Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.
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25 views
Entertaining video lectures of differential equation
this calculus lecture on coursera.com develops my interest in mathematics. I was very scared of math but after taking lectures of this course I just fall in love with math. I Now have Differential ...
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13 views
Differential Equations homogenous same degree with $\sec(y/x)$
How do you solve this equation?
$$\left(x\sec\frac yx-y\right)dx + xdy=0$$
They are homogeneous of the same degree but I don't know if I should use that method or not...
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0answers
44 views
Numerical or exact solutions of a system of differential equations
I need to solve the following system of differential equations:
$$x' + 3ax\sqrt{x+y} = -b\sqrt{xy}$$
$$y' + 3cy\sqrt{x+y} = b\sqrt{xy}$$
where $a, b, c$ are constant.
I appreciate any help.
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35 views
Hodograph transformation and implicit solution of a non-linear PDE
I am trying to understand how can one apply the Hodograph transformation to a non-linear PDE. I read that this transformation implies the representation of the solution in the implicit form . So, if I ...
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0answers
29 views
Solution of a ODE made by a series
Given a function $x(t)\in C^N$ and a linear differential equation:
$$\sum_{k=0}^Na_k\frac{d^k}{dt^k}x(t)=0$$
where the coefficients $a_k\in R$
is it possible to find the solution of this ODE vs. $N$?
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40 views
Existence of a unique solution with given initial value problems.
Directions: Find an interval centered about $x = 0$ for which the given initial-value problem has a unique solution.
$$(x - 2)y'' + 3y = x$$
Initial values: $y(0) = 0,\,\,y'(0) = 1 $.
My answer ...
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33 views
Importance of Riemann-Liouville fractional derivative from historical point of view
Why Riemann-Liouville fractional derivative is important from historical point of view than that of Caputo fractional derivative? As we know Riemann-Liouville fractional derivative is more theoretical ...
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38 views
van der pol and liapunov
i have attempted this question and done as much as i possibly could, any help regarding this question would be very helpful and appreciated.
a) show that the second-order differential equation for ...
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0answers
40 views
Cohomologies of $\mathbb R^n$ with rational differential forms
We can consider de Rham complex $0 \to \Omega^1 \to \Omega ^ 2 \to...$ on $\mathbb R^n$, where $\Omega ^r$ are $r$-forms on $\mathbb R^n$ with rational coefficients. What are homologies of this ...
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64 views
Solving a Forced Oscillations Differential Equation Problem
A building consists of two floors. The fi rst floor is attached rigidly to the ground, and the second floor is of mass m = 1000 slugs (fps units) and weighs 16 tons (32,000 lb). The elastic frame of ...
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102 views
Proof Strategy for a Dynamical System of Points on the Plane
I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
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102 views
Kernel of Fractional Differential Operator
Suppose we have a fractional differential equation:
$$\left[D^{nv}+a_{1}D^{\left(n-1\right)v}+\dots+a_{n}D^{0}\right]y(t)=0$$
where $\nu=\frac{1}{q}$ and $q\in\mathbb{N}$ and y is an analytic ...
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77 views
Solving ODE numerically with central difference quotient
I try to understand an old Fortran code that is not well documented. In this code the ODE
$$
\frac{dy}{dx} = -\frac{B(x - y)}{y}
$$
is solved numerically as an initial value problem from $x_0=0.99$, ...
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0answers
36 views
Continuity of the inverse Laplace Transform
If I know $Y(s)$,
can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions?
For example; I'm solving an ODE with the Laplace ...
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0answers
30 views
Find $N$ independent solutions through Laplace Transform
The Laplace Transform method gives us one solution of an Ordinary Differential Equation.
How can we use the same procedure to get $N$ independent solutions, being $N$ the order of the ODE?
Where can ...
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0answers
28 views
Monotonicity of Poincaré's Map
thanks for reading.
Consider a one-dimensional dynamical system $\dot{x} = f(t,x)$. Let's call $\phi(t,t_0,x_0)$ the solution passing through $x_0$ at time $t_0$ (where $t$ is the time argument of ...
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23 views
Unique continuation for elliptic operators
Consider the following system of linear elliptic equations:
$ \Delta s_i = \sum_{j=1}^{d} l_{ij} s_j $ for $ i=1,\ldots d $ where $ l_{ij} = l_{ji} $.
It should be true that the following unique ...
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0answers
43 views
Solution of Schrodinger's equation driving by a Heaviside function
Consider the the following equation with initial conditions:
$$
y''(t)+y'(t) \left(-\frac{f'(t,\tau)}{f (t,\tau)}+i~\omega \right)+y(t)~(f(t,\tau))^2=0\\
y(0,\tau)=0\\
y'(0,\tau)=1
$$
and
$$
...
