Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.
1
vote
1answer
12 views
Inverse Laplace transform after derivative of transform.
I've been using the following theorem in my intro ODE course:
If $F(s) = \mathscr{L}\left\{ f(t) \right\}$ and $n\in\mathbb{N}$, then $\mathscr{L}\left\{t^n f(t) \right\} = ...
2
votes
1answer
37 views
Questions on differential equations of matrices
I have a differential equation $$N'_x(x)=G(x)N(x)$$ where $N, G$ are $2\times2$ matrices depending on $x$, and $G$ satisfies $\sigma G+G\sigma=0$, $\sigma$ is one half of the pauli matrix, i.e. ...
1
vote
1answer
25 views
Radioactive isotopes differential equationa
I am having a hard time finding the correct differential equation to my problem. The problem is :
There's 2 isotopes: A and B. A is is transforming into B to a rate proportional to its quantity and B ...
2
votes
0answers
36 views
Solution to second order nonlinear ODE
I need to find and exact solution for the following ODEs $$y''=-3y'+2y+2x+3,\qquad y(0)=2$$ $$y(1)=-4+5\exp\left(-3/2+\left(\sqrt{17}\right)/2\right)$$ and $$y''=2y^3-6y-2x^3;$$ $$1\leq x\leq2;$$ ...
1
vote
0answers
22 views
How do you set up a system of ODE's for this problem?
The problem is as follows:
Black and White balls are being created inside an arbitrary volume at rates of $Q_{B}$ and $Q_{W}$. They also disappear from the volume at rates $\lambda_{B}$ and ...
1
vote
0answers
22 views
Schroedinger equation in cylindrical coordinates
How can one numerically solve the nonlinear stationary Schroedinger equation in cylindrical coordinates?
1
vote
0answers
32 views
Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems
Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz,
\begin{equation}
\begin{split}
\dot{x}_1 ...
0
votes
1answer
32 views
Find the first 5 terms of the expansion in a power series
Find the first 5 terms of the expansion in a power series
$$y′=xe^{x}+2y^{2}$$
I've got a riccati equation $$ x e^{x}+2y^{2}, y(0)=0$$
After solving: $$y=e^{x}(x-1)+\frac{2}{3}y^{3} - 1$$
And I ...
8
votes
4answers
94 views
Examples of nonlinear ordinary differential equations with elementary solutions.
I am looking for nice examples of nonlinear ordinary differential equations that have simple solutions in terms of elementary functions. (But are not trivial to find, like, for example, with ...
1
vote
1answer
24 views
Finding a strong enough solution to a specific PDE problem.
Let $U\subset \mathbb{R}^n$ with smooth boundary $\partial U$. And consider the expression
$$\Delta u = f.$$
$$\text{+"convenient boundary conditions"}$$
In my specific case $f\in H^2_0$. Under ...
1
vote
1answer
31 views
Inverse Laplace Transform. Computing the integral.
This question is related to this one, but I'm hereby taking a different approach.
Problem: Solve
$\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$.
Find the stationary points and examine their ...
0
votes
0answers
21 views
Second order nonlinear delay differential equation
I have to solve the following delay differential equation:
$$\ddot{x}(t)+A\sin(\omega x(t-\tau))=0$$
Can someone give me a hint on how to solve this equation?
Thanks
2
votes
0answers
27 views
What method can be used for finding Green function for Fokker-Planck equation?
Let's have an equation
$$
u_{t} - (xu)_{x} - \frac{1}{2}u_{xx} = 0, \quad u(x, 0) = g(x), \quad -\infty < x < \infty , \quad 0 < t < \infty .
$$
I need to find a Green function for it. ...
0
votes
0answers
19 views
General solution of diff.eq of order 3
please , why the general solution of $u'''(t)=e(t) , t\in [0,1]$ is given by
$u(t)=c_0+c_1t+c_2 t^2 +\frac12 \int_0^1 (t-s)^2 e(s) ds$
$e:(0,1)\rightarrow \mathbb{R}$, and $e\in L(0,1)$.
Thank ...
1
vote
1answer
22 views
Eigenvectors and differential equations
I was able to find part (a), and I got 4 and -1 for the eigenvalues and from these values I got eigenvectors of [1,1] and [-3,2], but I don't know what to do for part (b) and (c)
0
votes
0answers
15 views
Method of undetermined coefficient for the input functions associated with the unit step
I am trying to solve a second order non-homogeneous differential equation where $x(t)$ has $u(t)$, the unit step as a part. i.e. $ x(t)= f(t)u(t) $
I know how to 'guess' the particular solution for $ ...
2
votes
1answer
36 views
Differential equations math question?
We have the equation $xy' -4y - x^2 \sqrt{y} =0$
I modify this a little and I have $dx ( 4y-x^2 \sqrt{y}) -xdy=0$
I am thinking about solving this by integrating factors,but that is quite a long ...
1
vote
0answers
31 views
Defining the error derivative for Krasovskii-Lyapunov
I am attempting to work through "Shahverdiev, Sivaprakasam, and Shore (2002) Lag synchronization in time-delayed systems", and I'm missing something basic up front.
