determine all stable and unstable manifolds of the equilibria of $$ \dot{x}=x(1+x)(1-x). $$ Are there homoclinic/ heteroclinic solutions? Hey, just would like to know if I am ...
0
votes
0answers
9 views

Quasi-linear partial differential equations. Solving them.

This is what I have as a quasi-linear partial differential equation:$$u(x_1,...,x_n), \ \ \ \ \sum_{i=1}^{n}A_i(X,u) \frac{\partial u}{\partial x_i}=A_{n+1}(X,u) \ \ \ (1)$$ Then it says let ...
1
vote
2answers
25 views

How can I solve a first order ODE with $\pm$ signs by the Integrating Factor method?

I have the following first order ODE to be solved via the integrating factor method: $$\frac{\mathrm{d}z}{\mathrm{d}y}\pm z=-\frac12y\tag{1}$$ This is in the general form: ...
2
votes
0answers
15 views

Methods for first order PDEs in higher dimensions

What are the possible known methods for solving first order PDEs in higher dimensions? Is there anything else besides the method of characteristic curves? In particular, I have four first order, ...
1
vote
1answer
12 views

Show exponential stability quadratic form

Please help me with the following proof: Suppose $\dot x=f(x(t))$ and suppose that we have: $$ \frac{d}{dt}\left( x(t)^TPx(t) \right)\le -x(t)^TQx(t) $$ where $P$ and $Q$ are symmetric ...
0
votes
0answers
25 views

If a and b are negative , then can we use the same method we are taught for solving the equation y'' + ay' + by=0 ,

If $a$ and $b$ are negative , then can we use the same method that we are taught for solving the ODE which is $$y'' + ay' + by=0$$
1
vote
1answer
17 views

Solving Laguerre coefficients with Integral?

I'm having some difficulty understanding the solution to a particular Laguerre expansion. The problem reads "Expand the term $ e^{-x}$ as a Laguerre expansion, noting the orthogonality of $$ < ...
1
vote
0answers
29 views

Transient Terms in a General Solution

Find the general solution of the given differential equation: $$ (x^2-4)(\frac{dy}{dx}) +4y = (x+2)^2 $$ I found the general solution of the D.E and I got the following correct solution: $$ y = ...
-3
votes
1answer
23 views

PDE Proof that a linear combination of 2 solutions is also a solution [on hold]

Can someone please help? I've been trying to figure this for a few days now. Consider the first order PDE: $au_t + bu_x$ = 0, where a and b are constants. Show that if $u_1$ and $u_2$ are solutions ...
0
votes
0answers
13 views

How to integrate to solve a PDE with mixed partials in the integrand

Problem Statement: Determine the equlibrium temperature distribution inside a circular annulus $r_1\leq r \leq r_2$. If the outer radius is at temperature $T_2$ and inner radius at temp $T_1$. So ...
0
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0answers
16 views

What is the largest t-interval on which guarantees a unique solution? [on hold]

What is the largest t-interval on which guarantees a unique solution for this equation? $$y'' + y'+ 3ty = \tan t,\quad y(\pi) = 1,\quad y'(\pi) = -1$$
1
vote
0answers
26 views

Solving $\vec y'=\begin{pmatrix} 1 & -1 \\ 0 & 1\\ \end{pmatrix}\vec y$

I need to solve $\vec y'=\begin{pmatrix} 1 & -1 \\ 0 & 1\\ \end{pmatrix}\vec y$. The characteristic polynomial is $(r-1)^2$, so the only eingenvalue is $1$. I found ...
0
votes
1answer
27 views

Showing that if $y(x)$ is a solution, then $y(-x)$ is also a solution for a specific ODE

Given the ODE $(1-x^2)y''-xy'+\alpha^2 y=0$, I need to show that if $y(x)$ is a solution, then $y(-x)$ is also a solution. From what I understand, because $y(0)=y(-0)$, it means that all solutions are ...
1
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0answers
32 views

Different answer when using the 'method of undetermined coefficients' compared to Laplace transform

I have an ordinary differential equation: $$ \frac{\mathrm{d}^2u}{\mathrm{d}t^2} + u = \mathrm{e}^{-t}\cos(t)$$ with $u(0) = u_0$ and $\dot{u}(0) = v_0$, when using the method of undetermined ...
0
votes
1answer
20 views

How to calculate the variation of a matrix?

Suppose we have two diagonal matrices $$ A_{\mu \nu}=\left(\begin{array}{cccc} \rho(t) & 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0 ...
1
vote
1answer
17 views

Wronskian of two independent solutions equaling zero at a specific point only?

