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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1answer
262 views

A function where the area bounded by the tangent line and the two axes is constant.

Okay so I'm trying to find a function or an implicit solution that satisfies this criterion: The area bounded by any tangent to this curve and the two axes must equal a constant $A$, a positive real ...
2
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0answers
73 views

Pursuit Curve, Parametric Equation

So its a classic problem: Object $A$ starts at the origin $(0,0)$ and moves straight up the $y$ axis with a speed $v$. Object $B$ starts at point $(1,0)$, always moves towards object $A$ and has a ...
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0answers
83 views

a system with differential equations

I have a system which is described by the following differential equation. I want a closed form formulas to calculate $v_1(t)$ and $v_2(t)$ with the given parameters. In the following equations $p, k, ...
2
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0answers
61 views

Recursive differential equations

Suppose we took the odd solution to $y''+y=0$ which is $\sin(x)$. If we put this in place of $y$ in the differential equation we get the equation: $$y''+\sin(y)=0$$ the odd solution to which is an ...
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150 views

best book for practicing unsolved problems in differential equation and linear algebra

I started reading differential equation and linear algebra. Can anyone provide the link/book name where I may get many questions to practice. Generally, in the end of book only few problems are there. ...
2
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1answer
51 views

Is the Fourier transform a tame linear operator?

$\mathcal{F}:C^{\infty}_{0}(B^d)\to L_{1}^{\infty}(\mathbb{R}^{d},\mu,w)$ $\mathcal{F}(f)=\hat{f}$ I'd like show that $\left\|\mathcal{F}(f)\right\|_{n}\leq\left\| f ...
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62 views

Equilibrium points and linear stability

Consider the nondimensional amplitude equation for $A = A(t)$ where $t$ is time given by (1): $$ \frac{dA}{dt} = \sigma A - a_1 A^3 - a_3 A^5 = f(A) \text{ with } \sigma \in \mathbb{R}, a_1 < 0, ...
2
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1answer
43 views

example which does not satify Lipschitz condition, but has a unique solution

$$y'=1+\sqrt y\\y(0)=0 $$ Show that this IVP does not satisfy Lipschitz condition, but has a unique solution. I have shown the first way, like this: Let $f(x,y)=1+\sqrt y $. Then $\frac ...
2
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1answer
57 views

Solve a second order nonlinear equation

I have a second order nonlinear equation: $$-u''+ \frac{1}{4}(u')^2+au=x^2.$$ I am only interested in the solutions in $[0, \frac{x^2}{a}+\frac{1}{a^2}]$. One paper claims without proof that the ...
2
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0answers
51 views

Why does this nonlinear ODE solution not work?

I am relatively new to Python and trying to use it to solve a second order nonlinear differential equation, specifically the Poisson-Boltzmann equation in an electrolyte. $$\phi''(r) + \frac2 ...
2
votes
1answer
71 views

Differential Equation help: $\frac{dy}{dx}=\frac{y-3}{x^2 +y^2}$

The question is: solve for $y,$ $\dfrac{dy}{dx}=\dfrac{y-3}{x^2 +y^2}$ given it passes through $(0,1)$. I am struggling to find a way to separate the variables. Also as a side question, if you have ...
2
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0answers
270 views

Identifying Hamiltonian Systems with Phase Portrait

the following is a homework question (that isn't going to be graded) and I'm not sure how to do it. I know that the solution trajectories cannot cross the H(x,y)=constant curves, but I'm not sure ...
2
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1answer
28 views

Why can $1 + \cos(t+y) + \cos(t+y)(\frac{dy}{dt}) = 0$ be written in the form $(\frac{d}{dt})\lbrack t + \sin(t+y) \rbrack = 0$?

I'm reading Differential Equations and Their Applications by Martin Braun. In subsection 1.9, which deals with exact equations, the author writes: Example 1. The equation $1 + \cos(t+y) + ...
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2answers
42 views

$yy'=\sin(t),y(0)=1$ phase portrait

I need to draw a phase portrait for the equation $y(t)y'(t)=\sin(t)$ with the initial condition $y(0)=1$. So far i've found that $y(t)= \sqrt{3-2\cos(t)}$ and ...
2
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0answers
40 views

BVP eigenvalue problem

I am working on the following problem and I am completely stuck: Show that the eigenvalue problem $$ -u''+4\pi^{2}\int_{0}^{1} u(x)\,dx=\lambda\,u $$ with $u(0)=u(1)$ and $u'(0)=u'(1)$ has ...
2
votes
1answer
28 views

Laplace transform and differential equations

Given $\frac {d^2y(t)}{dt^2} + a\frac {dy(t)}{dt} = x(t) + by(t)$ Find: a) $ H(s) = \frac{Y(s)}{X(s)}$ b) ROC of the stable function and the correspond h(t) and determine if the stable system is ...
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0answers
33 views

Stability of an equilibrium solution with 0 denominator

I'm testing the equilibria of a differential equation and found that one has a 0 denominator. Example: $$\frac{dx}{dt}=2x^{(1/2)}-5$$ Which, when you try and evaluate the derivative at 0, you end up ...
2
votes
1answer
61 views