The question is to take a ...
0
votes
1answer
52 views
Liouville's formula
I have some questions concerning a proof of Liouville's formula:
$$W'(t)=\text{tr}(A) W(t)$$ where $W$ is the Wronskian of the homogenous ODE.
If the vectors in the columns of the fundamental matrix ...
1
vote
0answers
51 views
When is it justified to approximate a difference equation with its corresponding differential equation?
Consider the difference equation $f_{x+1}-f_x=a(f_x)$ and the differential equation $g'_x=a(g_x)$. When and Why is it justified to say "$f_x - g_x = o(1) $ hence we can solve the difference equation ...
1
vote
1answer
30 views
Solving Euler-equation alike 2nd order DE with disturbing RHS
For a homework problem, I have to solve
$$ t^2 \ddot{x} - 3 t \dot{x} + 3x = t^2 $$
which seems quite similar to the Euler Equation, which I would know how to solve, apart from the disturbing $ t^2 ...
0
votes
1answer
36 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
28 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:
$$
\frac{\partial f}{\partial t} = k \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right),\quad ...
5
votes
1answer
53 views
Having trouble using eigenvectors to solve differential equations
The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix}
5 & 4 \\
-1 & 1\\
\end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\
x_2 \\ \end{pmatrix}$$
I went ...
1
vote
1answer
15 views
How do you determine the particular solution to a non-homogeneous DE by undetermined coefficients?
I am asked to solve
$y'' +2y' = 2x + 5 -e^{-2x}$
I can find the general solution easily, but the particular solution in this case is hard to find. Here's the answer. I don't know why they got $Ax^2 ...
0
votes
1answer
26 views
Differential equation (2nd order) with divergent coefficients.
I have this equation:
$$x(x-1)y''+6x^2y'+3y=0$$
I try to get the series for the solution around $x=0$, using Frobenius (however it's written). the first solution must be of the form:
...
2
votes
1answer
38 views
Differential equations math help?
I have the equation $y' \sin x=y \ln y$. I told my teacher that we can solve if with separate variables method but he told me that we cant do that. He didn't explain why. Can you tell me why?
4
votes
1answer
61 views
Linearization of $ m \dfrac{dy^2}{dt^2} = u(t) - C_d \left( \dfrac{dy}{dt} \right)^2-mg $
$$ 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&= ...
3
votes
1answer
48 views
Using Fourier series techniques to solve $x'' + 3x = 7$ with $x'(0) = x'(5) = 0$
$$x'' + 3x= 7$$
Given conditions $x'(0)=x'(5)=0$.
I checked the list and I went through three books. I am doing intro to differential equations. I just don't know how to get the extensions... I was ...
1
vote
2answers
46 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
25 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 ...
3
votes
2answers
50 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 ...
6
votes
0answers
67 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 ...
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
40 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
39 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
26 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 ...
6
votes
1answer
61 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 ...
0
votes
1answer
47 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.
...
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 ...
1
vote
0answers
106 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:
$$\frac{\partial}{\partial t} F(x,t)= - i (x+ v_1 t) F(x,t)-\alpha^2 ...
0
votes
1answer
21 views
solve non linear differential equation: $y'\cdot\alpha+y+\beta\cdot e^{\delta\cdot y}+\theta = 0$
Could somebody help me to solve the non linear differential equation, where $y$ is a function of the time and starts with $y(0)=0$
$$
y'\cdot\alpha+y+\beta\cdot e^{\delta\cdot y}+\theta = 0
$$
It will ...
0
votes
2answers
28 views
Why does the differential equation $y' = y + 1$ have solution $y(x) = Ce^x - 1$?
I was watching a video on differential equations for a class that I'm taking. I took calculus so long ago that I can't seem to figure why the differential equation $y' = y + 1$ has solution $y(x) = ...
0
votes
1answer
36 views
Second Order Non-Linear ODE involving Bessel Functions
I'm trying to solve this but I'm getting nowhere. Does anyone know step-by-step solution? or at least the general techniques to use? I do know that the solution involves the Bessel functions.
$y'' + ...
0
votes
0answers
36 views
System of Differential Equations proof question
For the system of equations
$t \dfrac{d\vec x}{dt}=A\vec x$, where $A$ is an $n×n$ matrix and $\vec x=[x_1,x_2,\ldots,x_n]^T.$ Assuming that $\vec x =\vec w t^r$, where $\vec w$ is a constant vector ...
1
vote
1answer
26 views
Differential equations basic problem
I know this is a basic Physics problems but somehow I can't solve it.
We have the differential equation: $2x''x^2 - 4 x^2x' - 2 x^3 = 0$
We have to conclude that the system:
$x' = y $
$y' = 2y + ...
0
votes
1answer
34 views
How to make a unit step function?
I am trying to make a unit step function.
I have this function (the equation of an ellipse, not centered at the origin):
$$
f(x,y) = \frac{(x-X_c)^2}{a^2}+\frac{(y-Y_c)^2}{b^2}
$$
What I would ...
0
votes
1answer
29 views
continuity and differentiability and L'Hopital's Rule
Let
$$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 $
...
