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

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-3
votes
1answer
18 views

linear homogeneous constant coefficient systems [on hold]

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
votes
0answers
11 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
votes
1answer
10 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 ...
12
votes
2answers
458 views

Fourth Order Nonlinear ODE

I was looking at an ode $w^{(4)} + w^3 = 0$ with initial conditions $[w'''(0),w''(0),w'(0),w(0)]=[1,0,0,0]$. I can see via maple that there is a blowup around 3.7. I was wondering if there was a way ...
0
votes
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 ...
3
votes
2answers
51 views

How to address multiple cases in this BVP? (Laplace equation in quarter-annulus)

The original problem: $$\nabla^2 u =0 \ \ \ \ for \ \ \ 0<a<r<b\ \ \ ,\ \ \ 0<\theta <\frac \pi 2$$ $$u(r,0)=0,\ \ u(r,\frac \pi 2)=f(r),\ \ u(a,\theta)=u(b,\theta)=0$$ My ...
0
votes
1answer
24 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
votes
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: ...
1
vote
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, ...
2
votes
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 $y=\frac{t^2}{2}+t\sin t+\cos t+c$, $c\in\Bbb R$. ...
2
votes
2answers
45 views

First order differential equation integrating factor is $e^{\int\frac{2}{x^2-1}}$

So i got the first order ode $$(x^2-1)\frac{dy}{dx}+2xy=x$$ I divided both sides by $x^2-1$ $$\frac{dy}{dx}+\frac{2}{x^2-1}xy=\frac{x}{x^2-1}$$ in the form $y' + p(x)y = q(x)$ So that means the ...
2
votes
2answers
27 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, ...
-2
votes
2answers
15 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 $y_0 > 1$. ...
0
votes
0answers
48 views

Confusion regarding dF/dx=0, F=constant

I thought I found a theorem Given a curve in the $(y,x)$ plane defined by DE $\frac{dy}{dx} = f(y(x),x)$ and if there exist a directional derivative of $F$ along this curve satisfies relation $g = ...
3
votes
1answer
60 views

How to solve this 2nd order ODE

Consider $$\epsilon y''+yy'-y=0$$ with boundary conditions $y(0)=0$ and $y(1)=3$. I showed that the outer solution is $y_{in}(x)=x+2+O(\epsilon)$. Than for the inner solution, I wish to solve the ...
-1
votes
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} + ...
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
votes
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 ...
4
votes
1answer
70 views
+50

Behaviour of solutions to ODE near singular points

I am having trouble understand how to classify what happens to solutions of ODE near singular points. For example, I have a question that is about the ODE; $$(x^2-36)y''+(6-x)y'+(x^2+12x+36)y=0$$ ...
0
votes
0answers
18 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 ...
2
votes
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 ...
1
vote
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 ...
0
votes
1answer
51 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 ...
4
votes
2answers
58 views

How to solve linear, second order ODE with Frobenius method with a difficult recurrence relation?

The ODE in question is: $$4xy''+2y'+y=0$$ Shifting the power series of each term so that they are all raised to the power $(n+r)$ will yield this recurrence relation: $$a_{n+1}={a_n\over ...
0
votes
0answers
11 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 ...
1
vote
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
votes
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
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?
0
votes
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 ...
0
votes
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} ...
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 ...
3
votes
1answer
19 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 ...
0
votes
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 ...
-2
votes
0answers
10 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 ...
2
votes
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
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 ...
1
vote
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. ...
0
votes
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 $$\frac{dx}{dt}=x^2-y^2-z^2,$$ $$\frac{dy}{dt}=2xy,$$ ...
-2
votes
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} ...
1
vote
2answers
58 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
votes
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 ...
3
votes
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
vote
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 ...
0
votes
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
votes
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 ...
0
votes
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.
4
votes
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.
7
votes
4answers
315 views

What comes after Differential Equations?

First of all, please do excuse the lack of correct terminology, I've haven't learnt Differential Equations at school (yet) so this question comes from just a bit of research I did for my own ...
1
vote
1answer
28 views

What are difference among natural boundary, exit boundary, regular boundary and killing boundary??

In the paper i'm reading, they used the terminologies, natural boundary, exit boundary, regular boundary and killing boundary. I can't find the difference of them and definition of them. Tell me ...