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

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0
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18 views

Solving this Euler's ODE

As part of a tangent of my course, I've to pick up how to solve Euler's DE on the go. I have the equation $$\text{x $\phi $'' +$\phi $'+}\text{$\lambda $x}^{-1}\text{$\phi $=0}$$ Is this in the ...
0
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0answers
34 views

Solution for a differential equation

I am stuck in getting the solution for the following non-linear differential equation: \begin{equation*} x^2 + B\frac{dx}{dt} = A\sin(wt) \end{equation*} Is there any method to solve this kind of ...
0
votes
1answer
18 views

How does the solution of ODE $y'=F(t,y)$ extend to an open interval?

I'm trying to solve the above problem from Taylor's PDE I, and I'm supposed to use compactness of $K$. But how does it work?
0
votes
2answers
26 views

First order differential equation with initial conditions

I solved the differential equation $$\frac{dy}{dx} = \frac{x}{x^2+1}$$ to get the general solution $$y = \frac{ln|x+1| +c}{2}$$ Im given the initial condition $$yy' − 2e^x = 0, y(0) = 3$$ but ...
-1
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0answers
10 views
0
votes
1answer
33 views

Limit of the function $V(x,y)=x^4-x^2+2xy+y^2$

Let $$V(x,y)=x^4-x^2+2xy+y^2$$ Consider the coupled d.e.'s:$$\frac {\mathrm d x} {\mathrm d t} = - \frac {\partial V} {\partial x}, \qquad \frac {\mathrm d y} {\mathrm d t} = - \frac {\partial V} ...
1
vote
2answers
25 views

What is the proper DE for those questions?

A tank starts with 500 liters of water with 1 kg of salt dissolved in it. A salt and water mixture with concentration 0.1 kg/L is poured into the tank at a rate of 2 L/min. The mixture is drained at 4 ...
2
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1answer
15 views

How to calculate the continuum limit of a discrete system?

The question is based on the following excerpt from the book "Symmetries and Integrability of Difference Equations" Link: Book Excerpt Consider the discrete equation ...
-2
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0answers
8 views

Spectral Differentiation using FFT on an arbitrary domain( python) [on hold]

I am trying to write a python script for spectral differentiation on a domain of arbitrary length . The function I'm trying it on is the gaussian, $f(x)=e^{-x^2}$. The program works for the domain ...
0
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1answer
48 views

How to solve de differential equation $u'' + (1+x²)u=-1$?

How to solve the differential equation: $$\frac{\mathrm{d}^2 u}{\mathrm{d} x^{2}} + (1+x^{2})u = -1$$ with $\frac{\mathrm{d}u}{\mathrm{d} x}(0) = 0$ and $u(1) = 0$. I tried Laplace and Fourier ...
-1
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1answer
24 views

The system of differential equations is in steady state

We have a system of non-homogeneous differential equations $$X'=AX+B$$ What does it mean that the system is in steady state?? $X$ is the vector $\begin{pmatrix} x_1(t) \\ x_2(t) \\ ...
2
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1answer
25 views

Non-linear differential equation I

What is the solution to the non-linear differential equation $$ \frac{d^2 y}{dx^{2}} = \left( \frac{2 y -1}{y^2 + 1} \right) \, \left( \frac{dy}{dx} \right)^2\ \text{ ?} $$ I would suspect it has a ...
-2
votes
1answer
21 views

Second differential equations MATLAB [on hold]

I'm trying to resolve this equation with ODE: $y’’ + 4y = \sin^2(2x)$; Initial condition $\to y(\pi) = 0,\ y’(\pi) = 0.$ And compare with the analytical: $$ y = –\frac{1}{6} \cos(2x)+ \frac{1}{4} ...
0
votes
1answer
24 views

Does any numerical diff.eq. solver give correct results given small step-size?

I've seen that there are less stable numerical differential equation solving methods, like using plain Euler steps $y(x+h)=y(x)+hf(x)$. For a given $h$ there are better methods. But when solving ...
2
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0answers
34 views

How to calculate Gradient of a vectorized equation.

