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Questions tagged [ordinary-differential-equations]

For questions about ordinary differential equations, which are differential equations containing only derivatives w.r.t. one variable. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

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0answers
17 views

computing $\exp(At)$ in differential equations?

Find a fundamental matrix of the linear system $x'(t) = Ax(t)$ by computing $e^{At}$. (a) $A = \pmatrix{3& 1\\ 0 &3}$, (b) $A = \pmatrix{2 & −1\\ 1& 2}$, (c) $A = \pmatrix{2 & 0 ...
0
votes
2answers
31 views

which of the following functions can't be a solution of given ODE?

Which of the following functions can't be the solution of ODE $$y''+p(x)y'+q(x)y=0$$ for some continuous functions $p(x), q(x)$ are functions on some interval. $1)$ $e^{3x}$ $2)$ $x^{2}$ ...
2
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0answers
26 views

Let $f(x)=x^2$ and $g(x)=x^3,x\in[-1,1].$ Then which of the following are true?

Let $f(x)=x^2$ and $g(x)=x^3,x\in[-1,1].$ Then which of the following are true? (A) $W(f,g)(x)=0, x\in [-1,1]$ (B) $f$ and $g$ are linearly dependent. (C) $f$ and $g$ are the linearly independent ...
1
vote
2answers
54 views

How to solve $y^{\prime \prime}(x)= a \cos (y)$?

I was solving a physics questions and was stuck on this differential equation: $y^{\prime \prime}(x)= a \cos (y)$, where $a$ is some constant. I have no idea how to start. Please give a hint.
0
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0answers
14 views

Reference Request - Applied Math books and resources

I am second-year undergrad from Physics. I wanted to move to applied mathematics because I like the topics they study, like differential equations, fluid dynamics and related stuff. So I am looking ...
2
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1answer
26 views

How to change this Runge-Kutta method implementation from first order ode solver to system of ODEs solver?

I implemented 4-step Runge-Kutta method (k1..k4) ODE solver for a function $u'(x) = f(x,u(x))$ with initial condition $u(x_0) = u_0$ But it solves just ODEs of the first order. How could I change ...
0
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0answers
12 views

Show that for a regular S-L system, if q(t) is increased to q1(t) > q(t), each nth eigenvalue of the new system is larger than that of the old.

Consider the S-L equation $$\frac{d}{d t}\left[p(t) \frac{d u}{d t}\right]+[\lambda r(t)-q(t)] u=0$$ Show that for a regular S-L system, if $q(t)$ is increased to $q_{1}(t)>q(t),$ each $n$ th ...
4
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0answers
48 views

How to obtain recursion relations from this

I'm trying to solve a problem using the power series solution. Finally (and after substitution of differentations) I have come up with $$ -\frac1{2\mu}\sum_{i=2}^p i(i-1)a_i r^{(i+l-1)}+\frac1{2\mu}\...
0
votes
1answer
30 views

differential equation, general form

It's such a crazy question, but I just confused that how can I find general solution!!? $$ 2y''-y'=1, \qquad y(0)=0, \qquad y'(0)=0 $$ I tried with these forms but couldn't find! 1) $y=c_1c^x + c_2xe^...
3
votes
1answer
19 views

2nd order linear ODE non constant coefficient solution methods

I'm working on a project and I've come across a seemingly standard ODE, however I have no idea how to solve it. The equation in question is $$ \epsilon'' + \left[g(\tau) + \frac{\beta}{\tau}\right]\...
0
votes
1answer
25 views

find the value of y(x) at $\pi$

The value of $y(x)$ at $x=\pi$ when $y''+f(x)y=0$ and $$ f(x)=\begin{cases} -1&\;{\rm for}\;0\leq x\leq\frac{\pi}{2}\\ 1&\;{\rm for}\;\frac{\pi}{2}\leq x\leq\pi \end{cases} $$ given that $y(...
0
votes
1answer
22 views

Output response from closed loop transfer function using MATLAB

This transfer function is to control the position of Permanent Magnet DC motor. I was able to get the transfer function and now I need to analyze the output for the tuned closed loop for a given input ...
0
votes
1answer
33 views