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0answers
114 views
Questions about the Picard–Lindelöf theorem for an ODE
In the sketch proof of Picard–Lindelöf existence theorem for an
ODE:
It can then be shown, by using the Banach fixed point theorem, that the sequence of "Picard iterates" $φ_k$ is convergent and ...
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0answers
36 views
Constant of Integration of the Integrating Factor
I was reading my notes and at some point I write that in solving a first order linear DE using the integrating factor the final solution should have only one arvitrary constant arising from the ...
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76 views
Relating terms in differential equation with power series
Having problems with a task on a differential equation containing a power series.
Given
$$\frac{dx}{dt} = \lambda x + \sum_{n=2}^\infty b_n x^n$$
$$\frac{dy}{dt} = \lambda y$$
$$x(y) = y + ...
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21 views
Finding the linear representation of an ARMA process
I have the following $ARMA(2,2)$ process:
$X_t + 0.9X_{t-1} = Z_t + 1.3Z_{t-1}$
I'd like to write it in the form:
$X_t = \sum\limits_{i=0}^\infty\psi_iZ_{t-i}$
I just want to compute the first few ...
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0answers
36 views
unique holomorphic solution
Show that for any locally defined holomorphic function $g$ at $0$ $\in \mathbb{C}$, there is a unique holomorphic solution $f$ of the equation $f'=g \circ f$ on $D(0,r)$ with $f(0)=0$ for any ...
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99 views
George Simmons' “Differential Equations with Applications and Historical Notes” vs. “Differential Equations: Theory, Technique, and Practice”
I've heard much acclaim for George F. Simmons' "Differential Equations with Applications and Historical Notes" (2nd edition). I've noticed there's a newer book by Simmons and Krantz entitled ...
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34 views
Solutions of linear ODE with quadratic coefficients (reference request)
I am interested in the linear differential equation:
$$ \dot{x}(t) = (A t^2 + Bt + C) x(t)$$
where $A, B, C \in \mathbb{R}_{n \times n}$ and $x(t) \in \mathbb{R}^n$. Does anyone know about the ...
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60 views
Sturm-Liouville eigenvalues and the zeroes of a Bessel function
HINT ONLY PLEASE
I'm trying to show that the eigenvalues of the Sturm-Liouville problem
$xu''(x) + u'(x) + \lambda xu(x) = 0$
$0<x<1$
with $u(x)$ bounded as $x \to \infty$ and $u(1) = 0$, ...
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0answers
32 views
Stability Analysis, ODE, and Numerical Methods
Given the three equations, carry out stability analysis
This was an exam question I got wrong and still don't know how to do this question.
Is it possible someone can write the steps so that I can ...
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0answers
40 views
Planar circular restricted 3-body problem
Hi and sorry for my bad English, it's not my first language. I'm trying to find the equations of motion of the planar circular restricted 3-body problem. I did the gravitational force, but I have some ...
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62 views
Solving a Sturm-Liouville differential equation variationally
This is a problem from Haim Brezis' functional analysis book (Exercise 8.41). I solved parts of it, but am stuck on some parts/want confirmation on the method. The problem is as follows:
Let $q ...
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231 views
Simultaneously solution of four nonlinear ODE differential equations in MATLAB???
I want to solve the four nonlinear ODE differential equations simultaneously in MATLAB.
and I must tell that my unknown function is a matrix so how can I solve these equation in Matlab???
Thanks for ...
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54 views
Sturm-Liouville eigen value problem with one-dimensional eigenspace
Let $p\in C^1([0,1])$ with $p>0$ $\forall x\in[0,1]$ and $q\in C([0,1])$. Define the operator $L: C^2([0,1])\rightarrow C([0,1])$ by
$$
Lu = -(pu')' + qu,
$$
and define $L_{\lambda} = L-\lambda I$. ...
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54 views
Rescaling an ODE
I have a Cauchy problem
$$ \left\{ \begin{array}{l} \dot{x} = f(x) \\ x(0) = x_0 \end{array} \right. $$
with $x = x(t)$. Suppose to have $y(t) = a x\left(\frac{t}{b}\right)$, where $a$ and $b$ are ...
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114 views
Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$
Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...
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352 views
Replace the second-order ODE as two first-order ODEs and use Runge-Kutta method to solve
Replace the following second-order ODE by a system of two first-order ODEs. After writing the two first-order ODEs with the proper initial conditions, compute the corresponding values by using ...
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0answers
70 views
Unusual 2nd order inhomogeneous equation..
