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

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2
votes
1answer
427 views

Show $\nabla^2g=-f$ almost everywhere

let a continuous function $f(x,y,z)$ be absolutely continuous over every bounded region and let it be in $L^1$ that is $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} ...
0
votes
0answers
15 views

What does “order” exactly mean in numerical methods?

I am trying to understand the concept of order in solving numerical differential equations of the form $\frac{dx(t)}{dt}=f(t,x(t))$. Let's start from the local discretisation error at $t$: ...
1
vote
2answers
258 views

Proof with g(x) and f(x)

Let $g(x)$ be a function that is twice differentiable for all $x$ and let $f(x) = g(x) − g(1 − x)$. Prove that $f′′$ has a root in the interval $(0, 1)$.
2
votes
1answer
68 views

Bessel's Differential Equation - textbook queries:

In order to ask this question I must first give some background information as written in my text book: Given Bessel's Differential equation: $$x^2y^{\prime\prime}+xy^{\prime}+(x^2-p^2)y=0$$ ...
-1
votes
0answers
31 views

Initial value problem (Runge-Kutta)

The angle $\theta$ of a pendulum satisfies the initial value problem$$\frac{d^2\theta}{dt^2}=-\frac{g}{l}\sin\theta,$$$$\theta(0)=\frac{3\pi}{4}, \ \ \theta'(0)=0.$$ Write this as an initial value ...
0
votes
0answers
15 views

Differential equation Cauchy problem resolution.

I can't find my mistake in solving this problem \begin{cases} y'(t) = y(t)/t + 2t(y(t))^2 \\ y(1) = 4 \end{cases} I recognize this as a Bernoulli equation and thus apply the substitution $z(t) = ...
0
votes
0answers
29 views

How do I get these equations ($n$th-order deformation equation and its initial/boundary conditions)?

Earlier, I asked a question What does 'equating the like-power of $q$ ' mean? and I already got the answer about the meaning of a particular phrase in a certain context. However, my real question ...
3
votes
1answer
62 views

Linear ODE with non-constant coefficients

I have encountered some problem in computing the explicit solution for the following ODE: $$x^\prime(t) = (2t-1) x(t)-1, \quad x(0) =: x_{0}$$ The formula that I have used to solve it is: ...
0
votes
1answer
27 views

Check my work in this first-order nonlinear ordinary differential equation

Question, solve (and find my mistake): $$y'(x)=(1-y(x))(1+ny(x))$$ My work: $$y'(x)=(1-y(x))(1+ny(x))\Longleftrightarrow$$ $$\frac{y'(x)}{(1-y(x))(1+ny(x))}=1\Longleftrightarrow$$ ...
0
votes
2answers
40 views

what is the ODE that gives rise to the Laplace Transform.

It can be shown that the transform pairs can be obtained from a differential equation with some boundary conditions. See, for example, Keener's book chapter 7. For example. The Fourier transform can ...
1
vote
1answer
20 views

Find the flow for the following dynamical system

I have the following dynamical system: $\dot{x_1}= -x_2 + (x_1(1-(x_1^2+x_2^2)^2))$ , $ \dot{x_2}= x_1 + (x_2(1-(x_1^2+x_2^2)^2))$, $\dot{x_3}= \epsilon x_3$ . I am required to work out the flow ...
1
vote
1answer
27 views

Second order differential equation Cauchy problem.

I have the following Cauchy problem \begin{cases} y''(t) = \frac{(y'(t))^2 - 2}{2t y'(t)} \\y(1) = 4 \\ y'(1) = 1 \end{cases} I proceed by setting $v(t) = y'(y^{-1}(t)) $ to obtain the system ...
0
votes
1answer
11 views

Second order differential equation particular solution of a product

I read that when the right side $f(x) = e^x$, a suggested form of $y_{ps}$ is $Ce^x$, and for $f(x)$ is linear in $x$, a form of $y_{ps}$ is $Cx + D$. If $f(x)$ is the product of the two mentioned, ...
1
vote
0answers
27 views

unbounded solution, lim inf of trace,

Show that if $\lim \inf_{t\rightarrow \infty} \int_{t_0}^t \operatorname{tr}\left(A(s)\right)ds= \infty $ then the linear first-order system $x'(t)=A(t)x(t)$ where $A \in C\left(I, \mathbb{R}^{n\times ...
-1
votes
1answer
391 views

Asymptotic stability of an equilibrium solution of an autonomous differential equation

I'm having trouble understanding how I would prove the asymptotic stability of an equilibrium solution for the following problem. I understand what stability is and how to show it with other ...
1
vote
1answer
39 views

How to estimate the parameters of a logistic differential equation from the values of its solution at times 0, 1 and 2?

How do I solve this system of equations? I received these equations after letting Wolfram Alpha solve the logistic differential equation $$N'(t)=kN(t)(M-N(t)),\qquad N(0)=65,$$ that outputs: ...
3
votes
1answer
33 views

Linear Differential Equation but with piece wise function ( Cant Solve ) [on hold]

Find a continuous solution satisfying the DE $$y' + y = f(x),$$ where \begin{align} f(x) &= \begin{cases} 1, & 0 \leq x \leq 1 \\ -1, & x>1\text{.}\end{cases}\\ ...
0
votes
1answer
18 views

Steady state temperature distribution in a thin rectangular slab.

