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

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0
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1answer
11 views

Need help with Laplace transform of piecewise /step functions

Hi I am having trouble figuring out how to calculate the laplace transform for $f(t)$ where $$f(t)= \begin{cases} e^{4t} & \text{if $ 0 \lt t \lt 2 $} \\ 1 & \text{if $ t \gt 2 $} ...
-1
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0answers
24 views

Differential Equation ODE. [on hold]

Hello I have a problem with this differentials equations of first-order, im trying to do it with ode23 and ode23s. The differentials equations are the next one: y'+3y+z=0 z'-y+z=0 with this initial ...
1
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0answers
13 views

Why is this ODE solution only unique in either $(-\infty,0]$ or $[0,+\infty)$ and not in $\mathbb{R}$

Consider the following ODE: $$y'(t)=f(t,y)=e^{-t}+\log(1+y^2)$$ $$y(0)=0$$ You can clearly see the function is continuous on both variables, and the partial derivative with respect to $y$ is: ...
-3
votes
2answers
48 views

Differential equation $y''-4y = e^{-x}$ [on hold]

I need help with the following differential equation: $$y''-4y = e^{-x}$$ (no initial conditions given) Any help is appreciated.
-1
votes
0answers
9 views

Examples of ODEs with 3-dimensional function

I'm trying to test a numerical method program and I need some test cases, i.e. ordinary differential equations. I found some but in these examples the original Y funtion is unknown. I want to check if ...
4
votes
2answers
32 views

System of 3 differential equations

I'm trying to solve this system $$x'=x-3y+3z$$ $$y'=-2x-6y+13z$$ $$z'=-x-4y+8z$$ must be reduced to a single equation I tried to express the x 3 and substitute in the other two but then I have not ...
3
votes
1answer
35 views

problems with differential equation

i have problems solving eq. $$ u + \log(u-1) = \log (x); \quad u= \frac{y}{x}$$ which comes from solving diff equation $$x \frac{dy}{dx} - y= x\frac{y-x}{y+x}$$ any hints? thanks in advance
7
votes
4answers
226 views

solution to differential equation from deriving power series

Find the solution of the differential equation $$y'= 2xy$$ statisfying $y(0)=1$, by assuming that it can be written as a power series of the form $$ y(x)=\sum_{n=0}^\infty a_nx^n.$$ Im advised to ...
0
votes
2answers
15 views

A general solution of a partial differential equation with $f(x,y)$

I need to find a general solution to such a PDE: $$u_x-u_y=f(x,y)$$ I am able to find a solution if $f(x,y)=0$ or $f(x,y)=u$. But I have no idea how to get the general solution. Has anybody got any ...
1
vote
1answer
22 views

Differential equation where one solution induces a set of solutions

Consider a differential equation of the form: $$y' = f\left(\frac{y}{x}\right);\space\space\space x ≠ 0$$ where $f$ is any continuous function. I want to show that if $y(x)$ solves this equation, ...
0
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3answers
36 views

linearly independent (Linear algebra)

Show graphically that $y_1(x)=x^2$ and $y_2(x)=x|x|$ are linearly independent on $-\infty$ to $\infty$ but Wronskian vanishes at every point. The Wronskian is $$W = ...
2
votes
2answers
173 views

Laplace operator defined on a Sobolev space

Consider the Laplace operator $$A:W^{2,2}(\mathbb{R})\to L^2(\mathbb{R})\;\;\\A u = -u^{\prime \prime}$$ I want to know why this operator is closed (I'm using the closed graph theorem): Let ...
-2
votes
0answers
27 views

Differential equation with steps [on hold]

What steps are involved in solving this differential equation? I found the answer at Wolfram, but it didn't show me how to get to the answer. $$y''+ y = \sqrt{x+y+1}$$
1
vote
1answer
21 views

How do we deduce that the initial value problem has always a unique solution?

Theorem - General solution of $y''+p(x)y'+q(x)y=0, x \in I (\star)$ Let $y_1, y_2$ be linearly independent solutions of $(\star)$ in an interval $I$. Then if $y$ is a solution of $(\star)$ in $I$, ...
-3
votes
0answers
40 views

solve differential equation of $y''+y = \sqrt{x+y+1}$ [on hold]

I tried to solve this differential equation with no result, I even tried http://www.wolframalpha.com/ that showed no steps .. any help ?
0
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0answers
19 views

wave equation on a circular domain

Consider the wave equation for the displacement $$\text{u(r,$\theta $,t)}$$ in a circular domain $$\text{0 $<$ r $<$ a, -$\pi $ $<$ $\theta $ $<$ $\pi $}$$ How do I use the separation ...
3
votes
3answers
66 views

a linear differential equation with periodic coefficients

Let $$y' = a(x) y + b(x)$$ be a linear differential equation with continuous, periodic coefficients $a, b: \mathbb{R} \to \mathbb{R}$ that both have a period of $T > 0$. Also, we assume that ...
0
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0answers
24 views

What is the motivation for solving the Bessel equation.

My course is highly theoretical. For most part, we're taught to solve equations. But as a Physics student, I would very much like to know the motivation behind seeking the solution to the Bessel's ...
1
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2answers
24 views

Solving a two degree Differential equation (ordinary) with a variable coefficient

The question is to solve the following integral equation: $$y(x)=x-\int_{1}^{x}xy(t) dt; y \in C^1[1,\infty)$$ My try: I differentiated twice to get the ordinary equation $$y''(x)+xy'(x)+2y(x)=0$$ ...
-3
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1answer
19 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
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1answer
16 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
13 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
vote
1answer
23 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
votes
1answer
26 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
1answer
30 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
17 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
votes
2answers
28 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
24 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 $y_0 > 1$. ...
2
votes
1answer
28 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 ...
0
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0answers
21 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
13 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
votes
1answer
43 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
votes
1answer
25 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
votes
1answer
30 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
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1answer
23 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
2answers
67 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
votes
1answer
23 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
33 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
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0answers
11 views
0
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1answer
36 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
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1answer
22 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
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 ...
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 ...
0
votes
3answers
73 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
26 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
33 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
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 ...
1
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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. ...