Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.
0
votes
1answer
22 views
Finding the Extremals of a Functional J.
The functional $J$ is defined on smooth functions $y \colon [a,b] \to \mathbb{R}$ satisfying $y(a) = u$, $y(b) = v$ and is given by
$$J[y]=\int_a^b \sqrt{y} \sqrt{1+(y')^2}\, dx.$$
I have found ...
2
votes
2answers
23 views
Are there real numbers a and b such that $f(x,y,t) = x^a t^b$ satisfies the heat equation?
The question is in the title. The heat equation is as follows:
$$
\frac{\partial f}{\partial t} = k \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right),\quad ...
5
votes
1answer
44 views
Having trouble using eigenvectors to solve differential equations
The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix}
5 & 4 \\
-1 & 1\\
\end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\
x_2 \\ \end{pmatrix}$$
I went ...
4
votes
1answer
44 views
Linearization of $ m \dfrac{dy^2}{dt^2} = u(t) - C_d \left( \dfrac{dy}{dt} \right)^2-mg $
$$ m \frac{dy^2}{dt^2} = u(t) - C_d \left( \frac{dy}{dt} \right)^2-mg $$
where
$$\begin{align*}
y(t)&=\text{missile altitude}\\
u(t)&= \text{force}\\
m&= \text{mass}\\
C_d&= ...
0
votes
1answer
19 views
solve non linear differential equation: $y'\cdot\alpha+y+\beta\cdot e^{\delta\cdot y}+\theta = 0$
Could somebody help me to solve the non linear differential equation, where $y$ is a function of the time and starts with $y(0)=0$
$$
y'\cdot\alpha+y+\beta\cdot e^{\delta\cdot y}+\theta = 0
$$
It will ...
1
vote
1answer
14 views
How do you determine the particular solution to a non-homogeneous DE by undetermined coefficients?
I am asked to solve
$y'' +2y' = 2x + 5 -e^{-2x}$
I can find the general solution easily, but the particular solution in this case is hard to find. Here's the answer. I don't know why they got $Ax^2 ...
0
votes
1answer
23 views
Differential equation (2nd order) with divergent coefficients.
I have this equation:
$$x(x-1)y''+6x^2y'+3y=0$$
I try to get the series for the solution around $x=0$, using Frobenius (however it's written). the first solution must be of the form:
...
0
votes
2answers
51 views
Initial value problem uniqueness (Lipschitz)
Show that each of the following initial-value problems has a unique solution ($0 ≤ t ≤ 1 , y(0) = 1$).
$$y' = \exp(t-y)$$
Theorem 1: Suppose that $D=\{(t,y)|a≤t≤b, −∞< y<∞\}$ and that $f(t,y)$ ...
2
votes
1answer
35 views
Differential equations math help?
I have the equation $y' \sin x=y \ln y$. I told my teacher that we can solve if with separate variables method but he told me that we cant do that. He didn't explain why. Can you tell me why?
0
votes
1answer
35 views
Second Order Non-Linear ODE involving Bessel Functions
I'm trying to solve this but I'm getting nowhere. Does anyone know step-by-step solution? or at least the general techniques to use? I do know that the solution involves the Bessel functions.
$y'' + ...
6
votes
1answer
60 views
Nonlinear first-order differential equation with a simple parametric solution.
I have to solve the nonlinear first-order differential equation
$$\frac{a-y'}{\sqrt{1+y'^2}}e^{-a \arctan y'}=bx+c,$$
where $a,b,c$ are constants, and $y$ is a function of $x$.
Obviously, there is ...
1
vote
2answers
43 views
Proper constants for $\alpha, \beta$
Here is the problem:
For what values of $\alpha$ and $\beta$, the function
$$\mu(x,y)=x^{\alpha}y^{\beta}$$
is an integrating factor for the OE $$ydx+x(1-3x^2y^2)dy=0.$$ I am working on it just ...
0
votes
1answer
24 views
Integration factor differential equations?
I have the equation: $$2y\:\mathrm{d}x- (\ln{y}+2x-1)\:\mathrm{d}y=0$$
I have to solve this.So,I noticed that $\frac{\partial P}{\partial y}$ is different from $\frac{\partial Q}{\partial x}$ so I ...
5
votes
0answers
62 views
Invariant submanifolds
Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
3
votes
2answers
49 views
What's the difference between an initial value problem and a boundary value problem?
I don't really see the difference, because in both case we need to determine y and the values of the constants. The only difference is that we give the value of y and y' in the former and the value of ...
0
votes
1answer
45 views
Prey-predator question
Suppose we have 2 systems
i) $\dfrac{dR}{dt}=2R-1.2RF$
$\dfrac{dF}{dt}=-F+0.9 RF$
ii) $\dfrac{dR}{dt}=R(2-R)-1.2RF$
$\dfrac{dF}{dt}=-F+0.9RF$
R= Population of prey, F= population of predator.
...
1
vote
1answer
35 views
Can someone clarify this implication
I'm reading a finance book, and I saw this implication that I don't understand. I mean where this g function come from? If someone can clarify this I would appreciate. Thanks.