Lyapunov Stability is a problem for me

Let be $ \dot{X} = F(X) $, $F \in C^1( \mathbb{R}^n)$, $P \in \mathbb{R}^n$ isolate singular point. Suppose there exists a family $S_{{r}_i}$ with $ i \in \mathbb{N}$ such that: $S_{{r}_i} = \left ...
2
votes
1answer
61 views

Integral form of this IVP

How do I show that the following initial value problem $$ xu''+u'+xu=0,\quad u(0)=1,\quad u'(0)=0 $$ has the following integral form: $$ u(x)=1+\int_{0}^{x} t\ln(t/x)u(t)\,dt $$ I am stuck because if ...
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0answers
132 views

Stuck trying to solve wave equation in $n$-dimensions.

Solving the wave equation $u_{tt} = c^{2} \Delta{u}$ subject to $u(0,x) = f(x)$ and $u_{t}(0,x) = g(x)$ gives us d'Alembert's formula. I'm looking to solve the wave equation, subject to these same ...
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0answers
57 views

solving this differential equation for $y$, Is it even possible?

Lets say I have the following: \begin{gather} \frac{(y')^3 + 3 y' y'' + y'''}{(y')^2 + y''} = \sqrt{1+(y')^2}\\ \frac{((y')^3+3y' y'' + y''')^2}{((y')^2 + y'')^2} = 1+(y')^2\\ \frac{(y')^6 + 6 (y')^4 ...
2
votes
1answer
97 views

Riccati differential equation

The Riccati differential equation, $y'=x+y^2$ is special equation. I know that how can I solve it, but my problem is that I don't have initial conditions, and I firstly need a particular solution. How ...
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0answers
36 views

Initial conditions to solve an ODE?

Given is the following inhomogenous linear ODE (4th order): $$q_0\cdot\sigma + q_1\cdot \dot\sigma + q_2\cdot \ddot\sigma + q_3\cdot \dddot\sigma + q_4\cdot\ddddot\sigma = p_0\cdot\epsilon + ...
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0answers
45 views

Solving $2y'''(t)+3t\ y(t)=0$.

For a certain problem, I am trying to solve the ODE $$2y'''(t)+3t\ y(t)=0$$ I am pretty clueless what to do here, any hint would be appreciated. Thank you very much.
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0answers
30 views

Differential Equation. VERY small problem

I want to ask a question later, after I show you this TESTING: x^2 = 1 Differentiate both sides 2x = 0 TESTING: x = 1 Differentiate both sides dx/dx = 0 1 = 0 So when can I differentiate both ...
2
votes
1answer
76 views

Frobenius method for linear second order differential equation

I am trying to solve $x^{2}y''+xy'-9y=0$ using Frobenius' method. Plugging in the derivatives of $y=x^{s}\sum ^{\infty }_{i=0}c_{i}x^{i}$ to the equation, I get, for the lowest power $x^s$, the ...
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0answers
25 views

How to find Laplace transform of a differential equation?

$y′′ + 3y′ + 2y = f$ , $y(0) = 0$ , $y′(0) = 1$ where $f$ is given by $f(t) = \sum_{n=1}^\infty \delta(t−n)$; find a 1-periodic function $y_*$ with $\lim_{t\rightarrow \infty} |y(t)−y_*(t)| = 0$. I ...
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How to solve system of stochastic differential equations?

I have the following two SDEs $$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$ $$dN_2=(2pN_1-\mu N_2)dt+\alpha_2 N_2dW_2$$ $W$ is the standard Brownian motion/Weiner process. This isn't homework, I'm just ...
2
votes
2answers
152 views

Closed orbits of vector fields under perturbation

Consider a vector field $V$ on an annulus $U$, say. Also, assume that the vector field $V$ has a closed orbit. I am looking for a reference that gives stability results of the following type: If the ...
2
votes
1answer
30 views

Laplace transform of a differential equation?

$y′′ + 2y′ + 2y = δ(t − \pi) + aδ(t − T)$ , $y(0) = y′(0) = 0$ $a$ and $T$ are positive numbers and $T > \pi$. I need to find values for $a$ and $T$ such that $y(t) = 0$ for all $t \ge T$? I just ...
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0answers
117 views

Lyapunov function for a damped pendulum

The question is about damped pendulum. There are two statements I don't understand or I'm not sure if my justification for them is correct. Could you say if I'm right? The example is from a German ...
2
votes
1answer
45 views

Consider ODE ; find solution(s).

$x^{2} y''+ 6xy' + 6y = 4e^{2x}$ ; $y'(0)=1, y(0)=1$ Solutions? Thank you so much
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82 views

Second order differential equation, power series method

Solve the differential equation $$(x+2)y''-xy'+(1-x^2)y=0 ; \quad X_0=1$$ using the power series method about the point $x_0=1$. I get to this step after deriving the derivatives of the ...
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0answers
18 views

How is it possible to continue solutions for a differential equation along t?

