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

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Estimating upper bound

Let the following Cauchy Problem be $\displaystyle\cases{ y'(t)=f(t,y(t)) & \cr y(0)=\eta }$ for $t\in[0,T]$ Define the approximation $y_n$ of $y(t_n)$ as: ...
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1answer
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Linear Second order ODE with oscillating solutions

I encountered the following second-order ODE while tutoring recently, and struggled with the proper approach: $x^2y''+2xy'+\alpha y = 0$ The problem is: for which values of $\alpha$ do solutions ...
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Derivation of higher order bessel function in terms of lower order functions

I am really stuck trying to prove this.. ((x^-p)Jp(x))’ = -(x^-p)Jp+1(x) ---(1) Can someone please help how to actually prove this step by step, because whichever notes i see, they prove ...
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why equilibrium points are important in ODE theory

Why equilibrium points are important for the study of differential equations $\dot{x(t)} = h(x(t)$? There can be arbitrary sets which are stable, why stable "equilibrium point"s are important ?
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A basic question on equilibrium point of coupled differential equation

The system of ordinary differential equations given by $$ \dot{x_1}(t)= k + \sin(x_1 + x_2) + x_1$$ $$ \dot{x_2}(t)= k + \sin(x_1 + x_2) - x_1$$ do not have any equilibrium point for $k >1$. Why ...
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Why choosing small values of Alpha and Beta gives inefficient Runge Kuttan method

I have been trying my hand at a past exam paper and one of the questions is as follows: The second-order Runge-Kutta method to solve the equation $ \frac {dy} {dx} = f(x,y)$, $y(x_0)=y_0$ at the ...
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Solving Duffing equation by Matlab ode23

How can I use Matlab to solve numerically this duffing equation with known $\kappa, \Gamma, \omega$..thanks.. $$x'' +\kappa x' +x -x^3 =\Gamma \cos\omega t$$ I have only few knowledge of Matlab..
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Constructing Lyapunov function for system of ODEs

Background: I have been working on this problem for my research for months now, and I am in dire need of help. That is why I have come here to seek help. I have a system of nine ODEs that describe ...
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1answer
21 views

how to show this manipulation in the integral

Let we have: $$G(t)=y_1(t)\int y_2(s)ds$$ when we take the limits as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds$$ then is it possible to write it as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds=\int^t_{t_0} ...
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2answers
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Question in undetermined coefficient method for ODE

How should I formulate particular solution of this ODE? I want to use method of undetermined coefficients. $$ y'' - y = e^x \\ y_H = C_1 e^x +C_2 x e^x $$ $y_H, y_P$ are homogeneous and particular ...
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On Nonlinear Autonomous system of two equations if the eigenvalues of the Jacobian matrix are 0.

Suppose we have a non-linear autonomous system of two equations: $$\begin{cases} x'(t) = F(x,y) \\ y'(t) = G(x,y) \end{cases} $$ and we obtain a fixed point for this equation, but the eigenvalues of ...
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1answer
17 views

Solution of an ODE (Show equation)

Here are the preliminaries of my question: Let $\Omega$ be a compact metric space, and suppose a flow $(\Omega,\mathbb{R})$ is given. Let $A\colon\Omega\to M^2$ be continuous. Here $M^2$ is ...
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19 views

Wronskian, Linear Independence

Show that the following functions are linearly independent: $$e^t\begin{bmatrix}1\\1\\0\end{bmatrix}$$ $$e^{2t}\begin{bmatrix}0\\2\\1\end{bmatrix}$$ $$e^{-t}\begin{bmatrix}1\\0\\1\end{bmatrix}$$ ...
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Showing that a collection of m solutions is linearly independant

Show that a collection $ \Phi_1 .. \Phi_m $ : I-->R of continuous functions satisfying $ \\ $ $ \int_I(\Phi_J(t)\Phi_k(t)dt $ =1 when j=k , 0 when j$\neq$k $ \\ $ is linearly independent. Multiply the ...
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36 views

How do you solve ODE $x' = x^2$? [on hold]

How do you find all the solutions for that ODE?
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1answer
31 views

Explicit formula for the implicit Euler method

Given the problem; $\displaystyle\cases{ y'(t)=y^2(t) & \cr y(0)=1 }$ for $t\in[0,1]$ Using the implicit euler method, find an explicit formula to get $y_{n+1}$ HINT: The ...
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On a linear 3x3 system of differential equations with repeated eigenvalues.

I have the following system: $$\begin{cases} x'= 2x + 2y -3z \\ y' = 5x + 1y -5z \\ z' = -3x + 4y \end{cases} $$ $$\det(A - \lambda I)= -(\lambda - 1)^3$$ the eigenvector for my single eigenvalue ...
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26 views

Convert time derivative to a function of time

Physics: I am asking for help to derive a general expression for the total amount of energy lost as a function of time from a radiating object. I'll simplify my problem like this: Say for example ...
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1answer
29 views

First-order nonlinear ordinary differential eqauation

Can someone help me to solve the equation $y'=\dfrac{y}{x}\left(\dfrac{xy + 1}{xy - 1}\right)?$ I have been trying a few methods. Thanks. $P=xy^2+y,$ $Q=-(x^2y-x)$ I tried to make it exact ...
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1answer
21 views

polynomial solution of second order differential equation

Find the polynomial solution $$u_n(x) = x^n + a_1x^{n-1}+...+a_n$$ of the differential equation $$u_n'' + xu_n' - nu_n = 0$$ satisfied by u_n(x). Note that this is entry-level calculus, so in my ...
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1answer
30 views

