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

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1answer
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Canonical form and fundamental solution of pdf

Can someone help with these two PDE problems? Thank you. Reduce to Canonical form and find the fundamental solution if possible. $$y^2u_{xx} + x^2u_{yy} = 0.$$ What type of transformation should I ...
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2answers
33 views

Seperation of variables is not rigorous enough.

So I have started ODE's and have been told that solving initial value problems by using separation of variables(like I had been taught) is nothing but a leap of faith. Now we must use something along ...
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1answer
18 views

Ex ODE: $y'=4t \sqrt y- \lambda(y-(1+t^2)^2)$

How to solve the following equation? $y'=4t \sqrt y- \lambda(y-(1+t^2)^2)$ $y(0)=a$ Show those cases where a numerical method will solve this equation exactly. $(a,\lambda) \in {\Re}^2$
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2answers
25 views

Solution Set of System of ODE.

I am trying to find the solution of the system $$\begin{bmatrix}x_1\\x_2\end{bmatrix}'= \begin{bmatrix}1&3\\3&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$$. I am given that ...
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0answers
24 views

How to calculate inverse laplace of $e^{a\sqrt s}$?

I was using Laplace to find solutions for $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions $$u(0,t)=1 \\ u(1,t)=1 \\ u(x,0)=1+ \sin \pi x$$ I used ...
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0answers
16 views

Stability of gradient flow $x'(t) = -\nabla E(x)$, $E(x)$ is convex function.

Let $E: R^d \to R$ be convex and continously differentiable. The IVP $x'(t) = -\nabla E(x), x(0) = x_0, t>0$ is called the gradient flow. Show that gradient flow has following properties a) ...
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2answers
23 views

Explicit Euler Method's Order

I am seeing everywhere that the order of the Explicit Euler Method is 2 but I can't prove it on my own. The textbook that I found the method says that the proof is very easy so it is up to the reader ...
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1answer
29 views

How to use the 3rd and 4th boundary conditions in this?

I was solving $$ \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}$$ All the boundary conditions are as follows:- $$u(0,t)=0 \\ u(\pi ,t)=0 \\ u(x,0)=\sin x \\ u_t(x,0)=x^2$$ ...
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0answers
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Eigenvalue problem for $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue $E$ in the complex plane of $x$ for one dimensional Schrodinger equation $$ −\psi''(x) − (ix)^ N \psi(x) = E\psi(x). $$ where $N$ can be any real number, the boundary condition ...
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4answers
63 views

Why cannot $A\sin\alpha x +B\cos \alpha x$ be zero?

I was going through solving wave equations using fourier and I came across a note saying $A\sin\alpha x +B\cos \alpha x \neq 0$ I believe this applies to $\alpha ,A,B\neq 0$ I was solving $$ ...
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2answers
14 views

If $g(t)=d$ (constant), show that all solution of $L[y] \to \frac{d}{c}$ when $t \to \infty$

Let the differential equation $L[y] = a y'' + by' + cy = g(t)$, where $a$, $b$ and $c$ are strictly positive numbers. If $g(t)=d$ (constant), show that all solution of $L[y] \to \frac{d}{c}$ when $t ...
2
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1answer
59 views

Doomsday Prediction

I have a calculus problem I can't seem to figure out. Any help would be appreciated! Doomsday prediction. In $1960$, three electrical engineers at the University of Illinois published a paper in ...
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1answer
15 views

Show that $Y_1[t]- Y_2[t] \to 0$ as long as $t \to \infty$ - Differential equations

Let the differential equation $L[y] = a y'' + by' + cy = g(t)$, where $a$, $b$ and $c$ are strictly positive numbers. If $Y_1(t)$ and $Y_2(t)$ are solutions at the $L[y]$ equation, show that $Y_1[t]- ...
2
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1answer
17 views

Turing criteria for Sel'kvo glycolysis model

I have the Sel'kov reaction diffusion model for glycolysis as follows: \begin{eqnarray} u_t=D_uu_{xx}-u+av+u^2v\\ v_t=D_vv_{xx}+b-av-u^2v \end{eqnarray} How can I obtain the values for $D_u$ and ...
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0answers
12 views

Searching for second singular solution of a differential equation

I was given the following equation $\frac{dy}{dx}=\sqrt{y-4}$ and was asked to solve it and find a family of solutions. I've solved the equation so that $y=\frac{x^2+16}{4}+c$ or ...
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1answer
20 views

Separable solution to a nonlinear parabolic PDE

I seek a separable solution to the nonlinear parabolic partial differential equation, $\frac{\partial u}{\partial t} = u \frac{\partial u}{\partial x^2} + u^2.$ The physics of the problem allow ...
3
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1answer
30 views

