All questions with tag [math: ordinary-differential-equations]


Quasi-modified equation for harmonic oscillator

Could you aid me with this inquiry please? Find seemingly - changed formula of 2nd order for remedy of harmonic oscillator formula with semi - specific Euler (additionally called symplectic Euler) system.
2022-07-25 20:46:22

solving Differential Equation

I have the formula below: $$(t^2 + 1)dx=(x+4)dt$$ Where $x(0) = 3$ I am attempting to make use of splitting up of variables, and also I wind up here: $$\ln(x+4)=\arctan(t)+C$$ Trying to streamline it more: $$x=-4+\ln(\arctan(t)+C)$$ Is this proper? I assume I should make use of $x(0) = 3$ to find value of the constant, just how can I do t...
2022-07-25 20:46:10

ODE flow of the Chafee-Infante problem

I'm currently listening to a lecture on dynamical systems. Unfortunately I am lacking some of the requirements for that course and thus ran into some problems with the latest problem set. Show that the ODE flow $$ \frac{d^2}{dx^2}v + f(v) = 0 $$ for the Chafee-Infante nonlinearity $f(v)=v(1-v^2)$ is not global. [...] So far I thought t...
2022-07-25 20:46:03

Solve the boundary-value problem for the Laplace operator in domain.

i require a little aid with these Laplace eq or smt which resembles this. I'll be gratefull for any kind of aid. Instances or tutorials will certainly be handy also. $$ \left\{\begin{matrix} \Delta u = 0, 2\leq \rho\leq 3\\ u|_{\rho=2} = 1\\ u|_{\rho=3}=4cos(\varphi) \end{matrix}\right. $$
2022-07-25 20:45:52

Initial-boundary value problem for PDE

I need a little help with solving IBVP for hyperbolic and parabolic equations like these: $$ hyperbolic: \left\{\begin{matrix} \frac{\partial^2u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}+1\\ u(0,t) =u(1, t) =0\\ u(x,0) = 0\\ u_t(x,0)=x \end{matrix}\right. \\parabolic: \left\{\begin{matrix} 3\frac{\partial u}{\partial t}=4\frac{\partial^...
2022-07-25 20:45:31

Trouble with the constant when finding a function $x(t)$ such that $\dot{x}(t) = x(t)^2t$

I'm attempting to consider a function $x(t)$ such that $$\dot{x}(t) = x(t)^2t.$$ I exercised for my self that it has something to do with the $x(t) = -2t^{-2}$ yet I could not exercise just how to place the constant in to make sure that the function still functioned. Ever since I've been informed that the function is $x(t) = \frac{2}{c-t^2}$ a...
2022-07-25 20:45:12

How to manage the absolute value on a differential equation $|T(x)'+A(T,x)+B(T,x)| = f(T,x)$

Hi every person I require to address a formula of this type: $|T(x)'+A(T,x)+B(T,x)| = f(T,x)$ with borders problems. The absolute value is my trouble. Certainly without it, the remedy of these is well dealt with in the literary works. After search in the inquiries I located this: So, can I do the very same procedure? Or there is an additio...
2022-07-25 20:41:51

Application of method of continuity in partial differential equations

Consider a differential operator $$L_t:= (1-t)(\Delta-\lambda) + t L,\qquad t\in[0,1].$$ For any $u\in C^2_0(\mathbb{R}^2)$, we have $$\lambda^2 \|u\|_2^2 + 2\lambda\sum_{i}\|u_i\|_2^2 + \sum_{i,j}\|u_{i,j}\|_2^2 =\|\Delta u-\lambda u\|^2_2\leq \frac{1}{\mu^2}\|Lu\|_2^2$$ where $u_{i} = \frac{\partial u}{\partial x_i}$, $u_{i,j} = \frac{\partial...
2022-07-25 20:41:11

Finding value of a constant in Differential Equations

I have the adhering to ODE Where offered is $x(0)=1$: $$(t+3)dx=4x^2dt$$ After splitting up of variables I obtained this: $$\frac{-1}{x} = 4\ln(t+3)+C$$ I assume this streamlines extra as: $$x=\frac{-1}{\ln((t+3)^4)+C}$$ Please inform me if this is proper, I additionally have trouble searching for C in this instance, MapleTA does decline m...
2022-07-25 20:40:38

