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

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2
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2answers
42 views

Differential Equation -how to solve for the position of the rock on the given problem.

A $2 kg$ rock is thrown up from a $180 m $ high cliff with an initial velocity of $20 m/s$. Assume the force due to air resistance is equal to $0.5 v$, where $v$ is the instantaneous velocity, ...
0
votes
1answer
24 views

Does $\lim_{x \rightarrow \infty}y(x)$ exist for this nonlinear ode?

Consider the (very) nonlinear IVP $ f(x) = \left\{ \begin{array}{lr} y'=\displaystyle\frac{(1+x^2)y}{(1+x^2+y^2)^{\frac{1}{2}}} \\ y(0)=\eta \end{array} \right. $ Where ...
1
vote
2answers
40 views

Solutions for differential equation

The motion of a particle moving along the x-axis obeys the differential equation: $ \ddot x - 4 \dot x + 4x =-t^2 $ Find the solution for $ x(t)$, given $ x(0)=0 $ and $ \dot x (0) = 0 $. Can ...
0
votes
0answers
11 views

Example Of Initial Value ODE For Stability Check Of Linear Multi-Step Method

We are attempting to provide an example of an initial value ordinary differential equation to show that the following "linear multi-step method" is unstable. ...
6
votes
2answers
83 views

If $\int_0^xf(t)dt=[f(x)]^2$ but $f(x)\neq 0$, what is $f(x)$?

From the problems plus in Stewart Calculus 6e, it asks if $f$ is a differentiable function such that $f(x)$ is never $0$ and for any $x$, $\int_0^xf(t)dt=[f(x)]^2$, then what is $f(x)$? I figured ...
2
votes
0answers
55 views

Derivatives of solution to Schrödinger equation

Consider the differential equation (Schrödinger, but rewritten to be pleasing to Lie algebraic eyes): $\frac{d U(t)}{dt} = c(t)U(t)$ where $c(t)=a+w(t)b(t)$, $a,b \in \mathfrak{su}(n)$ and $w$ is a ...
2
votes
1answer
39 views

Are Friedmann equations linear or nonlinear?

I'm trying to improve my understanding of cosmology, and these 2 equations are basic . You can find them here: https://en.wikipedia.org/wiki/Friedmann_equations Also, if you could tell why they are ...
0
votes
1answer
49 views

How can you solve $y_t=Dy_{xx}+1$?

Suppose we have $y=y(x,t)$ defined on $x\in [0,L] $ where $$y_t=Dy_{xx}+1$$ And on $x=0,L$ respectively we have $$-Dy_x=a(y-y_0)$$ $$-Dy_x=b(y-y_L)$$ How can this be solved? Ideally with a fourier ...
2
votes
2answers
80 views

Numerical solution to a system of secon order differential equations

I'm writing a sort of physical simulator. I have $n$ bodies that move in a two dimensional space under the force of gravity (for instance it could be a simplified version of the solar system). Let's ...
2
votes
0answers
22 views

Difficulty understanding Floquet multipliers wrt Mathieu equation

We have the system $$\begin{pmatrix}y\\z\end{pmatrix}' = \begin{pmatrix}0 & 1 \\ a-2\epsilon \cos t & 0 \end{pmatrix}\begin{pmatrix}y\\z\end{pmatrix}$$ and from Abel's formula we have that ...
0
votes
0answers
11 views

Floquet multipliers of Mathieu pendulum

We have a linearized version of a pendulum system: $$y'' = (a-2\epsilon \cos t)y$$ or equivalently $$\begin{pmatrix}y\\z\end{pmatrix}' = \begin{pmatrix}0 & 1 \\ a-2\epsilon \cos t & 0 ...
0
votes
1answer
43 views

Splitting a 2nd order PDE into a system of first order PDEs/ODEs

Consider a standard wave equation: $ \frac{\partial^2 p}{\partial t^2} = c^2 \frac{\partial^2 p}{\partial x^2} $ The question is how to formulate this as a first order system: $ \frac{\partial ...
5
votes
1answer
139 views

This theory proof about instability of a point of equilibrium is not understandable for me, any help?

