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

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Two problems from Avner Friedman's PDE book.

The problems are as follow: Prove that if $Lu=0$ for any $u\in C^m(\Omega)$, then $L\equiv 0$ - that is all the coefficients of $L$ vanish identically. Prove that the assertion of the previous ...
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0answers
11 views

Eigenvalue interpretation with direction field

I am running into some trouble with respect to some direction field plots of different eigenvalues. I am working with a system given as follows: $$ \overrightarrow{y'} = \begin{pmatrix} 0 & -5 ...
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0answers
10 views

Implicit Runge Kutta (second order)

I'm familiar with explicit numerical methods for solving ODE including Euler's method, and even Runge-Kutta methods (2nd and 4th order). But I'm really confused when it comes to implicit methods. I ...
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2answers
26 views

Find the function satisfying the differential equation

Find the function satisfying the differential equation $$ f'(t)-f(t)=-7t\;\;\;\;\; f(3)=5 $$ For some reason I've never completely understood what f'(t) and f(t) or function notation very well. Is ...
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16 views

What is the saddle-point approximation? [on hold]

I want to take advice which books are useful to understand saddle point approximation. Can you give suggestion about that ? Also, if you explain what is the saddle point approximation, I will be so ...
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0answers
35 views

Picard iteration for second order ODE

Any help for the following two questions will be much appreciated: Question 1: Consder the ODE with IVP $y''(t)+g(t,y)=0, \ y(0)=y_0,\ y'(0)=z_0$ where g is continues on some region $D$, ...
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0answers
22 views

The solutions of second-order ODE $y''+ay=0$ with a negative coefficient are hyperbolic functions

I am trying to solve a partial differential equation where $$u(a,\theta,z)=0,\quad -\pi \leq \theta \leq \pi, \quad 0\leq z \leq b$$ $$u(r,\theta,b)=0,\quad 0 \leq r < a,\quad -\pi\leq \theta ...
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16 views

Inconsistency between analytical and numerical solution for differential equation and trouble with interpreting it

I am a bit confused over a result that I am getting for a differential equation solved analytically and numerically along with the direction plot. Basically, it the system is as follows: $$ ...
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2answers
37 views

Finding temperature after time. [on hold]

A thermometer is taken from a room where the temperature is $20^\circ\,\mbox{C}$ to the outdoors, where the temperature is $-7^\circ\,\mbox{C}$. After one minute the thermometer reads ...
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1answer
25 views

Can every separable differential equation be rewritten to potentially be exact (or NOT exact)?

Let's say an ordinary linear DE is separable. Then $$\frac{dy}{dx} = P(y)Q(x) \Leftrightarrow \frac{1}{P(y)}dy = Q(x)dx \Leftrightarrow Q(x)dx + R(y)dy = 0$$ is in exact form where ...
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17 views

Differential equations with dense solutions

Consider the differential equation $P(y',y'',y''',y'''')=0$ on $\mathbb R$, where $P(x,y,z,w)$ is the homogeneous polynomial of degree $7$ given by $$ ...
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1answer
15 views

Orthogonal trajectories - why is it necessary to isolate the parameter

For orthogonal trajectory, I realized that I need to express the parameter of the given family of curves in terms of x and y, in order to get the right answer. e.g. in $y = kx$, $k$ is the parameter ...
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0answers
35 views

Functional use of Differential Equation

Another model for a growth function for a limited pupulation is given by the Gompertz function, which is a solution of the differential equation $$ \frac{\mathrm{d}p}{\mathrm{d}t} ...
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36 views

Receiving different answers

Ok, so im following a tutorial on how to calculate a limit numerically and when the tutor plug'd in the number $(-1.1)$ into the equation $\frac{(t^6 -1)} {(t^3 + 1)}$ HE gets −2.331 as the ...
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0answers
24 views

Expressing a solution to a differential equation in a more compact form

We know that $Ae^{ikx}$ is a solution for $k \geq 0$ and $Be^{ikx}$ is a solution for $k \leq 0$. So does the $N$ here depend on whether or not $k \geq 0$ or $\leq 0$? Also I do not understand how ...
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0answers
34 views

show that the Power series solve the differential equation

$$f(t)=\sum\limits_{k=0}^{\infty}\frac{ (-1)^k\,(t/2)^{2k}}{(k!)^2}$$ show $t\, f''(t)+f'(t)+t\, f(t)=0$ By differentiating it, I got $$f'(t)= \sum\limits_{k=0}^{\infty}\frac ...
3
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0answers
47 views

Proving that $\lim\limits_{t\to\infty} e^{At}x_0 + \int\limits_0^\infty e^{A(t-s)}b(s)ds=\vec{0}$

Consider $x'=Ax+b(t)$, a system of differential equations. Given that $A$ has negative real parts in all its eigenvalues, and that $\lim\limits_{t\to\infty} b(t) = \vec{0}$, I need to prove that ...
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1answer
76 views

Solution of the Beast in Disguise $y'=x-y^2$ [on hold]

