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

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Meaning of the Maximal Interval of existence

What does it mean to say that a solution to a differential equation has a maximal interval of existence?
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Solving laplace equation and decomposing

I want to solve $$ 9U_{xx}+U_{yy}=\sin (2\pi x) + \sin(2\pi y) \label{eq:1}\tag{1} $$ with $U=0$ on the boundary of the unit square. I know you would have to decompose the problem to satisfy each of ...
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1answer
39 views

Show that $y_1(x) = \int_c^x f(x-t) R(t) dt$ is a particular solution of $L(y) = R$

This is problem 14 from Chapter 6.15 of Apostol Calculus, Volume 2 (p. 167): If $L(y) = y'' + ay' + by$, where $a$ and $b$ are constants, let $f$ be that particular solution of $L(y) = 0$ ...
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3answers
33 views

ODE) $y'' + 2y' = 1 + t^2 + e^{-2t}$

I'm stuck with the ODE problem: $y'' + 2y' = 1 + t^2 + e^{-2t}$ This problem is in "judicious guessing" chapter of Braun's "Differential Equations and Their Applications". The trick he taught in ...
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1answer
26 views

Qualitative Ordinary Differential Equations Problem

Find some continuously differentiable function $f:\mathbb{R} \mapsto \mathbb{R}$ and some number a such that the unique solution to the initial value problem $\dot{x}(t)=f(x(t))$, $x(0)=a$ is ...
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Why is solving this differential equation by separation of variables and using undetermined coefficients not the same?

Given $\frac{dx}{dt}=r(x_{0}-x)$, I was able to solve the equation by separation of variables but it is not coming out the same if I were to use undetermined coefficients. Why is that?
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1answer
136 views
+50

Rigorous separation of variables.

Let $I \subseteq \mathbb{R}$ denote an open, non-empty subinterval of the real line. We're given functions: $$f : I \rightarrow \mathbb{R}, \;\;g : \mathbb{R} \rightarrow \mathbb{R}.$$ Now suppose ...
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1answer
33 views

Eigenvectors question

$x'=x-2y$ $y'=4x-x^3$ Equilibrium points are $(2,1),(-2,-1),(0,0)$ Consider equilibrium point $(2,1)$: Let $X=x-2$ and $Y=y-1$. Subbing this into the main and eliminating all the nonlinear terms ...
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1answer
47 views

Prove that the solution of an ODE can be prolonged to $\infty$

I need an help understanding some general techniques in ordinary differential equations. I've never attended a course on ODE, so I'm quite confused on the argument, but I'm trying to improve my ...
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1answer
9 views

Question regarding $\mathcal{L}\{t*\mathcal{U}(t-2)\}$

I'm working on a problem for homework (* is multiplication not convolution): $\mathcal{L}\{t*\mathcal{U}(t-2)\}$ I understand that $\mathcal{L}\{(t-a)\mathcal{U}(t-a)\}=e^{-as}F(s)$ The first step ...
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2answers
35 views

Solving inhomogeneous PDEs with ODEs

I am trying to understand a general method of solving an inhomogeneous PDE. I have begun with the Heat equation but am stuck with the last step. For instance, solving ...
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473 views

Spring Calculation - find mass

A spring with an -kg mass and a damping constant 9 can be held stretched 2.5 meters beyond its natural length by a force of 7.5 newtons. If the spring is stretched 5 meters beyond its natural length ...
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1answer
18 views

Having trouble with discretization and boundry value problems

I have the following homework question: Consider the boundary value problem $y''(x) + 5y'(x) − (2 + x)y(x) = e^x$ on $x ∈ (0, 2)$ with boundary conditions $3y(0) + y'(0) = 5$ and $y'(2) = 7$. ...
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3answers
38 views

$\frac{d^2y}{dt^2}+4y=t\sin(2t)$'s particular solution

Finding the particular solution of : $\frac{d^2y}{dt^2}+4y=t\sin(2t)$ Hey everyone! My professor recently went over this problem and I can't seem to find where he derived a particular equation. ...
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1answer
42 views

Is this differential equation separable?

$$x\frac{dy}{dx}-y^2 = \frac{dy}{dx}+5$$ I have found that this equation is differentiable as shown in the following. $$x\frac{dy}{dx}-\frac{dy}{dx} = y^2+5$$ $$dy(\frac{x}{dx}-\frac{1}{dx}) = ...
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Question about this ODE? $\frac{dy}{dx} = \frac{2x-y}{x+2y}$

Am I being dumb, or is this question actually hard? I made the substitution $u=y/x \implies y = ux$, so then I get: $x \cdot \dfrac{du}{dx} + u = \dfrac{2x-ux}{x+2ux} \implies x \cdot \dfrac{du}{dx} ...
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3answers
36 views

Derivation with respect to a derivative

Let's consider a following equation: $$x''+x'+x=5$$ My question is: how should I differentiate it with respect to x'? I've read somewhere that you only differentiate x' and take both x and x'' as ...
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1answer
31 views

Solve an initial value problem using the directional derivative

In my notes there is the following example of solving an initial value problem using the directional derivative. The problem is the following: $$u_t(x,t)=u_x(x,t), x \in \mathbb{R}, t>0 \\ ...
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0answers
16 views

Why do we take the odd extension?