If a have a function ...
0
votes
0answers
38 views
Unique solution first order differential equation
I have a differential equation given by $ \frac{1}{c^2}=f(\beta)(f'(\beta)^2+1)$, where c ist a positive constant and we have that at some point $\beta'$, we have $f(\beta')=y>0$. Now the question ...
2
votes
1answer
33 views
solve $y(x)=\cos \left(y'(x)\right) + y'(x)\sin (y'(x)), y(0)=1$
solve $$y(x)=\cos (y'(x)) + y'(x)\sin (y'(x)), y(0)=1$$
with wolfram alpha I got that a solution is $y(x)=x\arcsin x+\cos (\arcsin x)$
but I have no idea how to find it.
I tried transforming into ...
1
vote
2answers
21 views
Find $y$-Lipschitz constant
$$f(x,y)=x^3e^{-xy^2}, 0\leq x\leq a, y\in \mathbb R, a>0$$
I need to find $K>0$ such that $$|f(x,y_1)-f(x, y_2)|\leq K|y_1-y_2|$$ for all $0\leq x\leq a$ and $y_1,y_2\in \mathbb R$
I did this ...
0
votes
2answers
23 views
Second order differential equation question.
Consider the equation
$\dfrac{d^2y}{d^2t}+k\cdot\dfrac{y}{m}=0$
a) let $y(t)= \cos (at)$. Under what conditions on $a$ is $y(t)$ a solution?
b) What initial condition in the yv-plane corresponds to ...
0
votes
1answer
45 views
finite difference equations
i havent had a response to this question in a while, could someone please help me. Im struggling to understand the concepts of forward/backward/central differences on finite difference equations.
i ...
1
vote
0answers
89 views
“Two-speed” linear integro-differential equation
Working on a problem of many-electron dynamics in quantum dots I have arrived to an a following integro-differential equation:
$$\frac{\partial}{\partial t} F(x,t)= - i (x+ v_1 t) F(x,t)-\alpha^2 ...
2
votes
1answer
27 views
What is the time-integral of motion for first order differential equations?
For a second order differential equation (many physical systems) in one variable, I know "procedures" to compute the energy. Given $$q''(t)=f(q(t),q'(t)),\ \ q(0)=q_0,\ \ q'(0)=v_0,$$
if we're lucky ...
1
vote
1answer
153 views
Solution of a differential matrix equation
Given a differential matrix equation, ie $X'=A(z)X+B(z)$ where both $A$ and $B$ are matrix of size $n\times n$ with coefficients that are holomorfic functions in a convex open set $\Omega$ and ...
4
votes
2answers
395 views
Matrix Differential Equation with a Skew-Symmetric Matrix
From a bank of masters exams:
Say the position of a particle moving
in $\mathbb{R}^n$ is given by a smooth
vector-valued function $\vec{x}(t)$.
Suppose that $\vec{x}(t)$ satisfies a
...
0
votes
2answers
28 views
Why does the differential equation $y' = y + 1$ have solution $y(x) = Ce^x - 1$?
I was watching a video on differential equations for a class that I'm taking. I took calculus so long ago that I can't seem to figure why the differential equation $y' = y + 1$ has solution $y(x) = ...
2
votes
3answers
67 views
Express differential equations as system of first order equations
Express the differential equation $$y'''-6y''-y'+6y=0$$
as a system of first order equations i.e. a matrix equation of the form
$$A(\vec x)'=0$$
where $$\vec x\text{ is the vector }\left[ ...
0
votes
1answer
64 views
Isolated Versus Non-Isolated Fixed Point, 2D Dynamics
I am trying to understand the classification of fixed points in a dynamical systems context (fixed points of a system of two linear differential equations are places where both $x_1' = x_2' = 0$).
...
0
votes
0answers
36 views
System of Differential Equations proof question
For the system of equations
$t \dfrac{d\vec x}{dt}=A\vec x$, where $A$ is an $n×n$ matrix and $\vec x=[x_1,x_2,\ldots,x_n]^T.$ Assuming that $\vec x =\vec w t^r$, where $\vec w$ is a constant vector ...
1
vote
1answer
30 views
Solving ODE using frobenius method. 3 coefficients
I'm trying to learn frobenius method by solving some problems (ODEs).
For example:
$$xy''+(2x+1)y'+(x+1)y=0$$
Let $y=\sum\limits_{n=0}^\infty a_nx^{n+r}$. Then, I took derivatives and put into the ...
1
vote
1answer
26 views
Differential equations basic problem
I know this is a basic Physics problems but somehow I can't solve it.
We have the differential equation: $2x''x^2 - 4 x^2x' - 2 x^3 = 0$
We have to conclude that the system:
$x' = y $
$y' = 2y + ...
0
votes
1answer
30 views
Existence of Phase Flow
Hi there I'm wondering if anyone can clear up my confusion: What is the proof of the local existence of a phase flow for a differentiable vector field?
0
votes
1answer
31 views
How to make a unit step function?