For some research Im doing, I've derived an equation of the form below for $C(r)$
$$C'' + \frac{2}{r}C' = W + \frac{f}{C}$$
Or, if you prefer,
$$CC'' + \frac{2}{r}CC' - W\cdot C = f$$
This has the ...
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0answers
51 views
Block Diagram Algebra - Sine Function
What is the transfer function of a system which computes the sine of its input (i.e. with input = $\theta$, output = $\sin (\theta)$?)
We have only been using linear systems so far in class.
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35 views
Generating Function for rational sequence
I'm trying to compute the generating function for the function defined for $1\le N < \beta$, $C_N = \frac{\beta}{N(\beta-N)}$. I think my math so far is correct, but I don't know how to solve the ...
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0answers
83 views
Whats the purpose : Hilbert's problems in measure space
This may sound a very newbie question, anyway I would like to ask here and to make it more clear for me.
I've got an assignment to consider boundary problems in space of finite measures W, where the ...
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0answers
102 views
finding an integrating factor for the nonlinear non-autonomous ODE $ (xy)y'+y\ln y - 2xy = 0 $
I though i might refresh my ODE knowledge a bit and decided to try the following exercise from "Advanced mathematical methods for scientists and engineers" by C. Bender and S. Orszag
Solve the ...
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22 views
Boundaries- regularity and local parametrization
Suppose we have a bounded domain $\Omega \subset \mathbb{R}^3$ with $C^2$ boundary.Let $x_0 \in \partial \Omega$. We choose a $X_1,x_2,x_3$- coordinate system such that the $x_1,x_2$-plane is ...
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18 views
Cost Function Neural Network With Weight derivation
If you have this cost function of a single-layerd neural network with a sigmoid function. I have figure out how to differentiate the left side of the + but the left side I need some help. a is a ...
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0answers
59 views
Invariant Subspaces and Differential Equations
Given
I'm given a marginally stable system, $\dot{x}(t)=Ax(t)$, where$A=\begin{bmatrix} -1 & -10 & -10\cr 1 & 0 & 0\cr 0 & 1 & 0 \end{bmatrix}$, and $x(0)=x_o$.The eigenvalues ...
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0answers
46 views
ODE: continuous dependence on parameters
Is it true that the solutions of the problem:
$$\begin{cases} \frac{\text{d}}{\text{d} s} [s^{2-2/N} u^\prime (s)] + \frac{\lambda}{c_N^2}\ u(s)=0 \\
u(\bar{s})=1\\ u^\prime ...
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0answers
96 views
Solving $y''(x)=-\frac{k}{y^2}$
I was trying to solve this second-order, autonomous, nonlinear, rational equation for
$$y:I=[0,\infty)\to\mathbb{R}$$
$$y''=-\frac{k}{y^2}\qquad y(0)=p_0\quad y'(0)=v_0$$
With $k>0$ constant, for ...
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0answers
71 views
new method for solving ODE
Can you please help me with this question?
In order to build the new method for solving ODE $y '(x)=f(x,y(x))$, we use an integration formula:
$\int_{a}^{b}f(x)dx\approx (b-a)f\left ( \frac{a+b}{2} ...
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0answers
21 views
Show eigenvalues of boundary problem are real
I have the boundary problem $$(xy)''+\lambda xy=0$$ $$y(1)=0$$ $$||y'(x)||<M$$
defined on the interval [0,1], and am asked to show its eigenvalues are real.
What I have so far: I can transform ...
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0answers
85 views
How can I solve this system of equation analytically?
$$
\begin{align}
i x'(t) &= c A(t)(e^{-i q(t)}) y(t)\\
i y'(t) &= c A(t)(e^{i q(t)}) x(t)\\
\\
A(t)&=a cos(wt-r)\\
q(t)&=b+dt+k sin(wt+r)
\end{align}
$$
where all of $a,b,c,d,k,w,r$ ...
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0answers
113 views
Analytic solution at an irregular singular point of linear ODE
It is easy to construct examples of linear ODEs with a regular singular point $z_0$ and a regular at $z_0$ solution. For instance, if one of the characteristic exponents at $z_0$ is a nonnegative ...
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0answers
117 views
Cylindrical waves
I am trying to solve the general equation for cylindrical symmetric waves:
$$\frac1{c^2}\frac{\partial^2u}{\partial t^2}= \frac1r\frac{\partial}{\partial r}(r\frac{\partial}{\partial r}u)$$ with $u = ...
1
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0answers
36 views
Half-stable fixed point on a circle
On a line graph, it's clear that a half-stable fixed point is the limit of moving the unstable fixed point towards the stable fixed point. Some solutions go to infinity depending on the initial ...