Conditions on The Characteristic Equation for an ODE

In a previous post on the Fourier Transform of Airy Equation, I assumed that the characteristic equation could be taken for an ODE of the form $$iy'(x)-k^2y(x)=0, \tag{1}$$ where $k\in\mathbb{R}$ and $...
3
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1answer
247 views

Fourier Transform of Airy Equation

I am trying to find $Y(k)$ of the equation $y''(x)-xy(x)=0$ and hence show that $$y(x)=\sqrt{\frac{2}{\pi}}\int_0^{\infty}\cos\left(\frac{k^3}{3}+kx\right) \ dk,$$ given $Y(0)=1$. Here, we use the ...
3
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1answer
25 views

ODE in $\mathbb{R}^n$ defined by the gradient of a function

I'm studying for an exam and I got stuck in this question: Let $x: I \to \mathbb{R}^n$ be a differentiable parametrized curve (I is an interval) in $\mathbb{R}^n$ and $f: \mathbb{R}^n \to \mathbb{R}$...
-1
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0answers
20 views

On a second order non-linear differential equation

How can one solve following non-linear differential equation (in term of $z$)? $$∂^2 y/∂z^2 +2ib ∂y/∂z -d y^3 =0$$ If I knew the boundary conditions $y=a$ and $y'=0$ at $z=0$ and I knew the ...
1
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0answers
17 views

Separable equations and implicit differentiation and differential form.

I'm a tad confused about some of the symbolic notation in my book regarding separable equations: I thought when you integrate an equation (let's call it a) and differentiate it, you wind up with a ...
1
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1answer
14 views

How is stability for a numerical solution generalized to a system of ODEs?

Consider the system of ODEs $$y' = \begin{bmatrix}-6&4\\4&-6\end{bmatrix}y, \quad t\in[t_0, t_e], \quad y(t_0)=y_0$$ I'm asked for what stepsize the explicit Euler method generates a stable ...
1
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0answers
20 views

Fokker planck equation, simple change of variables

I’m studying this fokker planck equation $\frac{\partial}{\partial t} p + a \frac{\partial}{\partial x} p = D \frac{\partial^2}{\partial x ^2}p $ where p is a fuction of x and t. a, D constant. Now ...
2
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0answers
17 views

Stability region of explicit midpoint method [duplicate]

Consider the explicit midpoint method, i.e $$y_{n+1}-y_{n-1} = 2hf(y_n).$$ I'm asked to apply this method to the linear test equation, $f(y_n) = \lambda y_n,$ in order to find the method's stability ...
0
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0answers
19 views

variation of parameters 1st order ODE Version

I have been given a variation of parameters formula for a first order ODE $x'(t) = a(t)x(t)+b(t)$ and have been asked to differentiate it, the formula is: $$ x(t) = Ce^{\int_{t_0}^{t}a(s)\,ds} ...
0
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0answers
16 views

help - A system consists of springs A and B and objects C and D

Hi all, I have completed the question up to part c). Using matlab, I have found the eigenvectors/eigenvalues but from there I am unsure as to how to find the solution which would comprise of the below ...
4
votes
2answers
210 views

Can we cancel the equality mark here?

Problem Let $f(x)$ satisfy that $f(1)=1$ and $f'(x)=\dfrac{1}{x^2+f^2(x)}$. Prove that $\lim\limits_{x \to +\infty}f(x)$ exists and is less than $1+\dfrac{\pi}{4}.$ Proof Since $f'(x)=\dfrac{1}{x^2+...
0
votes
2answers
32 views

Find number of functions $f$ which satisfy given conditions

Let $\ln (x)$ denote the logarithm of $x$ with respect to the base $e$. Let $S ⊂ \mathbb R$ be the set of all points where the function $\ln(x^2 − 1)$ is well defined. Then the number of ...
0
votes
0answers
18 views

Construct a differential equation whose solution in parametric form is the butterfly curve.