When we have the initial and boundary value problem $$u_{tt}(x,t)-c^2u_{xx}(x,t)=0, x>0, t>0 \\ u(0,t)=0 \\ u(x,0)=f(x), x \geq 0 \\ u_t(x,0)=g(x), x \geq 0$$ can we apply Green's theorem or ...
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How to solve C'=-a*N'?

I am a Molecular Biologist and I had been watching this video on Bacterial growth. At 9:25, the teacher solves differential equation by integrating: $$ C'= -\alpha*N' $$ Integrating both sides of the ...
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1answer
30 views

Conditions for Deriving $R_0$ for SIR Model Using Survival Function Method

I'm taking a look at the SIR model given by the system of differential equations \begin{align} \frac{dS}{dt} & = - \beta S I \\ \frac{dI}{dt} & = \beta S I - \gamma I \\ \frac{dR}{dt}& ...
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Find all differentiable functions $f$ such that $f(f(x))=f'(x)$ [duplicate]

Here is a problem I made up: Find all differentiable functions $f$ from the reals to the reals such that $f(f(x))=f'(x)$ for all real $x$.
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1answer
35 views

Finding coefficients for a differential equation.

Suppose i have an equation that satisfies the following first order differential equation: $$2x[f(x)]^2=e^xf(x)+e^xf'(x)$$ Is there a way to determine the coefficients of a higher order equation of ...
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2answers
26 views

Looking for an intuitive approach of ODE

I started reading up on the topics of differential equations and tried to solve certain problems to get used to those kind of equations, in particular I tried to understand every "$=$" and ...
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24 views

Implicit function theorem and derivatives

I'm a bit confused about determing $y'$ and $z'$ If I differentiate both equations wrt $x$ I get: $2x+2y\frac{dy}{dx}+2z\frac{dz}{dx}=0$ and $y+(x+z)\frac{dy}{dx}+y\frac{dz}{dx}=0$ Now because ...
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32 views

Solve a non-linear ODE problem from an old text book

In an old Differential Equations text book, dated in 1954, in the chapter corresponding to - First order non linear equations on $y'$ - , there are a list of proposed exercises, the first of which ...
2
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2answers
26 views

Simple equation re-arrangement

I have a simple re-arrangement of an equation which I can't seem to solve, help would be much appreciated. I'm trying to re-arrange the equation: $e^{-3t}\frac{dy}{dt} - 3e^{-3t}y = C$ where $C$ ...
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0answers
20 views

Another integral equation question

Suppose that $z = \int_{- \infty}^z f (y) d y$. If $f$ were continuous, we can differentiate both sides to get $f(y)=1$. But what if $f$ does not have to be continuous, is this still true or are there ...
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1answer
31 views

Difficult exercise on unicity of solutions for an IVP

Suppose $f$ and $g$ are continuous and $g$ is odd and strictly increasing function. I have to prove that the IVP $$y'=f(x)g(y)$$ $$y(0)=1$$ has a unique solution if and only if $$\lim \limits_{u \to ...
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2answers
25 views

I can't find the general solution (Second Order Linear Equations)

I'm trying to find the general solution to: y'' + 4y = t^2 + 7e^t The actual problem wants me to find the initial value problem with y(0) = 0 and y'(0) = 2 but I'm confident that I can find the IVP ...
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2answers
36 views

Why don't we write general linear second order ODEs as $P(x)u''(x) + Q(x)u'(x) + R(x)u(x) = g(x)$ instead of $u''(x) + q(x)u'(x) + r(x)u(x) = g(x)$?

The general form of the second order linear ODE is: $$u''(x) + q(x)u'(x) + r(x)u(x) = g(x)$$ In particular, note that any function dependent on $t$ multiplied by $u''(x)$ has been divided out, to ...
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1answer
305 views

Find all of the equilibrium points and describe the behavior of the $x' = sin(x), y' = cos(y) $.

Find all of the equilibrium points and describe the behavior of the $$x' = \sin(x), \quad y' = \cos(y) .$$ It has been a while since I took DE...Do we first need to set $x' = y'$ to solve for their ...
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1answer
41 views

Solutions of the constant coefficient Helmholtz equation via the Fourier transform

When $f$ is a rapidly decaying Schwartz function, $$ g(x) = \frac{1}{2\lambda} \int_{\mathbb{R}} \sin \left(2\lambda\left|x-y\right|\right) f(y)\ dy $$ is an element of ...
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1answer
26 views

Do we suppose that $y$ is the variable at which $f$ is differentiable in $\mathbb{R}$ ?