Second Order Diffrential Equation

$y'' - 4y' + 5y = 25x - 3e^2x$ Where $y(0)=0$ and $y'(0)=0$ This is what I have done so far:- $r^2 - r + 5 = 0$ $\sqrt{b^2-4ac} < 0$ ...... I get $2\pm i$ Thus $y(x)=e^2x(A\cos x + D\sin x)$ How do I find $A$ and $D$? Substitute $y(0)=0$ into the equation? What about the right hand side equation? This is what I did for that... $g = 25x-3...
2022-07-25 20:21:02

Linearization for differential equation

I'm trying to make linearization for the following system: $y_1'=(1+y_1)\sin y_2, y_2'=1-y_1-\cos y_2$ in the critical point $y_1=y_2=0$. I'm not sure that I know what I need to do. I believe that I need to solve the homogeneous system: $(1+y_1)\sin y_2=0, 1-y_1-\cos y_2$=0 so I got that $y=-1$ and $\sin y_2=0$, but the first option is not possi...
2022-07-25 20:18:20

Irregular singular point of a DE

In my notes it claimed that ${d^2y\over dx^2} -xy=0$ has an uneven single factor at $x=\infty$. It was highlighted by doing a transformation $y(x)=w(t)$ with $t=\frac{1}{x}$. Yet I do not recognize, just how did they get ${dy\over dx} =-t^2 {dw\over dt}$ and also ${d^2y\over dx^2}=t^4 {d^2w\over dt^2}+2t^3 {dw\over dt}$? And also just how did...
2022-07-25 20:17:50

Links between difference and differential equations?

Does there exist any kind of document in between distinction formulas and also differential formulas? Specifically, can one cast some courses of ODEs right into distinction formulas or the other way around?
2022-07-25 20:15:42

an odd question about solving ODE by MATLAB

Why MATLAB occasionally can not address the reasonably certain instance yet can address the reasonably basic instance? For example: I attempted to input (x^2 - 1) * D2y+0 * x * Dy+1 * x * y = 0 in MATLAB and also MATLAB still can not address. So I attempted the instances of various integer values of a in MATLAB and also the outcomes are sur...
2022-07-25 20:14:58

Runge-Kutta 4 for systems of equations

This question is part of an assignment in numerical methods class. I am supposed to find the position and velocity of a spaceship flying around the Earth and Moon. I am given initial values of the position and speed, and functions that describe the acceleration of the spaceship, so this can be solved using the Runge-Kutta methods. Description Th...
2022-07-25 20:14:47

Distinction between error estimator and error indicator

When addressing differential formulas numerically one can sustain discretization mistake and also one can construct a posteriori mistake approximates to approximate truth mistake. There is a difference usually made in between "error estimators" and also "error indicators". What is the distinction in between both?
2022-07-25 20:13:26

Show velocity of a particle during its flight at time $t$

I'm completely stuck, I think I have to use Newton's second law but I have no idea where to start, any help would be appreciated! At time $t=0$ a particle of unit mass is projected vertically upward with velocity $v_0$ against gravity, and the resistance of the air to the particle's motion is $k$ times its velocity. Show that during its flight t...
2022-07-25 20:05:57

Probabilistic interpretation of $\sum_{n \geq 0}\mathrm P_{n}(t)= 1$

The following is a problem from Spiegel's Applied Differential Equations: The probability $\mathrm{P}_{\mathrm n}(t)$ that a counter (such as a Geiger counter) will register exactly $\mathrm n$ nuclear particles in a time $t$ is determined by the system of differential equations: $$\mathrm P_{\mathrm n}^\prime(t)=\lambda \left[\mathrm P_{\mathr...
2022-07-25 20:02:05

Classify the fixed points at the origin

Consider the straight system $$\frac{dx}{dt}= -3x+2y, \frac{dy}{dt}= ax+6y, a \neq -9$$ identify the set factor at the beginning? Is the proper strategy to explore the constant states and also just how these factors will certainly transform according to the factor a, so take into consideration the array from - $\infty$ to $\infty$. Several m...
2022-07-25 19:57:28

Implications of given solutions

This has been solved! Thanks to everyone who read and thought about it Suppose lines of the form $(x_0,y)$ and $(x,y_0)$ for any given $x_0,y_0\in \mathbb R$ are solutions to the system of differential equations Let $A,B,C,D$ be functions of $(x,y)$ and $x,y$ are functions of $t$ $$\begin{cases} {d\over dt} (A\dot{x} +B\dot{y})={1\over 2} (A_x\...
2022-07-25 19:43:54