I am solving a huge optimization problem in Matlab. I am now required to obtain gradient of objective function,and nonlinear constraint along their hessien matrix. ...
0
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1answer
27 views

system differential equation 11

The system in the symmetric form is given by $$\frac{dx}{x^2-y^2-z^2}=\frac{dy}{2xy}=\frac{dz}{2xz}.$$ Rewrite using the derivatives $$\frac{dx}{dt}=x^2-y^2-z^2,$$ $$\frac{dy}{dt}=2xy,$$ ...
3
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2answers
35 views

Find all the solutions of the differential equation

I want to find all the solutions of the differential equation $y'+2y=b(x), x \in \mathbb{R}$ where $$b(x)=\left\{\begin{matrix} 1-|x| &, |x| \leq 1 \\ \\ 0 &, |x|>1 \end{matrix}\right.$$ ...
1
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1answer
34 views

Solving second order nonhomogeneous linear equation

So i have the equation $$\frac{d^2y}{dt^2} + y = \sin(t)$$ I know the first step is to find the corresponding homogeneous equation, which i think would be: $$r^2+1=0$$ giving real roots and therefore ...
0
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0answers
20 views

Laplace vs. non-Laplace Solution of ODE

Consider the following equation: $$ y' + y = u(t - 1);\qquad y(0)=0 $$ Using Laplace transform technique one obtain the following solution: $$ y(t) = \left(1 - e^{-(t - 1)}\right)u(t - 1)$$ If we try ...
5
votes
0answers
42 views

Finding a Lyapunov function for a given system

I need to find a Lyapunov function for $(0,0)$ in the system: \begin{cases} x' = -2x^4 + y \\ y' = -2x - 2y^6 \end{cases} Graph built using this tool showed that there should be stability but not ...
2
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1answer
35 views

the solution of $\lambda u''(x) = u(x)$ is $u(x)=\{sin(n\pi x)\}_{n=1}^\infty$

in my text it says: the solution of $\lambda u''(x) = u(x)$ is $u(x)=\{sin(n\pi x)\}_{n=1}^\infty$with boundary condition u(0)=u(1)=0 how do I know that this set contains all solutions? What if their ...
4
votes
3answers
73 views

How do I solve the following differential equation

$$\frac{d^2y}{dx^2}=x^2y$$ Solving it by writing out a characteristic equation is not helping me find the solution to the above equation. Any help would be appreciated thanks.
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1answer
39 views

Working out rent? [on hold]

I have a question that I'm trying to work out and the explanation behind the answer would be great: A company has paid rent of £16,000 during the year including all this years rent and a payment for ...
1
vote
1answer
25 views

Looking for tip/procedure of series solutions to ODE

I have been having a few questions about series solutions to ODE and I found an example that can illustrate my question. It is just a simple example, say we consider the ODE $$ y''-xy'-y=0$$ around ...
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0answers
26 views

Calculating the magnetic vector potential

I've calculated A as proportional to r^2 for ra but I really don't think is correct. If someone could take me through the calculation I would really appreciate it.
-1
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1answer
24 views

Is ODE essentially different from PDE or actually PDE is the generalization of ODE? [on hold]

Is ODE essentially different from PDE or actually PDE is the generalization of ODE? If so, how are they essentially different from each other?
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1answer
23 views

Solve the initial-value problem by Laplace [on hold]

I must solve the initial-value problem $$ \begin{cases} 2y'' + 5y'- 3y = 0\\ y'(0)= 31\\ y(0)= -1 \end{cases} $$ How am I supposed to do this?
2
votes
3answers
43 views

$f(x) = x \tan^{-1}(x\ln(x))$ find $f'(e)$

$f(x) = x \tan^{-1}(x\ln(x))$ find $f'(e)$ my work $f'(x)=\tan^{-1}(x\ln(x)) *1 + x$ ---> stack here I know $\tan^{-1}(x)'= \frac{1}{1+x^2}$ so $\tan^{-1}(x\ln(x)) = ???$ I need help to solve ...
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0answers
10 views

Applying different boundary conditions in a quasi Helmholtz problem.

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, Consider the two-dimensional eigenvalue problem $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...
0
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0answers
10 views

Differentialequation with Eulers method

I have a problem with a differential equation that can be used Euler method in a digital manner. I use a program that is designed to excel. The entire task looks like this: Differential equations y ...
2
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0answers
29 views

Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= ...
0
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1answer
27 views

How to plot a phase portrait for system of differential equations in mathematica or R?

Please, help me. I'd like the phase portrait for this system: If anyone can make this portrait and post a print screen here, I would thank you very much.
0
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0answers
22 views

System of ODEs: Boundary Value Problem in Matlab [on hold]

I want to finde a numerical solution for the following 3d system of time dependent ODEs \begin{align} \dot{y}_1 &= y_3 + \frac{y_1}{20} + \frac{1}{y_3+y_1} - \frac{1}{1-y_2} -\frac{1}{y_3} - ...
2
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0answers
18 views

Solving differential equation with Fourier-series-inhomogenity

Let $\lambda$ be a real number , $(c_k)$ a complex sequence with $\mid c_k \mid \leq C(1+\mid k \mid)^{-2}$ for all k with a constant $C \geq 0 $. Find all periodic, two times differentiable ...
-3
votes
2answers
44 views

Differential equations/ 4 [on hold]