Given $y_1(x)=\sin(x^2)$ and $y_2(x)=\cos(x^2)$, I constructed a linear, homogenic ODE of order 2 by solving: $$ \begin{vmatrix} y & y_1 & y_2 \\ y' & y_1' & ...
1
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0answers
19 views

Find all the solutions of the initial value problem for a first order non-linear equation

I am trying to solve the initial value problem: $$ y'= \frac{10}{3}xy^{2/5}, \qquad y(0)=0 \qquad \qquad (1) $$ where $ x\in \mathbb{R} $. The first order equation is not linear in the form: $$ ...
0
votes
1answer
26 views

What is the dimension of set of all solutions to $y''+ay'+by=0$?

$$y''+ay'+by=0,\quad y(0)=y(1)$$ where $a$ and $b$ are positive real numbers. Let $V$ be the set of all the solutions of this equation. Then the dimension of $V$? The answer to it that I think is ...
0
votes
1answer
23 views

Non-monotonically decreasing flow whose limit is $\vec{0}$

I'm trying to come up with $x'=Ax$, which is a system of linear differential equations, whose flow satisfies $\lim\limits_{t\to\infty} \lvert e^{tA}x\lvert = 0$ for all $x\in \mathbb{R}^n$, but ...
0
votes
1answer
36 views

Compare analytic model with numerical, mass spring system.

So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance. I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and ...
0
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1answer
32 views

Why do difference of squares partial fractions have to be decomposed this way?

Why do you have to factor out $-1$ here? $$\frac{2000}{(10-h)(10+h)}$$$$=\frac{A}{10-h}+\frac{B}{10+h}$$ Decomposing this finds A annd B to be 100, which is wrong. Symbolab and Wolfram Alpha factor ...
0
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0answers
21 views

method of characteristics for non-linear PDE

I'm trying to solve the PDE $u_x^2-u_y^2=8u$ with initial conditions $u(x,x)=f(x)$. I have that $F(x,y,u,p,q)=p^2-q^2-8u$, with $p=u_x, q=u_y$, and then \begin{equation*} \begin{array}{ll} ...
0
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0answers
17 views

Peano theorem expression. [on hold]

How is the Peano theorem expressed in analogous terms to the Lipschitz condition: $|f(x,y_2)-f(x,y_1)| > M|y_2-y_1|?$
-1
votes
0answers
33 views

Let $y$ be the solution of $y^\prime +y= \mid x\mid,~~x \in \mathbb{R},~~ y(-1)=0$ then $y(1)=$ [on hold]

Let $y$ be the solution of $y^\prime +y= \mid x\mid,~~x \in \mathbb{R},~~ y(-1)=0$. Then $y(1)=$ (a) $ \frac{2}{e}- \frac{2}{e^2}$ (b) $ \frac{2}{e}- 2e^2$ (c) $2- \frac{2}{e}$ (d) $2-2e$
0
votes
2answers
37 views

Solution of differential equation with complex coefficients

Given $$\dfrac{d^2y}{dx^2}-(3-2i)y=0,\quad y(0)=1,\quad y(x\rightarrow \infty)\rightarrow 0$$ then what is $y(\pi)$ ? The answer given is $-e^{-\pi}$. But I cannot understand how its solution can ...
0
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0answers
37 views

Solve this system of nonlinear differential equations

Although I have a brief understanding of solving linear ODE systems, I got stuck with this non-linear system: \begin{align*} \left(x'(t) \right)^2 + k_1 x + k_1 y = 0\\ \left(y'(t) \right)^2 + k_2 x ...
-4
votes
0answers
52 views

solve $4x^2y'' + y=0$, $y(-1)=2, y'(-1)=4$

This would require taking the $\ln(-1)$, which Zill solved in the 7th edition of diff eq $4.7$ problem $37$ by substituting $t$ for $x, y(1)=2, y'(1)=4$. Then substituting $-x$ for $t$ in the final ...
3
votes
1answer
31 views

Solve the differential equation : $0.5 \frac{dy}{dx}=4.9-0.1y^2$

The question is to solve the differential equation : $$0.5 \frac{dy}{dx}=4.9-0.1y^2$$ What I have attempted: $$0.5 \frac{dy}{dx}=4.9-0.1y^2$$ $$ \frac{dy}{dx} = \frac{4.9-0.1y^2}{0.5} ...
0
votes
1answer
28 views

System of differential equation (Matrix form)

I'm trying to solve this system $$ M\ddot{X}(t) = KX(t) $$ where M is a known diagonal matrix and K is a symmetrical known matrix. I'm asked to do the ansatz $Y(t) = M^{1/2}X(t)$ where $M^{1/2} = ...
1
vote
1answer
25 views

A specific question regarding a proof in Hassan Khalil's book, Nonlinear Systems

I am trying to understand the proof of a Lemma in the book 'Nonlinear Systems' by Hasaan Khalil (3rd edition). In the Proof of Lemma 3.1, about Lipschitz continuity of vector valued functions, I am ...
0
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2answers
24 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
votes
1answer
29 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 ...
0
votes
2answers
43 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
votes
3answers
37 views

How might I go about forming a general solution for the following differential equation? [on hold]

$\frac{df}{dt}+t^kf=t^k$ 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.
0
votes
1answer
23 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
votes
2answers
37 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
37 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
votes
1answer
40 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
votes
1answer
39 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))'=-q(x)f(u(x)) \end{equation} \begin{equation} (p(x)v'(x))'=-r(x)g(v(x)) ...
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
29 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
25 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
42 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
31 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
42 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
30 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
60 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 $ ...