-This theory is irritating me, because I don't understand it's logic. Theorem: If in some neighboorhood $\mathbb O (0)$, exists a continuous, differentiable function $V(X), V(0)=0,$ such that the ...
0
votes
1answer
40 views

Converse to Wronskian Condition ODEs

Let $u_1(x),\dots, u_n(x)$ be solutions to the $n$th order, linear, homogeneous differential equation $y^{(n)} + p_1(x)y^{(n-1)} + \dots + p_n(x)y = 0.$ Let $W := W[u_1, \dots, u_n]$ denote the ...
0
votes
2answers
38 views

How to calculate matrix exponential of a $2\times 2$ matrix with repeated e values

Specifically, I am trying to calculate the matrix exponential, $e^{At}$, where A = $\begin{bmatrix}-1 & 1\\-9 & 5\end{bmatrix}$. I calculated the the E values to be 2 with a multiplicity of 2 ...
2
votes
0answers
43 views

Does this type of differential equation have a name?

Does a differential equation of the form: $$y''(x)+\delta(x)y=Ay$$ where $\delta(x)$ is the Dirac Delta function and $A$ is a constant have a specific name?
0
votes
0answers
13 views

Modelling a solute in reactor by concentration or total amount?

I'm trying to obtain a model of the concentration of a solute, S (g/L) in a reactor with the variable volume V (L). A solution of S is continuously added to the reactor with the flow F (L/min), since ...
1
vote
1answer
46 views

System of differential equation with variable coefficent

How to solve this system of differential equations $x'(t)=\frac{a+s}{(1-t)d}x(t)-\frac{b}{(1-t)d}y(t)$ and $y'(t)=\frac{a}{(1-t)d}x(t)-\frac{(s+b+(1-t)c)}{(1-t)d}y(t)$ where a,b,c,d and s are ...
0
votes
1answer
33 views

Kinematics Problem (Differential Equation) : Particle moving in circle such that Radial acceleration=Tangential acceleration

A particle is moving on a circle of radius R such that at every instant the radial and tangential accelerations are equal in magnitude.If velocity of the particle be $v_0$ at t=0,then what ...
2
votes
0answers
14 views

Step response for different definitions of step function

I was thinking about the solution of the known problem of determining the step response for the concentration leaving a CSTR tank. The differential equation is: ...
1
vote
1answer
43 views

Solution to Second order ODE with variable coefficient

Is there any general method to solve a Second order ODE with variable coefficient of the form $a(x)y''(x)+b(x)y'(x)+c(x)y(x)=0$
0
votes
3answers
47 views

Why can't the particular solution have factors found in the general homogeneous equation in it?

Consider the following differential equation: $$y''-4y'+4y = g(x)$$ The general solution of the homogeneous equation is $e^{2x}(C_1+ xC_2)$. Find the particular solution when $g(x) = e^{2x}$ and ...
0
votes
1answer
47 views

Why does $dy/dt=5y^{4/5}$ not satisfy the Uniqueness theorem?

So I'm just wondering if my answer is correct. Why does $$\frac{dy}{dt}=5y^{4/5}$$ not satisfy the Uniqueness theorem? My thinking is since every number and its corresponding negative result in the ...
2
votes
1answer
25 views

Solving inhomogenous continuous-time system with non-diagonalisable system matrix

I have an exercise where i have to find the general solution to this problem: $$ X'=\left( \begin{matrix} 2&-1\\ 4&-2 \end{matrix} \right)X + \begin{pmatrix} 2\\1 \end {pmatrix}. $$ ...
0
votes
0answers
20 views

Existence and Uniqueness ODE

I have this ODE: $\theta''(t)+ \frac{g}{l} \sin(\theta(t)) =0$ The question is: "What do you know for certain about the existence and/or uniqueness of this ODE?" I can solve it so a solution exists. ...
0
votes
1answer
30 views