I am trying to make a unit step function.
I have this function (the equation of an ellipse, not centered at the origin):
$$
f(x,y) = \frac{(x-X_c)^2}{a^2}+\frac{(y-Y_c)^2}{b^2}
$$
What I would ...
1
vote
3answers
197 views
differential equation : non-homogeneous solution, finding YP
hi i have a problem for this Differential Equations :
$$
\frac{d^{3}y}{dx^3} - 9\frac{dy}{dx} = 10 - 4x
$$
i know first we must solve the homogeneous equation:
and my result is : $C_1 + C_2e^{3x} + ...
3
votes
1answer
54 views
minimization problem on differential equations - optimal control
I am trying to minimize an time-integral of a linear function with respect to differential equations. The problem is formally defined as follows:
Given $\lambda< \mu_1, \mu_2$ fixed ...
0
votes
1answer
29 views
continuity and differentiability and L'Hopital's Rule
Let
$$f_n(x) = \begin{cases}
0 & x < -\tfrac{1}{n} \\
\tfrac{n}{2} & -\tfrac{1}{n} \leq x \leq \tfrac{1}{n} \\
0 & x>\tfrac{1}{n} \\
\end{cases},$$
$n=1,2,3,\ldots$.
Let $g(x)$ be a ...
1
vote
0answers
25 views
Invariant relation in ODE
It is well known that if function $g(x)$ is an invariant relation under ODE $\dot x = f(x)$ then $\frac{\displaystyle d}{\displaystyle dt}g = \lambda g$.
More precisely. Let ...
0
votes
1answer
16 views
System of Differential Equations Question Assistance
The following question has just left me confused with no real decent avenue of attack so any assistance on this would be appreciated.
For the system of equations
$t {\frac{d \vec x}{dt}} = A\vec x $
...
0
votes
1answer
21 views
Are these ODEs equivalent?
I have the following set of ordinary differential equations:
\begin{equation}
\left\{
\begin{array}{l}
\dot{a} = f_1(a, b, c, d) \\
\dot{b} = f_2(a, b, c, d) \\
\dot{c} = f_1(c, d, a, b) \\
\dot{d} = ...
1
vote
0answers
24 views
A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$
Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$.
$F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$.
$P_1(F)$ is the set of all ...
1
vote
0answers
39 views
Approximating the modified Bessel’s function with a sum of exponentials
I am looking for an approximation for modified Bessel’s function $I_\alpha(f(t))$ (specially $I_0(f(t))$ or at least $I_0(t)$) with a sum of exponential functions. I mean I want to approximate the ...
2
votes
1answer
426 views
Rewrite matrix equation for Euler method and Improved Euler method
Consider a system of the form:
(1) $x' = Ax + g$
For appropriate matrices $x'$, $A$, $x$, and $g$.
If we let $y_n$ be the approximation to the solution of (1) at time step $t_n$, what matrix ...
1
vote
0answers
40 views
Unusual jump condition for Green function
This question is related to a previous question I posted a while ago.
Imagine that I'm computing the Green function of a linear operator $L$, such that:
$$LG(x,s)=\delta(x-s).~~~~~~~~~~~(1)$$
Now, ...
0
votes
1answer
24 views
Euler's method for second order differential equation
Not really homework but sample exam.
The question is to use Euler's Method to approximate Y:
$Y''(t) = Y'(t) - 2Y(t)$
$Y'(0) = Y(0) = 1$
with $t_0 = 0$ and $h=0.2$
So what I did:
First ...
0
votes
2answers
28 views
stability and asymptotic stability: unstable but asymptotically convergent solution of nonlinear system
Consider nonlinear systems of the form $X(t)'=F(X(t))$, where $F$ is smooth (assume $C^\infty$). Is it possible to construct such a system (preferably planar system) so that $X_0$ is an unstable ...
1
vote
1answer
36 views
Simple Diffy-Q problem
So as a fun project, I'm trying to work my way through Kreyzig's "Advanced Engineering Mathematics". But I've gotten to a really simple problem:
$$xy' = 2y$$
where I know the solution is $x^2$ but ...
0
votes
1answer
18 views
If $\lim_{t\to\infty}\gamma(t)=p$, then $p$ is a singularity of $\gamma$.
I'm trying to solve this question:
Let $X$ be a vectorial field of class $C^1$ in an open set
$\Delta\subset \mathbb R^n$. Prove if $\gamma(t)$ is a trajectory of
$X$ defined in a maximal ...
1
vote
1answer
20 views
Bifurcation value and description
Find the bifurcation of $a$ and describe the bifurcation that take place at each value
$\displaystyle dy/dt=e^{-y^2}+a$
I let $\displaystyle e^{-y^2}+a=0$ then solve for y. I got $y^2=-\ln(a)$ What ...
0
votes
1answer
15 views
Better than Runge-Kutta-Fehlberg 4(5) at high order?
I wonder what are currently the best numerical solvers of ODE for high-accuracy computations. I need an efficient and accurate method to solve ODE that are not pathological (all is smooth) using ...
