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

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1answer
6 views

linear homogeneous constant coefficient systems

Solve the following LHCC system by finding the eigenvalues, eigenvectors and generalised eigenvectors. Give a fundamental set of solutions and show that the set is independent. $$x'= \left[ ...
0
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0answers
6 views

Ordinary point of a Bessel DE

The Bessel DE: $$z^2\frac{\text d^2f}{\text{d}z^2}+z\frac{\text{d}f}{\text{d}z}+\left(z^2-m^2\right)f = 0.$$ The Bessel DE can be rewritten as: $$\frac{d^2f}{\text{dz}^2} + a(z)\frac{df}{ dz } + ...
0
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1answer
8 views

Reducing a Bessel's differential equation to a more 'useable' form

Suppose the given equation is: $$r^2\frac{\text d^2f}{\text{d}r^2}+r\frac{\text{d}f}{\text{d}r}+(\lambda r^2-m^2)f = 0$$ My text demonstrates the following: Let $$\text{z = }\sqrt{\lambda }r$$ So ...
1
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1answer
18 views

Question in regard to solving for inverse laplace transform

I am having some confusion when it comes to solving for the inverse laplace transform. ( We are allowed the tables with the common values by the way). Il give an example. Take, ...
0
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1answer
23 views

Differential equation for the logistic map

From the Wikipedia article on the logistic map I find the following definition as a recurrence relation: $$x_{n+1} = rx_n(1 - x_n) \tag{1} $$ Then, in another article, I see how to derive from this ...
0
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1answer
24 views

Change of variable of system of ODE [on hold]

I have one problem with the change of variables of this system: \begin{cases} 2y’ + z’ –y + 2z = 0 \\ y’ + 3z’ –3y +z = 0 \end{cases} with initial values $y(0) = 1$, $z(0) = 0$ I've made this ...
0
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0answers
15 views

System of ordinary differential equations, Fundamental Matrix

Let $\Phi(x,x_0)$ be a principal fundamental matrix of the system: $$u'=A(x)u$$ in an interval J. i.e. $$\frac{\partial \Phi(x,x_0)}{\partial x}=A(x)\Phi(x), \Phi(x_0)=I $$ Prove that: ...
2
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2answers
26 views

Proving a differential equation is a circle

So, I have solved the differential equation, to find the general solution of: $$\frac{y^2}{2} = 2x - \frac{x^2}{2} + c$$ I am told that is passes through the point $(4,2)$. Using this information, ...
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2answers
14 views

Differential equation maximal interval and solution [on hold]

Consider the differential equation $y' = 1 - y^2$. First, is $y(x) = 1$ the only constant solution? I now want to solve the equation for the initial value problem $y(0) = y_0$, with $y_0 > 1$. ...
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0answers
16 views

How to solve the linear bi-harmonic equation using a fourier transformation?

Let $D$ be and interval in $\mathbb{R}$ or a rectangle in $\mathbb{R}^2$, e.g. $D = [0, d_1] \times [0, d_2]$. For given $f : D \to \mathbb{R}$ With $\Delta = {\partial^2 \over \partial x_1^2} + ...
2
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1answer
27 views

Differential equations application problem

I am studying differential equations, and I saw this interesting problem in another question (here): A destroyer is hunting a submarine in a dense fog. The fog lifts for a moment, discloses the ...
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0answers
17 views

Solving 2nd order ODE with 2 independent parameters(over finite intervals), with bounds on solution

I have a 2nd order ODE of the form: $\ddot {x} + 2c \dot {x} + 39Ex = 0 $ $Initial$ conditions being: x(0) = 0 and $\dot {x}(0)$=0.1 Where c is in the interval [1,5] and E is in the interval ...
1
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0answers
11 views

Characteristics and additional conditions for differential equation

I need to solve such a DE: $$(1+x^2)u_x+u_y=0$$ And then I need to draw its characteristics. The second part of the task says: Write three additional conditions such that this equation: Has one ...
2
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1answer
40 views

Explicit solution of parametric solutions of an ODE

I need to find the explicit solution of the following ODE: $y'+\sin y'=x$, $y=y(x)$. I have found these two parametric solutions: $x=t+\sin t$ and $y=\frac{t^2}{2}+t\sin t+\cos t+c$, $c\in\Bbb R$. ...
0
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0answers
10 views

What is the definition of ``2nd-order quasilinear parabolic'' ? for partial differential systems?

I have to know why the mean curvature flows are 2nd-order quasilinear parabolic. Let $\Omega\subset\mathbb{R}^n$ be a bonded domain (or a smooth manifold of $n$ dimensional) and $N\geq 2$. When the ...
2
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1answer
25 views

What is meant by a linear SDE?

I am sure this is a ridiculous question, but I can't seem to find a definition. I know the definition of linear ODE or PDE just by saying that the differential operator should be linear, but how does ...
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0answers
20 views

Light attenuation through water at an angle

I know that light intensity decreases exponentially governed by \begin{equation*} \frac{dy}{dx} = -ky \end{equation*} where $y$ is the intensity and $x$ is the distance. Now what happens when light ...
0
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0answers
21 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
votes
1answer
49 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
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1answer
20 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?
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2answers
32 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 ...
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0answers
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0
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1answer
34 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} ...
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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 ...
3
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1answer
18 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 ...
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0answers
9 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
55 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 ...
0
votes
2answers
46 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 ...
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1answer
26 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
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1answer
26 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 ...
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0answers
38 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. ...
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1answer
31 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
35 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 ...
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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 ...
6
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0answers
45 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
36 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
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3answers
76 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
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1answer
27 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
28 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.
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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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votes
1answer
25 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 $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...
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0answers
11 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
28 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.
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0answers
23 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} - ...