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

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What does omega limit sets have with invariant sets?

What does omega limit sets have with invariant sets? I was thinking of omega limit set as the limit of a sequence inside the invariant set. But... if I look at the definition of Invariant set, it's ...
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Trouble finding solution to a higher order differential equation

I'm attempting to solve for $y$ as a function of $x$ if $y''''-12y'''+36y''=256e^{-2x}$, with initial conditions: $y(0)=5, y'(0)=-1, y''(0)=40, y'''(0)=-8$. I've tried to solve this multiple times ...
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Help unpacking this system of differential equations

The geodesic equation is given as $~~\ddot{\alpha} + (\dot{\alpha} \cdot N \dot{\circ} \alpha)(N \circ \alpha) = 0$, where $N(p), \alpha(t) \in \mathbb{R^{n+1}}$ and $p \in \mathbb{R^{n+1}}$, $t \in ...
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How to solve coupled differential equations?

I am trying to solve question for what is the best angle to launch a ball for it flies the biggest distance (taking into account air resistance). I came up with these equations: $$ \frac{d v_y}{dt} = ...
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Calculate a state not by its derivative in ODE

I have a system with state space representation. This system intakes input $u_1$ and $u_2$. There are two nonlinear blocks. One generates state $x_3$ and two internal states $x_1$, $x_2$. And the ...
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solve 2nd order differential equation

Consider $$ x^2 y''+y'=x-1, $$ where $y=y(x)$ and $x>0$. I've tried the power method but obtained a divergent series. Any help would be appreciated. Update The equation can be rewritten as $$ ...
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Steady state reaction-diffusion $A+B \leftrightarrow C$. How to prove solution uniqueness?

Consider the 1D steady state reaction-diffusion equations $A+B\leftrightarrow C$ with no-flux boundary conditions: $ -k_{on}AB + k_{off}C + D_A \frac{d^2 A}{dx^2}=0 \\ -k_{on}AB + k_{off}C + D_B ...
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28 views

Eulers method to obtain approximate solutions for a differential equation

dy/dt=(1-y)^3 initial value: y(0)=4 Use Euler's method to obtain an approximate solution for the model using a step size of .6. What is happening here? Will starting at a different initial value make ...
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25 views

Canonical representation of a differential equation.

Consider the differential equations: $$a_{11}u_{xx}+2a_{12}u_{xy}+a_{22}u_{yy}=0 $$ how can I put the equation in its canonical form (that is with no mixed derivatives), there exist an elegant ...
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Solve a equation involved differential operator in Matlab

I want to solve $F$ in the following equation in Matlab: $$ F + \partial_x F = Y, $$ where $F$ is a matrix (image), $\partial_x$ is the differential operator of $x$ axis, and $Y$ is a known matrix. ...
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Existence of solutions to this ODE arising from Faedo-Galerkin method?

Let $\{w_j\}$ be a basis of $H^1_0(\Omega)$ and let $\phi(x) = \frac{x}{|x|^{1-{\frac 1p}}}$ (for $2 < p < 3$). Define $$v_m(t) = \sum_{i=1}^m \zeta_i(t)w_i$$ where the coefficients ...
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27 views

Solve of $y''-2y'+y=\frac{3e^t}{1+t^2}+7$

Solve the following DE $y''-2y'+y=\frac{3e^t}{1+t^2}+7$ I can solve for the homogeneous equation, that isn't a problem. However, I don't know how to approach the particular solution. I would try the ...
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Simulation of an ODE model with non-constant parameter

I have a model, I can formulate the model using ordinary differential equation with parameter $P$. I want to simulate the model, but instead of using a fixed constant $P$ for the parameter, I want to ...
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34 views

Derive the recursion relation

Consider the nonhomogeneous linear equation $y' = 2y/(1-x) + f(x)$. It is singular at $x=1$, of course, but it is regular at $x=0$ if (the known function) $f(x)$ is analytic there. Assume $y(x) = ...
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At what rate is the tip of her shadow moving? At what rate is her shadow lengthening?

A woman 5 ft tall walks at the rate of 3.5 ft/sec away from a streetlight that is 12 ft above the ground. At what rate is the tip of her shadow moving? At what rate is her shadow lengthening?
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Minimising error in iterative method

Consider the method $$y_{i+1}=y_i+h\left(\frac{1}{3}f(y_i)+af(y_i+hbf(y_i))\right)$$ as applied to $y'=f(y)$. How should $a$ and $b$ be chosen to minimise the local error? I thought maybe the best ...
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Rabbit population in regards to wolves eating them once they reach a certain population.

A rabbit population increases exponentially with growth rate $k=0.12 months^{-1}$ When population reaches $R=300$ at, for example, time $t=0$, the wolves begun eating rabbits at a rate of $r$ rabbits ...
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About roots of multivariable complex polynomials.