How to solve this differential equation: $$ \frac{ydy+zdz}{\sqrt{y^2+z^2}}+\frac{ydz-zdy}{y^2}=0$$ I gave similar but then nothing happens maybe this is exact differential equations?
-4
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1answer
58 views

Assume that $f(t)$ is a known continuous function on $[0,\infty)$and $\lim_{t\to\infty} f(t)=2005$ [on hold]

Assume that $f(t)$ is a known continuous function on $[0,\infty)$and $\lim_{t\to\infty} f(t)=2005$ Consider a 1st order differential equation $dy/dt + 409y = f(t)$ a)Solve and write the general ...
2
votes
1answer
46 views

Verify solution to ODE

I am given the ODE $$\left(f''(x)+\frac{f'(x)}{x} \right) \left(1+f'(x)^2 \right) = f'(x)^2f''(x)$$ and I already know that the solution to this ODE is given by $$f(x)= c \cdot arcosh \left( ...
1
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1answer
25 views

Differential Equations: Stable, Semi-Stable, and Unstable

I am trying to identify the stable, unstable, and semistable critical points for the following differential equation: $\dfrac{dy}{dt} = 4y^2 (4 - y^2)$. If I understand the definition of stable and ...
3
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2answers
60 views

How to determine generalized eigenvectors of $\begin {bmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 &1 \\ 0 & 0 & 0 & 2 \end{bmatrix}$

I want to calculate the general solution of this DE-system: $$ \frac{d \vec x}{d t}= A \vec x,\text{ with }A = \begin {bmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 ...
0
votes
1answer
22 views

Why aren't numerical solutions (Euler method) to Lotka-Volterra system (all parameters equal to 1) periodic? [on hold]

Why aren't numerical solutions (Euler method) to Lotka-Volterra system (all parameters equal to 1) periodic? Any help or just tips will be appreciated, thanks.
1
vote
1answer
45 views

how to show that all solutions tend to zero?

Here is our nonlinear first order ode: \begin{equation*} y'(t) +2y(t)+y^3(t)=e^{-t} . \end{equation*} We want to show that all solutions tend to zero as $t$ goes to infinity. Attempt: Multiply both ...
-1
votes
2answers
31 views

System differential equations 0 [on hold]

System of nonlinear differential equations $$y'= -\frac{4y}{x+4}+\frac{y^2x}{4t}, $$ $$ x'= \frac{x^2}{t^2}-\frac{9x}{t}+24 $$ help if I understood correctly you need to express $x$, but I can't
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2answers
57 views

Sane solution for an ODE with physical interpretation

I have an object which is being subjected to a continual force that is a quadratic function of the object's velocity, ie, $F=f_0+f_1 v + f_2 v^2$ for arbitrary but given constants $f_0$, $f_1$, and ...
0
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1answer
20 views

Question about phase shift on multiple-scale analysis

Consider the following ODE $$y''(t) + y(t) + \epsilon y^2(t) y'(t) = 0$$ for $t>0$ with boundary condition $y(0)=1$ and $y'(0)=0$ I have found the leading order asymptotic expansion, that is ...
-1
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0answers
30 views

first-order nonlinear ordinary differential equation0 [on hold]

How to solve this differential equation: $$(x^{2}+\ln(y))\cos(2x)+\sin(2x)(xdx+\frac{dy}{2y})=0 $$ I tried to rearrange the equation to the form $\frac{dy}{dx}$ but I couldn't.
0
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1answer
14 views

Learning spectral methods in numerical analysis

I'm trying to learn the theory about spectral methods without any specific ties to a particular program like MATLAB. I tried to search for some lecture videos but it seems very limited and I'm not ...
0
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0answers
26 views

limit problem-equation

H, I have this problem $$c^2 U''(x)=F(x),\quad U(0)=A,\quad U(\ell)=B$$ $F$ is done, and $0 < x < \ell$ I read that we must found that $$U(x) = A + (B-A)\frac{x}{\ell} + \dfrac{x}{\ell} ...
0
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3answers
38 views

Particular solution of y'' -3y' + 2y = e^t

I'm trying to find a particular solution of $$y'' -3y' + 2y = e^t$$ My fundamental set is: $$y_1 = e^{2t}\\y_2 = e^t$$ So I chose $y_p = A t e^t$, which gives me:$$y_p' = Ae^t + Ate^t\\y_p'' = 2Ae^t ...
1
vote
1answer
25 views

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$ given that $z(x,2x)=2x$. I want to explain to you how we were taught to solve these at class, and this method seemed to work with other ...
0
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0answers
11 views

Do you know how get differential equations of HSIR model of propagation malware? [on hold]

I have differential equations but I don't know how get it? thank you for help me.