Given the equation $y' = e^{\sin y+t} + t\cos(y)$. I rewrote it as $$ y(t) = y(0) + \int_{0}^{t}ye^{\sin y+t}+t\cos y $$ I'm asked to prove that every solution can be continued for every t. I know ...
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0answers
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Study of a system of differential equations

I'm asked to study everything that is possible to know about the sytem$$\begin{cases}x'=x^2-y^2\\y'=2xy\\z'=-z\end{cases}$$ My questions here is, how much can be know about it?, how do I know I ...
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1answer
72 views

Solvability of system of differential equations

Given $a_i:\mathbb{R}^n \to \mathbb{R}$ $(1\leq i \leq n)$, I am trying to find the conditions under which the equations $$ \frac{\partial f}{\partial x^i}=a_i(x_1,...,x_n) $$ $$ f(x_0)=z_0 $$ is ...
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43 views

Find an integrating factor such that $y'=\frac{1-x+y}{x-y}$ is exact

Yet another question of this sort, and hopefully the last. In the previous question I posted, we were lucky enough and the integrating factor was a function of only one variable, the ansatz $\mu_y=0$ ...
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0answers
56 views

Annoying differential equation involving composition

Upon trying to crack into a problem, I managed to end up with the following differential equation. $$ y = xy' - y'\circ y', \qquad\text{or}\qquad y(x) = x\cdot y'(x) - y'(y'(x)) $$ I haven't a clue ...
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3answers
58 views

Solving $y'(x) = \frac{y(x)}{3x-y^2(x)}$?

Solving $y'(x) = \frac{y(x)}{3x-y^2(x)}$ ? I'm trying to solve this first order non-linear equation. I've tried to plug in a couple of different things and would appreciate if anyone could point me in ...
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0answers
66 views

Solution of differential equation $x'=Ax$ where $A=PJP^{-1}$

Let $A$ be a $n\times n$ matrix. Suppose $A=PJP^{-1}$ being $J$ the Jordan form of $A$. Prove that if $x(t)=(x_1(t),\dots,x_n(t))$ is a solution for $x'=Ax$, then ...
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0answers
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Find the 3rd order DE whose general solution is $ y= C_1e^{2x} + C_2\cos x + C_3 x\sin x $

My attempt $$ \begin{matrix} y &=& C_1e^{2x} &+& C_2\cos x &+& C_3x\sin x\\ y' &=& 2C_1e^{2x} &-& C_2\sin x &+ &C_3(\sin x &+& x\cos x)\\ ...
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0answers
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A basic ODE question

Let $G \subset \Bbb R^d$ be open and let $ V: G \to [0, \infty)$ be such that $\dot{V} = \nabla V.h : G \to \Bbb R$is non-positive. We assume that $H=\{x: V(x) =0\}$ is equal to the set $\{x: ...
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0answers
163 views

Famous parametric curves that are solutions to differential equations

I know that the cycloid satisfies the differential equation $ \left( \frac{dy}{dx} \right)^2 - \frac{2r}{y} + 1 = 0. $ Are there other famous plane curves that are also solutions to a differential ...
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1answer
150 views

Tough second order differential equation

I can't figure out this diff equation (in cylindrical coordinate). How can I solve it ? Any comments appreciated $$ \frac{1}{r}\frac{d}{dr}(r\frac{dE}{dr})+\frac{d^2E}{dz^2}+(\epsilon_0 ...
2
votes
1answer
42 views

Differential equation (inhomogeneous )

I have been trying to solve this equation for a while. Is there anyone who can help me to solve this ? Any comment appreciated. $$\frac1r \frac{\partial}{\partial r}\left(r\frac{\partial E}{\partial ...
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0answers
72 views

Yet Another Differential Equations Problem

I come from a non mathematical background, so solving differential equations is something that I have to acquire on the go. I hope the following makes sense. I want to chose a nonnegative ...
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0answers
175 views

Fundamental Matrix

Determine $\phi(x,0)$ for $A(x)=\begin{pmatrix} -1 & \cos(x) \\ 0 & -1\end{pmatrix}$, where $\phi(x,0)t_{0}$ is a solution of $\frac{d}{dx}t(x)=A(x)t(x)$. I am not entirely sure as to ...
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0answers
71 views

Fundamental Matrices for Linear ODE

Why is the following statement true?: For a matrix ODE: $\mathbf{x'=Ax}$ with special fundamental matrix, $\Phi (t)$ or $e^{\mathbf{A}}$, where $\Phi(t_0) = I$, and fundamental matrix containing the ...
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0answers
104 views

An Ordinary Differential Equation with time varying coefficients

Let $A$ and $B$ be complex numbers, let $\beta_1$ be real and $\beta_2=2$. Consider a following Ordinary Differential Equation: \begin{equation} \frac{ d^2 r_t}{ d t^2} + \left(\frac{A}{t^{\beta_1}} + ...
2
votes
1answer
59 views

Finding Solution of 2nd order ODE near a regular singular point

I am having trouble solving this problem and could use some hints, ideas or a solid walk through that I could use to clear up the foggy areas. $$x^2y''+3xy'+(1+x)y=0$$ I have proven that $X_0=0$ is ...