Is it possible, and if so, does anyone know how to construct a differential equation whose solution on parametric form is the butterfly curve: $$x=\sin (t)\left(e^{\cos (t)}-2 \cos (4 t)-\sin ^{5}\...
1
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1answer
31 views

How can I find the control for a finite system by definition?

Currently I am working on control theory, precisely in controllability but still on the basics, in the following example by E. Zuazua: It says, consider the following problem \begin{equation} \...
0
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2answers
26 views

Finding solution to a non-linear differential equation

I am given the equation below, where b is the average number of births (b=8), and d is the average number of deaths (d=3), and I am given an initial condition: P(0) = 500. Given the above values, I ...
2
votes
0answers
23 views

Why does the rotating wave approximation work?

Consider two coupled oscillators with position coordinates $X_a$ and $X_b$. In general, the motion is described by a system of coupled first order linear differential equations: $$ \frac{d}{dt} \begin{...
0
votes
1answer
28 views

Paul's Online notes example seems to be wrong? Variations of parameters diff equation

Paul's Online Note Example As far as I know, the complementary solution only consists of cos and sin when the roots are complex. From what I see, the roots for this equation are r=0 or r=-9. Not ...
1
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1answer
29 views

Differential equations error of magnitude question

Let $x = x(t), y = y(t)$ be the solution to the initial-value problem $$\frac{dx}{dt} = -x - y, \hspace{1em} \frac{dy}{dt} = 2x - y, \hspace{1em} x(0)=y(0)=1.$$ Suppose that we make an error of ...
5
votes
2answers
137 views

How to remove this numerical artifact?

I am trying to solve a differential equation: $$\frac{d f}{d\theta} = \frac{1}{c}(\text{max}(\sin\theta, 0) - f^4)~,$$ subject to periodic boundary condition, whic would imply $f(0)=f(2\pi)$ and $f'(0)...
3
votes
1answer
41 views

How to prove $\lim_{n\to\infty}\sup A(t)\le\frac{a}{b}$?

Suppose $A(t)>0(t\ge 0)$, $a, b>0$, let $$ A'(t)\le aA-bA^2. $$ Prove $\lim_{n\to\infty}\sup A(t)\le\frac{a}{b}$. Using Taylor formula $$ A(0)=A(t)-tA'(t)+o(t)\ge (1-ta)A(t) +tbA^2(t)+o(t). ...
5
votes
1answer
247 views

Proving that any solution to the differential equation of an oscillator can be written as a sum of sinusoids.

Suppose you have a differential equation with n distinct functions of $t$ where $\frac{d^2x_1}{dt^2}=k_{11}x_1+...k_{1n}x_n$ . . . $\frac{d^2x_n}{dt^2}=k_{n1}x_1+...k_{nn}x_n$ I want to show ...
2
votes
1answer
32 views

ODE with discontinuous vector field

Consider the ODE $$\partial_t \Phi(t,x) = \mathbf b(\Phi(t,x)), \qquad t \in [0,T], \quad x=(x_1,x_2) \in \mathbb{R}^2$$ $$\Phi(0,x) = x, \quad x \in \mathbb R^2,$$ where $\mathbf b = (0,\chi_{\{x_1 \...
0
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1answer
18 views

Solving a differential equation with an unknown relation

Say we have an unknown function $x(y)$. But we do know that $f(x) + g(y) = {{dx} \over {dy}}$, where $f(x)$ and $g(y)$ are known. Is it possible to find $x(y)$ like this? If not, what can we know ...
1
vote
2answers
54 views

Solving $\frac{dy}{dx}=\sqrt{3x+2y}-\frac{3}{2}$ without stuff from higher-order differential equations

I'm trying to solve this equation: $$\frac{dy}{dx}=\sqrt{3x+2y}-\frac{3}{2}$$ without using stuff from higher-order differential equations. I've tried using substitution $ w=\frac{y}{x} $, but ...
0
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0answers
21 views

Uniqueness of integral curves starting on boundary?