Evolution semigroups for differential equations

I would like to ask whether "evolution semigroups" are really useful (to discover something that can't be discovered in some other way?). There is a huge machinery to deal with them, but from my point ...
0
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1answer
14 views

Finite Difference Equation with Constant Co-efficient

I trying to find tutorials on the topic (Finite Difference Equation with Constant Co-efficient) but I can't get exactly what I want. The said Difference Equation has a ...
2
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3answers
30 views

Differential equations (second edition) - William E. Boyce & Richard C. Diprima (#$31$, page $142$)

In many physical problems, non-homogeneous term may differ from one time interval to another. By example, determine the solution $y = \phi(t)$ at $$y'' + y = \begin{cases} t ...
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1answer
19 views

Stable eigenspace of $x'=Ax$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, the solution is $x(t) = \begin{bmatrix} e^{-2t} & 0 ...
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1answer
19 views

How to find isoclines of the following system?

If I had a system of equations such as: $$\dot{x}=(1+2x+y)x$$ $$\dot{y}=(4+6x+2y)y$$ How would I find the horizontal and vertical isoclines of such a system? I know ...
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0answers
21 views

$\bf{x'}=Ax$ with eigenvalues of multiplicity greater than $1$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, if I solve it by first finding matrix $\bf{P}$ and then ...
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2answers
29 views

Solving this differential equation via dividing by $t^n$

I have this differential equation that I need to solve: $y'=\frac{(x^2-y^2)}{3xy}$ My teacher gave a hint to divide everything on the right hand side by $t^n$, but I do not see how this is helpful. ...
2
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1answer
33 views

Finding Fourier series constant and integral

I have been studying Griffith's Intro to Electrodynamics. I am studying differential equations and Fourier series. I am studying the problem discussed here: Why is this allowed? ("Fourier's ...
0
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1answer
17 views

Find the population from a model

$\frac{dp}{dt} = \frac{2p}{700(7-p)}$ $p > 0$ given $p(0) = 3$ find $p(57)$ I'm able to separate the variables and get $\frac{dp}{(2p \times 700)(7-p)} = dt$ I then separate to different ...
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1answer
42 views

$x'=\cos^5(x) +1$ has unique solution defined for all $t\in \mathbb{R}$

I would appreciate if someone could please give me a hint on how to do this problem. Or where to see some examples. Unfortunately, the sources that I have do not seem to actually explain it and show ...
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0answers
9 views

Galerkin method in Sobolev space

I've got this problem to solve: Using Galerkin method, prove that there exists a weak solution of this differential equation: $$-\Delta u = a(x) \circ \triangledown u - u_t +f(x)$$ on $\Omega$ $$u ...
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1answer
25 views

Mixing problems

A tank contains $70$ kg of salt and $1000$ L of water. A solution of a concentration $0.035$ kg of salt/liter enters a tank at the rate $5$ L/min. The solution is mixed and drains from the tank at the ...
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1answer
13 views

Question about a solution of a partial differential equation by separation of variables

I'm trying to understand this text: http://www.ekayasolutions.com/UCDMath/HeatCondSphere.pdf But I'm having problem with this part: Whe have to solve: \begin{equation} \dfrac{\partial ...
2
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1answer
28 views

ODE problem with a single function but two arguments

I have been trying to solve the following ODE with no success: $$ \frac{df(x)}{dx} = -x f(x) + 4xf(2x)$$ I even tried using Maple but it seems to only accept ODE's that are function of the same ...
0
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1answer
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Parameters for existence of Solutions for Inhomogeneous BVP Equation.

I've been studying the Fredholm Alternative recently and have become stuck on a couple of questions. What values of $A$ and $B$ will give a solution for the following BVPs? I know that $Q_1$ is ...
2
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2answers
45 views

Differential equation $y' + yx - 2 = 0$ [on hold]

I'd like to see different methods to solve the differential equation $$y' + yx - 2 = 0.$$ Thank you in advance. I tried substituting $y = uv$ or $u^v$ for $u, v$ functions but none worked.
1
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1answer
19 views

Determine whether $f(x,y) = |y|^{\alpha}$ satisfies the Lipchitz condition.