We need to solve the following : $\nabla^{2} u =0 ,\:\:\:0<x<a$ $u(x,0) = f(x) , \:\:\: 0\leq x \leq a$ $u(x,b) = 0 ,\:\:\: 0\leq x \leq a$ $u_x (0,y) = 0 , \:\:\: u_x (a,y) =0$ We use ...
1
vote
0answers
16 views

Solving or knowing something about a non-linear PDE which is “almost” linear?

Let $a>0$ be fixed. I have the following PDE: $u=u(t,x)$, $t\in [0,1]$, $x\in \mathbb{R}$, $$-\partial_t u = |\partial_x u| + \frac{1}{2}\partial_x^2 u, \quad ...
2
votes
1answer
938 views

Wrong answer for this differential equation temperature problem.

(a) An object is placed in a 68°F room. Write a differential equation for H, the temperature of the object at time t. ANSWER: dH/dt = -k(68 - H) (b) Give the general solution for the differential ...
0
votes
2answers
52 views

what is the solution of this differential equation [on hold]

What is the solution of this differential equation with the given initial values? Is it linear or not? $$y'' + 6y + \frac{3}{y^2} = 0 \quad , \quad y(0)=3,\, y'(0)= 2$$
0
votes
0answers
11 views

In which interval the solution exist and unique on the area $R=\{(x,y);|x|\le3,|y|\le2\}$?

for the initial value problem $y'=\frac x{y-1},y(0)=0$ in which interval the solution exist and unique on the area $R=\{(x,y);|x|\le3,|y|\le2\}$ $-2/3\le x\le2/3$ $-2/3\le x\le3$ $-3\le x\le3$ ...
0
votes
1answer
22 views

How can I understand $x(b)=x(a)+\int_a^{b}f(s,x(s))\,ds$?

I am trying to understand this integral form of the ordinary differential equation: $$x(b)=x(a)+\int_a^{b}f(s,x(s))\,ds\quad\text{for }a\leq t\leq b$$ I tried to pick a concrete example: Let ...
0
votes
2answers
54 views

To find the initial value problem $y(x) = 1 + \int_{0}^x (t-x) y(t)\,dt$ [on hold]

The initial value problem corresponding to the integral equation $$y(x) = 1 + \int_{0}^x (t-x) y(t)\,dt$$ is?
1
vote
1answer
24 views

Is this an isolated equilibrium point?

I've just been learning the definition of an isolated equilibrium point. From my understanding of this definition, I would expect (as an example) the point $x=1$ to be an isolated fixed point for the ...
5
votes
2answers
47 views

Drawing conclusions from a differential inequality

Let $f(x)$ be a smooth real function defined on $x>0$. It is given that: $f$ is an increasing function ($f'(x)>0$ for all $x>0$). $x \cdot f'(x)$ is a decreasing function. I am trying to ...
1
vote
1answer
22 views

What does 'equating the like-power of $q$' mean?

I am reading a book "Homotopy Analysis Method in Nonlinear Differential Equations" by Shijun Liao chapter 13 Applications in Finance: American Put Options. It is stated there that Substituting ...
0
votes
0answers
13 views

Applying Boundary Condition to Finite Element Matrix

Several times now I have seen the following done without justification and I cannot figure out why it can be done: Consider the 1 dimensional "pde" $-u'' = f, u(0) = a, u(1) = b$ over $[0,1]$. We ...
2
votes
4answers
126 views

Is the function $y(t)$ is a solution of the equation $y'=\sin(yt)$?

Is the function $y(t)$ a solution of the equation $y'=\sin(yt)$? any thought to start me up? I'm not sure what is the question asking. EDIT: Someone tell me if I'm correct or not . If I'm finding ...
0
votes
1answer
384 views

Runge Kutta Method Matlab code

So I have a programming assignment with the following instructions: Consider the nth-order differential equation $$Ax^n (t) = x ^{(n-1)}(t) + x^{(n-2)}(t) + ... + x(t)$$ where $A$ is a ...
2
votes
0answers
73 views

Pursuit Curve, Parametric Equation

So its a classic problem: Object $A$ starts at the origin $(0,0)$ and moves straight up the $y$ axis with a speed $v$. Object $B$ starts at point $(1,0)$, always moves towards object $A$ and has a ...
2
votes
3answers
70 views

How do I solve the differential equation $y' = (1-y)(1+6y)$?