We have a function $f : \mathbb{C}^2 \rightarrow \mathbb{C}$ such that, $f(z_1,z_2) = \prod_{i} (z_1 - a_i) = A(z_2-b)(z_2-c) $ where $a_i$ are known to be real. Now say $T$ is an operator which ...
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Concerning linearly independent functions

Let $f_1(x)$ and $f_2(x)$ be two linearly independent functions. If we have $g_1(x) f_1(x) = g_2(x) f_2(x)$ for some functions $g_1(x)$ and $g_2(x)$, which could also be linearly independent, then ...
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41 views

Fundamental set of solutions

Prove that if $y_1$ and $y_2$: a. vanish at the same point in the interval $\alpha < t < \beta$ then they cannot forma fundamental set of solutions on this interval. So I know that a ...
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Compute $W[y_1,y_2](t)$ for the bessel equation

Let $y_1$ and $y_2$ be solutions of bessel's equation $t^2y"+ty'+(t^2-n^2)y=0$ on the interval $0<t< \infty$ with $y_1(1)=1$ $y_1'(1)=0$ $y_2(1)=0$ $y_2'(1)=1$. Compute $W[y_1,y_2](t)$. So I ...
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Show that $y_1(t)=\sqrt {t}$ and $y_2(t)=1/t$ are solutions of the differential equation $2t^2y"+3ty'-y=0$ on the interval $0<t< \infty$

Show that $y_1(t)=\sqrt {t}$ and $y_2(t)=1/t$ are solutions of the differential equation $2t^2y"+3ty'-y=0$ on the interval $0<t< \infty$ a. Computer $W[y_1,y_2](t)$. What happens as t ...
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Show that the operator L define by $L[y](t)=\int_a^ts^2y(s)ds$ is linear, that is $L[cy]=cL[y]$ and $L[y_1+y_2]=L[y_1]+L[y_2]$

Show that the operator L define by $L[y](t)=\int_a^ts^2y(s)ds$ is linear, that is $L[cy]=cL[y]$ and $L[y_1+y_2]=L[y_1]+L[y_2]$ First I will show that $L[cy]=cL[y]$: ...
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Bounded solutions of 2nd order IVP

Consider the pair of initial value problems on the real line: $$y'' + (x^4 - \lambda)y = 0, \ \ (\ y(0) = 1, y'(0) = 0 \ ) \ \text{ or } (\ y(0) = 0, y'(0) = 1 \ ) $$ One can find numerically for ...
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Determine the stability of a fixed point

Consider $x'=f(x)$, where $f(0)=0$ and $f(x)=-x^3\sin\left(\frac{1}{x}\right)$ for every $x\neq 0$. How to determine the stability of the fixed point $x^*=0$?
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A different variation of parameters technique

I discovered a variation on the variation of parameters technique (I'll call it "VOP2") after a student asked me yesterday why we can make the assumption $u_1'(x)y_1(x)+u_2'(x)y_2(x)=0$. I didn't know ...
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There is no solution of the ODE at the inteval $[0,2]$

According to my notes: We take into consideration the initial value problem: $$\left\{\begin{matrix} y'(t)=y^2, \ t \in [0,1]\\ y(0)=1 \end{matrix}\right.$$ $$\frac{dy}{dt}=y^2 \Rightarrow ...
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Should Jacobian involve derivatives wrt independent variable?

I'm trying to numerically solve a BVP and am trying to compute the analytic Jacobian. Sorry if this is a naïve question, but should the entries in the Jacobian matrix should involve derivatives wrt ...
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how did they get this domain for this trig function?

Solve the ODE and give the largest interval I over which the general solution is defined. $\cos x \frac{dy}{dx} + (\sin x)y=1$ I got the solution $y=sinx+c \cdot cosx$ but how did they get the ...
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proving unique solution for initial boundary problem $y' = (x^2 - y^2)\sin y + y^2 \cos y, y(0)=0$

I want to show that the initial value problem $y' = (x^2 - y^2) \sin y + y^2 \cos y, y(0)=0$ has a unique solution $y(x) \equiv 0$ in the closed rectangle $S=\{(x,y) \in \mathbb{R}^2 : |x| \le a ...
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Third-order linear differential equation a la Sturm-Liouville

Can somebody help to find the solution of this ODE $$ f'''+3t^2f'+3tf=\pm if, $$ where $f=f(t)$? Without the right side, the equation is self-adjoint because $1/2(3t^2)'=3t$ but I'm not sure what to ...
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Wronskian and finding a function