Where $\alpha \in [0,1),y \in [-1,1]$ It seems to be satisfied, if I choose $\alpha = 1/2,y = 1/2,z = 1/3$ But I can't think how to begin $$||z|^\alpha-|y|^\alpha| \leq \text{???}$$ Could I have a ...
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3answers
62 views

Find the general solution of the ODE $xy′′ − y′ + 4x^3y = 0$ [on hold]

Can someone help me figure out this ODE, its driving me crazy. I dont need a full solution beacuse that would take hours but maybe just the final answer? Find the general solution of the ODE $xy′′ − ...
1
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2answers
79 views

Solving $\ddot y=\frac{1}{2}e^{-y}$

As the title states I am trying to solve $\ddot y=\frac{1}{2}e^{-y}$ with initial conditions $\dot y(t=0)=0$ and $\ddot y(t=0)=i$, I'm aware of the trick to evaluate this differential but I'm having ...
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2answers
52 views

How to solve the Sturm Liouville problem $y''-3y'+λy=0$, $y'(0)=0$, $y'(\pi)=0$?

Hi everybody I need to find the values of λ>0 and λ<0 to find the own values, I knowk that in the case λ=0 I obtain the trivial solution, but in the other cases I don`t know what to do, any help ...
0
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1answer
20 views

Find $c, M > 0$ such that $\lvert e^{tA}x_0\lvert \le Me^{ct}\lvert x_0\lvert$

In a system of differential equations $x'=Ax$, where $A$ is a constant matrix, and the equation is a sink (all eigenvalues of $A$ have negative real parts), I need to find constants $c,M>0$ such ...
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0answers
22 views

Difference Equations and Discrete Integral and Derivative

So I'm trying to learn difference equations, and the book that I'm using defines the following: The discrete derivative of a function $a_n$ of the integers is defined as: $$ D a_n = a_{n+1} - a_n $$ ...
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0answers
14 views

Undetermined coefficients when RHS is a polynomial

When applying undetermined coefficients to a non-homogeneous linear differential equation, we let $y_p=A_0+A_1x+\cdots A_n x^n$ when a degree $n$ polynomial is on the RHS. Doesn't this potentially ...
2
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1answer
36 views

Solve $(2x - 1)y'' - 4xy' + 4y = 0$

I am self-studying differential equations using MIT's publicly available materials. One problem set question asks us to first verify that $x$ is a solution to $(2x - 1)y'' - 4xy' + 4y = 0$ (with $2x ...
3
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1answer
41 views

Differential equation where Picard-Lindelöf can not be applied

My question is the following : Let $f:\mathbb{R}\to\mathbb{R}$ be continuous function and let $u:[a,b]\to\mathbb{R}$ be a $C^1$ function such that $$\forall t\in[a,b],u'(t)=f(u(t))\text{ and ...
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2answers
30 views

How to solve the differential equation $y' + \sec(x)*y = \tan(x)$

I am really struggling to solve the differential equation: $y' + \sec(x)y = \tan(x)$. If someone could point me in the right direction or give me a step by step plan it would be much appreciated! So ...
0
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2answers
32 views

Linear Stability Analysis of an ODE

Question: Find the equilibrium points of the following ODE and then use Linear Stability analysis to determine the stability. $$\frac{dy}{dt}= cy - \frac{y}{1+y^2}$$ My Attempt: I have worked out ...
1
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1answer
19 views

Confusion regarding the Cauchy-Euler equation

In my textbook it says that: An equation of the form $$a_2 x^2 \frac{\mathrm{d}^2y}{\mathrm{d}x^2}+a_1 x \frac{\mathrm{d}y}{\mathrm{d}x}+a_0 y=f(x)$$ is called ...
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2answers
27 views

Lyapunov Exponent sensitivity to initial conditions

I am plotting the Lyapunov exponent as a function of a parameter $r$ with an initial condition $x_0$. The equation looks like this: $$x_{n+1} =4rx_n (1-x_n)$$ When I try different initial conditions ...
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2answers
33 views

Find the general solution of $y'= a^{x + y}$ where y is the function

How should I approach this problem? Should I put ln logarithm on both sides and move factor of variable of lnx on right side in front of the ln, or do somehing else? I understand general solving and ...
0
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1answer
17 views

On the computation of the Hessian matrix.

I'm trying to compute the Hessian matrix of a data fit of an ODE model to some data. Below is a cut out of the instructions I'm following (which can also be found at ...
0
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1answer
23 views

Proving uniqueness using $\dfrac{\partial}{\partial y}$? [on hold]

I remember in the beginning of my undergrad linear differential equations class (while or before we were introduced to linear ODE's), we proved the uniqueness of a solution to an IVP by taking the ...
0
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1answer
39 views

Solving Secondary Linear ODE : $xy''+y'-xy=0$

The ODE is$$x\frac{d^2y}{dx^2}+\frac{dy}{dx}-xy=0$$ I thought that this equation looks very similar with Bessel's equation so I tried substitution $t=ix$. Then by $\frac{dy}{dx}=i\frac{dy}{dt}$ and ...
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1answer
22 views

differential equation with non differentiable non homogeneous part [on hold]

i am not able to solve this please if somebody could help