$y'=x-y^2$ Its a first-order nonlinear ordinary differential equation. The solution is given here http://www.wolframalpha.com/input/?i=solve+y%27%3Dx-y%5E2&x=0&y=0. But how is it obtained? ...
2
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1answer
32 views

How do the initial conditions change if I solve $\frac{dx}{dy}$ instead of $\frac{dx}{dt}$ and $\frac{dy}{dt}$

Good day, In differential equations it is often useful to compute $\frac{dx}{dy}$, I mean if you have System like $$\frac{dx}{dt}=xy, \frac{dy}{dt}=y, y(t_0)=y_0, x(t_0)=x_0~ (x_0,y_0>0)$$ These ...
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2answers
30 views

Coupled differential equation

$\ddot{y}=\omega \dot{z}$ $\ddot{z}=\omega (\frac{E}{B}-\dot{y})$ These are the coupled differential equation i came across . They have already been asked here How to Solve the Coupled Differential ...
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2answers
38 views

Unsure why ODE non-exact equation solution is wrong?

The question I'm trying to solve is $$\left(y-4y^6\right)=\left(y^4+5x\right)y'$$ where $y(0)=1$ I want to find the solution explicitly for $x$. I found the integration factor to be $u=y^-6$. ...
2
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1answer
53 views

f: R → R and $|f'(x)| ≤ |f(x)|$ [duplicate]

Let $f: R → R $ be a function such that $f'(x)$ is continuous and $|f'(x)| ≤ |f(x)|$ for all $x ∈ R$ , if $f(0)=0$ the maximum value of $f(5)$ is My Attempt: I proved that $f'(x)=0$ for $x ∈ [0,1]$ ...
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1answer
19 views

Solve the initial value problem

$2yy'+5=y^2 +5x$ with $y(0)=6$ To solve this, we should use the substitution $u=$ With this substitution, $y=$ $y'=$ After the substitution from the previous part, we obtain the ...
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1answer
16 views

Multiple eigenvalue solutions problem

In a problem regarding multiple eigenvalue solutions (defective eigenvalues, complete eigenvalues, the like) I have a 4x4 matrix with one complete eigenvalue, and another incomplete eigenvalue with a ...
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0answers
16 views

Fundamental solutions for pde [on hold]

I am looking for fundamental solution for the following PDE problem. $$2xy(u_{xx} + u_{yy}) + yu_{x} + xu_{y} =0$$ Any help will be appreciated. Thnak you.
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1answer
16 views

Linear independence and Wronskian - Proof or Counterexample

If $y_1(x) , y_2(x) ,\ldots,y_n(x)$ are linearly independent in $C[b,c]$ then they are Linearly Independent in $C[a,d]$, where $a<b<c<d.$ So I know if the Wronskian isn't zero for at least ...
0
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1answer
37 views

Show there are infinitely many distinct maximal solutions of $\frac{dx}{dt} = (3/2)x^{1/3}$ that pass through the point $(t_0,0)$

$$\frac{dx}{dt} = (3/2)x^{1/3}$$ Solve Show that given any point $(t_0,0)$ on the $t$-axis, there are infinitely many distinct maximal solutions that pass through the point. We are given: ...
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1answer
45 views

Differential equation including Dirac Delta function

I am trying to understand Dirac delta function. Here is a DE to solve: $f''(x) - 3f'(x) +2f(x)=k\delta(x-a)$ with intial conditions $ f(0)=f'(0)=1$ where $ k \in \mathbb{R}$ is constant. I know this ...
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0answers
34 views

Find Population from Model

First time I asked this it didn't get entered correctly so got wrong answers, which helped explain a different question I had. $$\frac{dp}{dt} = \frac{2p(7-p)}{700}$$ Given $p(0) = 3$ and $p \gt ...
2
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1answer
35 views

Jump Diffusion Infinitesimal generator

I have this difussion process $dX(t)=\mu X(t)dt+\sigma X(t)dW(t)+u X(t) dN(t),\qquad X(0)=x > 0$ where $W(t)$ is a Brownian Motion and $N(t)$ is a Poisson process. And I need to know the ...
2
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1answer
33 views

Maximal interval of solutions existence: $x'(t)=-x(t)+\sin x(t)+t^3$

$x'(t)=-x(t)+ \sin x(t)+t^3$ in $\mathbb{R}$ I consider the function: $$ f(t,x)=-x+\sin x + t^3 $$ $$\frac{\partial f}{\partial x}=\cos x-1$$ I see that: $$\left| \frac{\partial f}{\partial x} ...
2
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0answers
43 views

Solution of $x'=Ax$ is not what it is supposed to be

Consider a system of ODEs $x'=\begin{bmatrix} 0 &1 \\ -1&0 \end{bmatrix}x$. Wolfram Alpha says that the solution is $x(t) = \begin{bmatrix} \cos t &\sin t \\ -\sin t& \cos t ...
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28 views

Solving a delay differential equation

Is it possible to solve differential equations with composite functions, e.g. $f^{(n)}\circ (x-1)$ and $f^{(n)}\circ(x)$. I'm particularly interested in $$2f'(x-1)+(x-1)f''(x-1)-f''(x)=g(x).$$ Do ...
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1answer
14 views

Definition of trajectory

I am writing something that involves comparing the solutions of many different differential equations, and I need precise definitions of the terms trajectory and solution curve. Given a dynamical ...
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2answers
27 views

Canonical form and fundamental solution of pdf

Can someone help with these two PDE problems? Thank you. Reduce to Canonical form and find the fundamental solution if possible. $$y^2u_{xx} + x^2u_{yy} = 0.$$ What type of transformation should I ...
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2answers
42 views

Seperation of variables is not rigorous enough.