Find the Integrating factor of $y(x^2+y^2)dx - x(x^2+2y^2)dy =0$

Find the Integrating factor of $y(x^2+y^2)dx - x(x^2+2y^2)dy =0$ I've solved this a bunch of times but I still can't find the I.F. I often get stuck with the $\frac{M_y-N_x}{N}$ $\frac{\partial ...
3
votes
2answers
44 views

Find the Integrating Factor of $xdy-3ydx=\frac{x^4}{y}dy$

This is integrating factor by inspection, $xdy-3ydx=\frac{x^4}{y}dy$ I've been trying to look for the Integrating factor for this problem but I can't still get one right. I think I really need to use ...
0
votes
1answer
25 views

In proving existence and uniqueness of ODE, why do we consider rectangular regions instead of circular regions?

I had this question while reading a proof on existence and uniqueness of solution for ODE...example: http://www.math.uiuc.edu/~tyson/existence.pdf In the proof, function $y' = F(x,y)$ is assumed to ...
0
votes
1answer
49 views

Integrating Factor by Inspection $(x^3+xy^2+y)dx + (y^3+xy^2+x)dy=0$

$$(x^3+xy^2+y) \hspace{.1cm} dx + (y^3+xy^2+x)\hspace{.1cm} dy=0$$ So I tried to solve this problem but can't figure out my integrating factor all I can see here is if I distribute first I can get a ...
6
votes
4answers
178 views

$2^{nd}$ order PDE: Solution

I am trying to solve the following equation: $$\frac{\partial F}{\partial t} = \alpha^2 \, \frac{\partial^2 F}{\partial x^2}-h \, F$$ subject to these conditions: $$F(x,0) = 0, \hspace{5mm} F(0,t) = ...
1
vote
0answers
20 views

How To Graph This Hopf Oscillator

I tried using Wolfram to accomplish this with no success. The equation for an oscillator i is: $$\dot x_i = \alpha(\mu - r^2_i)x_i - w_iy_i$$ $$\dot y_i = \beta(\mu - r^2_i)y_i - w_ix_i$$ $$w_i = ...
1
vote
0answers
25 views

About an asymptote of Golomb's sequence and an asymptote of a sequence in general

In this question, I asked for the solutions to the differential equation $f(f(x))=\frac{1}{f'(x)}$ because I think that a solution to this equation is related to Golomb's sequence. Let me explain, ...
6
votes
1answer
51 views

Differential equation related to Golomb's sequence

While studying Golomb's sequence, the following differential equation arouse: $$ f(f(x))=\frac{1}{f'(x)} $$ I don't know much about differential equations so I am a bit clueless. Is there a way to ...
1
vote
0answers
20 views

$t^{\nu}K_{\nu}(\beta t)$ The solution to an unknown ODE.

Good day to you all, This question is fairly open, so I hope that I'm not in breach of any of the Stack Exchange rules. Clearly $f(t)=K_{\nu}(\beta t)$, where K is the modified Bessel function of ...
0
votes
0answers
22 views

asymptotic matched expansion with transiently blowing inner solution

I have been trying to solve following set of equations with method of matched asymptotic expansion, $\frac{dy(t)}{dt}=k z(t) - 3 \alpha y(t) - y(t)^2 + \mu (M-z(t))^2$ $\epsilon ...
3
votes
3answers
399 views

How to solve this seemingly simple initial value problem? $xy' - y = x^2$

$$xy' - y = x^2,\quad y(-1) = 0$$ I found this question in the old exams paper of a Calculus II course. I tried finding $y'(-1) = 1$ but it doesn't seem to be helpful at all. Can anyone help? Thanks ...
0
votes
0answers
32 views

Normal differential equations

In my textbook I read: "A $n^{th}$ order DE is called normal when it can be rewritten in the following form: $y^{n} = f(x, y,y', ... , y^{n-1}$)" I don't really get this. At first sight it seems all ...
2
votes
1answer
38 views

Non linear system of differential equations

Is there a specific name to the following type of non linear ODEs $\begin{array}{c} \dot{x}_1 &= c_1 \, x_2\, x_3 \\ \dot{x}_2 &= \, c_2 x_1 x_3 \\ \dot{x}_3 &= c_3 \, x_2 x_1 ...
0
votes
1answer
24 views

Convergence of Linear First-Order Differential Equations

Suppose $u$ is a twice continuously differentiable function with linear growth, $$\lim_{x\rightarrow \infty} u'(x)-\frac{1}{g(x)} u(x) = 0 $$ and $g$ is a Lipschitz continuous function with Lipschitz ...
2
votes
0answers
72 views

Do I have any hope with this PIDE?