In my notes there is the following: The solution of the problem $$u_{tt}-c^2u_{xx}=0, x \in \mathbb{R}, t>0 \\ u(x, 0)=f(x), x \in \mathbb{R} \\ u_t(x,0)=g(x)$$ is ...
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2answers
27 views

Finding an ODE given some of its solutions

Find $a, b, f(x)$ such that $$y''+ay'+by = f(x)$$ Is satisfied by $g_{1}=\sin x + e^x$ and $g_{2}=\sin x - e^{-x}$ What I tried to do: First, I used the fact that if $g_{1}$ and $g_{2}$ are ...
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1answer
24 views

Canonical form of the differential equation

In my notes there is the following: Find the canonical form of the differential equation $$4u_{xx}-12u_{xy}+9u_{yy}+u_{y}=0$$ $$\Delta=(12)^2-4^2 \cdot 3^2=0$$ The canonical form will be of ...
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1answer
27 views

How to guess a change of variable for an ODE?

My question is related about how to guess what change of variable to make in an ODE? For example, we have $$\varepsilon^2y(x)''=axy(x)$$ where $\varepsilon\ll1$, $a$ is constant and $x$ is the ...
5
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1answer
188 views

$f ' (x) = f(x - (x+1)^t + 1)$

Let $x > 0 $ and $c $ a given real $> 0.$ Let $t $ be between $0 $ and $1.$ How to find $f(x)$ or good asymptotics for $ f(x)$ such that $$ f ' (x) = f(x - (x+1)^t + 1) $$ And $ f(1) = 1 + ...
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1answer
37 views

What does it mean for a difference equation to be made dimensionless?

Given the logistic difference equation, what does it mean for it to be made "dimensionless" so that it's written in a simpler form? $${N_{n+1}-N_n\over \Delta t}=r\cdot N_n\cdot \left(1-{N_n\over ...
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2answers
36 views

Find $x(t)$ and $y(t)$ which satisfy the following differential equations

Find $x(t)$ and $y(t)$ which satisfy $3\dot x + \dot y +5x-y=2e^{-t}+4e^{-3t}$, $\dot x + 4\dot y -2x+7y=-3e^{-t}+5e^{-3t}$, subject to $x=y=0$ at $t=0$. This is how I tried it: If we multiply ...
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1answer
54 views

How can I solve like this exercise

Let we have the following initial value problem : $$y'=f(x,y)=e^y$$ With the condition $y(0)=0$ Find the largest interval $|x| \le a $ makes the initial value problem has an unique solution ...
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1answer
35 views

Differential Equations: Linear or Nonlinear

In my textbook, the authors said that a differential equation is linear if it can be expressed in the form $$a_0(t)y^{(n)}+a_1(t)y^{(n-1)}+\cdots+a_n(t)y=g(t)$$ According to the definition, why the ...
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0answers
75 views

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in Polar coordinate $ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial u} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 u} {\partial \theta^2}$ with ...
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2answers
23 views

What theorem can I use to decide if an ODE which admits separation of variables has a unique solution?

Suppose that I have the IVP : $$y' = f(x)g(y)$$ $$y(a)=b$$ It's easy to show that any solution of such an equation will satisfy the implicit formula: $$ \int \frac{1}{g(y)}dy = \int f(x) dx$$ I ...
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34 views

Separable ODE and singular solutions

In most introductory ODE textbooks we can find the following: A separable first-order ODE is the one of the form $$y'=g(x)h(y)$$ and if $h(y)\neq0$, then the general solution is found by integration ...
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0answers
26 views

Getting solutions for an ODE in different intervals

I have to obtain solutions for: $$ x(x-1)y' + y = x^2 -1$$ In the intervals $(- \infty, 0), (0,1), (1,+ \infty)$. I got the solution $$f(x) = \frac {x}{x-1}(C+x+\frac{1}{x})$$, where $f$ is defined ...
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1answer
22 views

Find A,B such that the given function is a solution to the given differential equation

The equation is: $x(t)=A\sin(t)+B\cos(t), \; x' - 3x = \frac{1}{2}\cos(t)$ I'm honestly just lost from where to start. I'd really appreciate any help. Answer: $A = \frac{1}{20}, \; B= ...
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1answer
59 views

Solution of differential equation by conversion to an integral equation [on hold]

Let us say we have the following initial value problem : $$ y''+g(t,y)=0$$ such that $y(0)=y_0$ And $y'(0)=z_0$ . And $g$ is a continuous function in the domain $D$ contains the point $(0,y_0)$. ...
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1answer
47 views

How can I solve the following differential equation [on hold]

How can I solve the following differential equation : $$w''+(\sin z)w'+(1+z^2)w=0$$ In two case : without use power series use power series near the point $z=0$
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1answer
33 views

The time interval a solution exists for

I understand Q1, however it is required for Q2 Let $J(x_0)$ be the union of all open intervals $J(x_0)∋0$ st. $x_{Ji}$ is a solution of $x′=f(x),x(0)=x_0$ (∗) so $J(x_0)$ takes the form ...