How can I solve this ODE? $$\frac{\mathrm{d}y}{\mathrm{d}x} = (1-y)(1+6y)$$ I tried using classical Runge-Kutta, but the results are not satisfying. Can anyone suggest some other method? Please ...
0
votes
1answer
26 views

a crucial doubt on an ODE

I have this Riccati ODE: $y'(x) = -\kappa y(x)-0.5\beta^2y(x)^2+\xi$ and I know the solution of the analogous one $y'(x) = -\kappa y(x)-0.5\beta^2y(x)^2+1$ leads to the closed-form solution $y(x) ...
1
vote
1answer
15 views

Solve the following distributional differential equation: $(xT_f)' \equiv H$

As stated in the title, I want to solve the distributional differential equation $(\star)$ $$(xT_f)' \equiv H $$ $T_f \in (C_0^\infty)^*$ is a distribution induced by an arbitrary $f \in ...
1
vote
1answer
301 views

Integral form of $\frac{dy}{dx} = \frac{d}{x} - \frac{x}{y}$

How can I write $\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{d}{x} -\dfrac{x}{y}$ in integral form not containing $y$? (Its solution represents the family of curves orthogonal to the family of curves ...
0
votes
0answers
20 views

Initial Value Problem for $(\cos x -x\sin x +y^2)dx + 2xy\,dy =0$, $y(\pi )=1$

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following question. If it's any indication of difficulty, the exercise is only Part 1-B of the sheet, ...
-1
votes
0answers
13 views

Alternative form of the general solution of a 2nd order ODE

The eq. is the following: $$d^2y/dt^2 + a dy/dt + by = 0$$ https://www.wolframalpha.com/input/?i=a+y%27%27(t)+%2B+b+y%27(t)+%2B+c+y(t)+%3D+0 is a solution, but is it $$y(t) = (c+dt)e^{-(a/2)t}$$ ...
0
votes
0answers
25 views

Representation of a real function through a Fourier Transformation

I 'm trying to do some calculations regarding some differential equations and I came across an interesting way to express a real function through a double integral of the form: ...
2
votes
0answers
27 views

Numerical methods for ODE: Implicit, explicit, stability, stiffness

Hy everybody! I am new to the subject "numerical methods for ODE". I read some basic literature but since most of the concepts and methods are new to me, I wanted to ask you, if you could give me ...
2
votes
0answers
23 views

Numerical methods for ODE: Taylor vs. Interpolation approaches

Hy everybody! I am new to the subject "numerical methods for ODE". I read some basic literature but since most of the concepts and methods are new to me, I wanted to ask you, if you could give me ...
1
vote
3answers
48 views

Finding the initial equation

Having trouble with this problem. Find the solution to the initial value problem $$\frac {dy}{dt} + y = t^2$$ Where $$y(0) = 0 $$ Can someone help me get started?
0
votes
1answer
61 views

Differential Equation: Solve $y''-4y'+3y=4e^x$

For this question it has an initial condition of $y(0)=5$ and $y'(0)=3$. I managed to get $y = Ae^x + Be^{3x} - 2xe^x$. Solving for $A$ and $B$ I would get $A = 7$ and $B = -2$. However, the real ...
4
votes
2answers
183 views

Method to solve $xx'-x=f(t)$

I would like to resolve this differential equation: $xx'-x=f(t)$ any suggestions (or any online texts on similar differential equation) please? Thanks.
6
votes
2answers
216 views

How to solve this differential equation for $y$ in terms of $x$ and $k$

$$yy'+\frac yx+k=0$$ How to solve this differential equation for $y$ in terms of $x$ and $k$ where $k$ is a parameter of $x$? $y(x)=y$ is a function and $x(k)=x$ is a gamma function
3
votes
2answers
140 views

Solve $f ' (x) + f '' (x)/2 = \sqrt f(x)$

How to solve the differential equation $$f ' (x) + f '' (x)/2 = \sqrt {f(x)}$$ Edit My efforts Assume $f(x) = a x^2 + b x + c$. Then we plug this into the differential equation $2 a x + b + a ...
0
votes
1answer
170 views

Using the Heaviside function to represent a given graph

The question is the following: The graph is zero between $0$ and $2$, is a straight line from the point $(2,0)$ to $(5,5)$, a straight line down from $(5,5)$ to $(4,0)$ and zero everywhere else. ...
0
votes
1answer
65 views

Proof that $Y=A\cos(px)+B\sin(px)$ is only periodic if $p=n$.

I am asked to show that a function $y=A\cos(px)+B\sin(px)$ can only be periodic if $p$ is an integer $n$, where $A$, $B$ are arbitrary constants. In other words $y(x)=y(x+2 \pi)$ I begin by solving ...
0
votes
1answer
32 views

Form of the particular solution for $g(t)=3\sin(2t)$

What is the "proper" form of the particular solution for $y''+4y=3\sin(2t)$? When I use $Y(t)=A\cos(2t)+B\sin(2t)$, I am not able to solve for the coefficient after subtitution of the ...
0
votes
3answers
48 views

Triangle of zero spherical excess on a sphere

Find differential equation of an (obviously non-geodesic) line on a sphere such that when three such lines intersect the sum of internal angles of the triangle so formed is $\pi$. EDIT 1: DE I get ...
0
votes
1answer
39 views

How to solve this differential equation: $\frac{dy}{dx}=\frac{x^3+y^2}{2xy}$?

Using homogeneous, it's changed to $\dfrac{dv}{dx}=\dfrac1{2v}+\dfrac v{2x}$ where $v=\dfrac yx$. I'm not sure how to solve this further on. I tried using symbol but the steps seem to be too ...