So I have started ODE's and have been told that solving initial value problems by using separation of variables(like I had been taught) is nothing but a leap of faith. Now we must use something along ...
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1answer
21 views

Ex ODE: $y'=4t \sqrt y- \lambda(y-(1+t^2)^2)$

How to solve the following equation? $y'=4t \sqrt y- \lambda(y-(1+t^2)^2)$ $y(0)=a$ Show those cases where a numerical method will solve this equation exactly. $(a,\lambda) \in {\Re}^2$
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2answers
34 views

Solution Set of System of ODE.

I am trying to find the solution of the system $$\begin{bmatrix}x_1\\x_2\end{bmatrix}'= \begin{bmatrix}1&3\\3&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$$. I am given that ...
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0answers
29 views

How to calculate inverse laplace of $e^{a\sqrt s}$?

I was using Laplace to find solutions for $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions $$u(0,t)=1 \\ u(1,t)=1 \\ u(x,0)=1+ \sin \pi x$$ I used ...
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Stability of gradient flow $x'(t) = -\nabla E(x)$, $E(x)$ is convex function. [on hold]

Let $E: R^d \to R$ be convex and continously differentiable. The IVP $x'(t) = -\nabla E(x), x(0) = x_0, t>0$ is called the gradient flow. Show that gradient flow has following properties a) ...
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2answers
28 views

Explicit Euler Method's Order

I am seeing everywhere that the order of the Explicit Euler Method is 2 but I can't prove it on my own. The textbook that I found the method says that the proof is very easy so it is up to the reader ...
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1answer
33 views

How to use the 3rd and 4th boundary conditions in this?

I was solving $$ \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}$$ All the boundary conditions are as follows:- $$u(0,t)=0 \\ u(\pi ,t)=0 \\ u(x,0)=\sin x \\ u_t(x,0)=x^2$$ ...
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0answers
48 views

Eigenvalue problem for $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue $E$ in the complex plane of $x$ for one dimensional Schrodinger equation $$ −\psi''(x) − (ix)^ N \psi(x) = E\psi(x). $$ where $N$ can be any real number, the boundary condition ...
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4answers
65 views

Why cannot $A\sin\alpha x +B\cos \alpha x$ be zero?

I was going through solving wave equations using fourier and I came across a note saying $A\sin\alpha x +B\cos \alpha x \neq 0$ I believe this applies to $\alpha ,A,B\neq 0$ I was solving $$ ...
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2answers
21 views

If $g(t)=d$ (constant), show that all solution of $L[y] \to \frac{d}{c}$ when $t \to \infty$

Let the differential equation $L[y] = a y'' + by' + cy = g(t)$, where $a$, $b$ and $c$ are strictly positive numbers. If $g(t)=d$ (constant), show that all solution of $L[y] \to \frac{d}{c}$ when $t ...
2
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1answer
62 views

Doomsday Prediction

I have a calculus problem I can't seem to figure out. Any help would be appreciated! Doomsday prediction. In $1960$, three electrical engineers at the University of Illinois published a paper in ...
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1answer
18 views

Show that $Y_1[t]- Y_2[t] \to 0$ as long as $t \to \infty$ - Differential equations

Let the differential equation $L[y] = a y'' + by' + cy = g(t)$, where $a$, $b$ and $c$ are strictly positive numbers. If $Y_1(t)$ and $Y_2(t)$ are solutions at the $L[y]$ equation, show that $Y_1[t]- ...
2
votes
1answer
20 views

Turing criteria for Sel'kvo glycolysis model

I have the Sel'kov reaction diffusion model for glycolysis as follows: \begin{eqnarray} u_t=D_uu_{xx}-u+av+u^2v\\ v_t=D_vv_{xx}+b-av-u^2v \end{eqnarray} How can I obtain the values for $D_u$ and ...
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0answers
14 views

Searching for second singular solution of a differential equation

I was given the following equation $\frac{dy}{dx}=\sqrt{y-4}$ and was asked to solve it and find a family of solutions. I've solved the equation so that $y=\frac{x^2+16}{4}+c$ or ...
0
votes
1answer
31 views

Separable solution to a nonlinear parabolic PDE

I seek a separable solution to the nonlinear parabolic partial differential equation, $\frac{\partial u}{\partial t} = u \frac{\partial u}{\partial x^2} + u^2.$ The physics of the problem allow ...