$\frac{\omega(1-\omega)}{N_1} \frac{\partial^2 f}{\partial x_1^2} + \frac{\omega(1-\omega)}{N_2} \frac{\partial^2 f}{\partial x_2^2} + \cdots + \frac{\omega(1-\omega)}{N_k} \frac{\partial^2 ...
0
votes
2answers
47 views

Looking for a Finite Difference scheme of the following form…

I'm having trouble deriving a finite difference scheme that calculates the second derivative of a function on the boundaries of a non-uniform grid and makes use of a known first derivative at the ...
0
votes
1answer
34 views

Conditions on a linear system of ODEs

Let $x:[0,T]\to\mathbb{R}^n$ and $y:[0,T]\to\mathbb{R}^n$ be solutions to an $n\times n$ system of linear ODEs. That is, $$\frac{dx}{dt}=A(t)x+b(t) \mbox{ and } \frac{dy}{dt}=A(t)y+b(t) \mbox{ for } ...
2
votes
0answers
27 views

two point block method for solving ODE

How to solve the ordinary differential equation $$y'(t) = -1000 y(t)+ 999 e^{-t}, \hspace{10mm} 0≤t≤5.$$ $y(t)=e^{-t}$, for $t<0$. Using two point block method $$hf_{n+1}= \frac{1}{3} (hf_{n+2} ...
2
votes
0answers
37 views

Relationship between solutions of two matrix differential equations

Given a ($4\times4$ in the important case) matrix differential equation: $\frac{d U_t}{dt}= A_t U_t$ where $U_t \in SU(n)$ and $A_t \in \mathfrak{su}(n)$. What is the relationship between the ...
0
votes
0answers
17 views

block method to solve ODE

How to solve the equation $$y'(t)=-1000y(t)+999e^{-t},\quad 0 \leq t \leq 5\\ y(t)=e^{-t},\quad t \leq 0 $$ using block method $$ hf_{n+1}=\frac{1}{3}\left(hf_{n+2} - 2y_n + 2y_{n+1}\right)\\ ...
1
vote
2answers
44 views

Question of Differentiation/Integration

An ice cube melts and decreases in volume at a rate of \begin{align} &10.8cm^3s^{-1} \end{align} and it's length decreases at rate of\begin{align} 0.5cms^{-1}\end{align} find its width. Ans: 2.68 ...
0
votes
1answer
19 views

Applying existence and uniqueness of ODE

At page no. $333$ of Spivak's Differential Geometry book , theorem $13$ says that fundamental existence and uniqueness theorem guarantees the existence of $\epsilon_1$ and $\epsilon_2 >0$ so that ...
2
votes
1answer
42 views

$y'=\sin^2(y-x)$

I want to solve $y'=\sin^2(y-x)$ and see how the solutions look like, but I have no experience with this short of trigonometric differential equation. Should I simply try to integrate it like this? ...
1
vote
1answer
18 views

Phase portrait of a $2 \times 2$ system of linear, autonomous differential eqns. with a zero eigenvalue

Let $\mathbf{Y} = \begin{pmatrix} x(t) \\ y(t) \end{pmatrix}$ and $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $a, b, c, d \in \mathbb{R}$. Now, consider the system $$ ...
2
votes
0answers
32 views

Solution to system of ordinary differential equation

Given the system: $\begin{cases}x''=2y \\ y''=-2x\end{cases} $ I found the (I think) equivalent linear equation $x^{IV}+4x=0$ First question: is the equation actually equivalent to the system? ...