I'm only aware of ODE uniqueness on open subsets of $R^n$, which is useful when we want to prove uniqueness of integral curves on a smooth manifold. Suppose we have an interal curve $f:[0,\delta]\to ...
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0answers
39 views

Prove that $f(x,0)=f(0,x)$ for all X (Hint:Use A1) Let f(x):{ f(x,0) if x≠0, Let f(x):{ 1 if x=0 [on hold]

Consider throwing a dart at the origin of the Cartesian plane. You are aiming at the origin, but random errors in your throw will producr varying results. We assume that: 1.) The errors do not depend ...
1
vote
1answer
30 views

Picture flow of ODE

Consider the ODE $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0, \quad x \in \mathbb R^2 \\ \Phi(x,0) = x, \quad x \in \mathbb{R}^2 \end{cases}$$ Suppose that the flow $\Phi$ ...
3
votes
1answer
51 views

Using a Lyapunov function to determine stability of equilibria

Given $$\left\{\begin{aligned} x' &= -x^3 + 7xy^2\\ y' &= -3x^2y+y^3\end{aligned}\right.$$ find $a, b > 0$ such that $L(x,y) = a x^2 + b y^2$ obeys $\frac{d}{dt}L \neq 0$ whenever ...
1
vote
1answer
16 views

What is the general solution of $xy=(x^2+4y)(dy/dx)?$

I have tried to use u-substitution to separate the variables, as well as distributing $dy/dx$ as y' to $(x^2+4y)$. However, I just can't seem to separate the variables no matter what methods I've ...
0
votes
1answer
42 views

Series solution about $x=0$ of $xy''-y'+4xy=0$.

I want to find at least one solution of the differential equation $$xy''-y'+4xy=0$$ about the point $x=0$. I identified that $x=0$ is a regular singular point and thus Frobenius Theorem is applicable. ...
0
votes
2answers
22 views

Long term behaviour of trig functions

In answering a differential equation question, the question asked me to solve the equation and then give the function to which $y$ approximates when $x$ is large and positive. To which I have no idea ...
0
votes
1answer
40 views

How to prove that the solution of $\frac{dy}{dx}=1+y^4$ with $y(x_0)=y_0$ cannot be extended to $\infty$ and $-\infty$

A friend of mine gave me a proof, but I didn't get it. If the solution can be extended to +∞ $$ \frac{dy}{1+y^4} = dx $$ Integrate $x$ from $x_0$ to +∞ So $$+∞ = \int_{x_0}^\infty \frac{ dy(x)}{1+...
0
votes
1answer
36 views

$3y''+4y'+y = 200 \cos x$ finding particular integral

$$3x''+4x'+x=200 \cos(t) $$ This is what I did: \begin{align} x &= \lambda \cos (t) + \mu \sin(t) \\ x' &= -\lambda \sin(t) + \mu \cos(t) \\ x''&= -\lambda \cos(t) - \mu \sin(t) \end{...
1
vote
1answer
66 views

How do you solve this kind of third order ODE? [on hold]

For this given equation: $y'''(x)+y(x)y''(x)=0$ How do you get the solution for this ODE?
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1answer
34 views

How do i solve this differential equation. [on hold]

I'm not really concerned with the answer, im just wondering how the last two lines are equal.
0
votes
2answers
18 views

variable co-efficient 2nd order linear ODE

I am trying to solve a variable co-efficient 2nd order linear ODE by using a transformation for the independent variable: $y'' + \frac{2}{4x} y' + \frac{9}{4x} y = 0$ with transformation $t = \sqrt{x}...
0
votes
0answers
28 views

Initial conditions of ODEs: Why is $\frac{𝑑𝑢}{𝑑𝜙} = \frac{1}{𝐷 \cdot 𝑡𝑎𝑛(𝛼)}$?

I've started reading this paper (kindly check out the paper for more context, figure 1 provides a visualization of the problem) where it describes light trajectories on a Schwarzschild spacetime and ...
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votes
0answers
40 views

Existence of a periodic solution?

The sign of the result will change only depending on how $y$ changes. I need it to always be negative or always be positive. Is that possible here? I'm looking for the existence or nonexistence of a ...