$W[e^{2x},h(x)]$= $\begin{vmatrix} e^{2x} & h \\ 2e^{2x} & h' \\ \end{vmatrix}$ => $h'e^{2x}-2he^{2x}=3e^{4x}$ $h'-2h=3e^{2x}$ Integrating factor is $e^{-2x}$ $e^{-2x}h(x)=3x+C$ ...
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33 views

Van der Pol oscillator method of multiple scales boundary conditions

Could someone explain to me where the boundary conditions right at the bottom of page 42 come from? ...
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33 views

about separable equations when C is complex no

Question: $$\frac{dy}{dx}= x[(1−y2)^1/2], \quad y(0)=0$$ right now I have that $f(x)=x$ and $g(y)=(1-y^2)^{1/2}$, then \begin{align} \int \frac 1{g(y)} \, dy&=\int x \, dx \\ ...
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Solving the production scheduling problem using optimal control

I want to solve the production scheduling problem with quadratic production costs and no inventory costs using optimal control without discounting. My work: The quadratic production cost is $c_{1} ...
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Lineal Differential Equation Questions

How $dy/dx = x-y/x+y$ Is not lineal? And $dy/dx +p(x)y = f(x)$ f(x) can be Cero(0) too?
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Solutions to Euler's equation on intervals not containing singular points

The question (paraphrased slightly) is Find a solution to the Euler equation $x^2y^{\prime\prime}+\alpha xy^\prime+\beta y=0$ that is valid on any interval not including the singular point. I ...
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Question about DE related to physics; includes Hooke's Law and Newton's Second Law as well as system of DE equations and solutions, and a phase plane.

I mainly need help with part A, and a little bit on B and C. Thank you in advance for your answer or any comment or edit that helps!!! A second-order DE can be sometimes solved with clever ...
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Uniqueness of the trivial solution

I am considering the following second order ODE with initial conditions: $$y''(3y+2x)^2=3(3y'-1)(9yy'+4xy'-y), y(0)=y'(0)=0.$$ We also have the conitions (1) the solution is of the class $C^2$. (2) ...
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Finding explicit Lipschitz constant

Let $\varphi_0\in (0,1)$. Note $\zeta(\varphi_0)<0$, so take some intial data $\alpha\in [-5,0)$ and consider the IVP \begin{align*}\tag{1} \begin{cases} ...
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Lie bracket equation

I need to solve the equation system of the Lie brackets of vector fields. So I want to find vector fields $X,Y,Z$ such that $F:(\mathbb R^3,\times)\to (V(\mathbb R^3),[.,.])$ , ...
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Rewriting a differential equation for expansion

I have a system as $$ x' = -y +\mu x + xy^2$$$$ y' = x+\mu y -x^2 $$ I am trying to convert it to a single expression to apply a Poincare-Lindstedt method on it. We were given a hint to use $ ...
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Proof the limit

I have to prove if m>1 and lambda<1 then lim as t goes to infinity for x(t) approeach to 1-lambda, provided that x(0)>0. please see my work below. I'm stuck in the last part.
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State space representation

Consider Newton's law, with input the force and output the position, $u(t)=m\ddot{y}(t)$ We choose the state $x(t)\in\mathbb{R}^{2}$ as $x(t)=\begin{pmatrix} y(t) \\ \dot{y}(t)\end{pmatrix}$. For ...
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Determining the solution of a closed loop system

Then applying this to our system, we get $u(t)=-2(q'(t)+\epsilon\sin{(\omega t)})-(q(t)+\epsilon\sin{(\omega t)})$ I want to determine the solution of this closed loop system and that the ...
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stability assessment

I have been asked to assess the stability of my numerical solutions to two different sets of transient differential equations that govern the same phenomena. I am not sure how I can assess and compare ...
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How to find a positive function that is constant on solutions

Consider the initial-value problem $\dot{w} = \frac{d}{dt}\begin{bmatrix} w_1\\w_2\\w_3 \end{bmatrix} = \begin{bmatrix} w_2w_3\\-w_3w_1\\-\mu w_1w_2 \end{bmatrix}, \ w(0) = w_0.$ For $\mu > 0$ and ...
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65 views

Charge distribution in a conducting sphere

This is a math problem that appeared in a physics one: "Evaluate the density of electrical charge in the surface of a hollow sphere such that the electric field is constant (in direction and absolute ...
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An differential problem is general

Does there exist a differential equation $g(f^{n},f^{n−1},...,f,x)=0$, such that the corresponding equation $g(x_n,x_{n−1},...,x_1,x_0)=0$ has solutions in $C^{n+1}$, but that itself has no solution?
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Several questions regarding existence of a unique solution for nth order differential equations

I am trying to understand this theorem in my Differential Equations book: Let $a_n(x), a_{n-1}(x),...,a_1(x),a_0(x)$ and $g(x)$ be continuous on an interval $I$ and let $a_n(x) \ne 0$ for every x in ...