Analytic solution for a type of PDE systems

Peace be upon you, I have the following system of partial differential equations \begin{align*} \begin{cases} \frac{\partial}{\partial a}S(a,b,c,d)=f_1(a)\\ \frac{\partial}{\partial ...
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25 views

Duffing equation of forced spring motion

The motion of a forced spring is described by the equation $$x'' + \kappa x' +x-x^3 =\Gamma \cos(\omega t)$$ We wish to investigate the stability of solutions of this equation having the forcing ...
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ODE -parabolic cylinder functions

How do we solve $\frac{d^2f}{dz^2} + \left(Az^2+Bz+C\right)f=0 \tag 1$ where $f(z),A,B,C$ are matrices of order $3 \times 3$.
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bibliography for weak solutions of ODE's

Some one could recommend to me some bibliography about weak solutions of ODE's, and solutions of ODE's that are not lipschitz??
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2answers
39 views

solving a second-order non-linear differential equation

Good day. Could you help me to solve the DE $$ y''=\sqrt{1+y'^2} $$ I have tried to write the equation in terms of $y'=z, y''=z'$, which results in the new DE $$ z'=\sqrt{1+z^2} $$ but then I got ...
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1answer
21 views

Show that the function is positive

Let $f:\mathbb R\to\mathbb R$ be a Lipschitz continuous, monotone increasing function, with $f(0)=0$, if a function $\phi$ satisfies; $\displaystyle\cases{ \phi'(t)=f(-\phi(t))-f(\phi(t)) ...
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Get a special form of an linear System of ODE (using polar form)

In this post Converting an ODE in polar form it is shown that a linear system of ODE $$ x'=\begin{pmatrix}a(t) & b(t)\\c(t) & d(t)\end{pmatrix}x $$ can be written in polar coordinates ...
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1answer
30 views

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$ I integrated it once to get, $2x + \sin x + C$, $C$ being a constant. Then I integrated it a second time to get $x^2 - \cos x ...
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1answer
18 views

Converting an ODE in polar form

Convert the ODE system $$ \dot{x}=\begin{pmatrix}a(t) & b(t)\\c(t) & d(t)\end{pmatrix}x $$ into polar form. You should get two equations $$ \frac{d}{dt}\Phi(t)=...\\ ...
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What are some planes (spaces) akin to the trace-determinant plane in other disciplines?

When studying basic differential equations, I found the trace-determinant plane incredibly illuminating. Similarly, I find it very helpful to see different kinds of conics as slices of a cone. What ...
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1answer
14 views

System of Linear differential equations with variable coefficients

Could someone please suggest a technique for solving the following linear system of ODEs: $$ \begin{array}{l} i\alpha \frac{{dx(q)}}{{dq}} = \left( {\beta - 2c\cos (q)} \right)x(q) - ig\,y(q)\\ ...
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Prove $x\to 0$ as $t\to \infty$ if we consider the system of equations $x'=(A+B(t))x$ where $B(t)\to 0$ and $A$ has negative eigenvalues.

Consider a matrix $A$ such that all of its eigenvalues are negative. Consider $B(t)$ where $B(t)\to 0$ as $t\to\infty$. Then consider the system of equations $$ x'=(A+B(t))x$$ Prove that any ...
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1answer
15 views

Solution of a second order ODE

I want to solve the following ODE \begin{align} f^{''}(t) + \frac{1}{t} f^{'}(t) &= 0 \\ f(1)&=0 \end{align} Is this an Euler-type ode? In oder to find the solution, i have to rearrange the ...
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Solve the following ODE

Solve the following ODE $$(y-x)\left(1+x^2 \right)^{\frac{1}{2}}\dfrac{\mathrm{d}y}{\mathrm{d}x}=n\left(1+y^2 \right)^{\frac{3}{2}}$$ I have tried substituting $y=\tan \theta$ and $x=\tan \phi$ ...
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Nonhomogeneous DE with constant coefficients , reduction of order [on hold]

Please solve this equation using the Reduction of order method .. $$ y''-4y' + 3y = x $$
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If $\limsup_{t\to \infty} \int_{0}^{t}Tr(A(s))ds = \infty$ then $\limsup_{t\to \infty} |x(t)|=\infty$

For a homogeneous linear system of differential equations: $x'=Ax$ : Suppose that $\limsup_{t\to \infty} \int_{0}^{t}tr(A(s))ds = \infty$ ($tr(A):=$ trace of the matrix A). Then there exists solution ...
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A strange 3rd order ODE

This is the original ODE: $ y^{1/2}y'''+e^{-x}(y'')^{2+c}-(\frac{xy}{x+1})y'=x $ with c is a positive number. $y(0)=1,y'(0)=0,y''(0)=1$ $1st$ question: If x is large, then $ y^{1/2}y'''$ and ...
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40 views

Prove that solutions to linear system form a vector space of dimension $\geq 2$

I accept & appreciate any form of help with the following problem: $B_{nxn}$ "periodic matrix" with period $T$ such that $B(t+T) = B(t)$ for all $t\in \mathbb{R}$. Assume that the system $x' = ...
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Limit of solution of linear system of ODEs as $t\to \infty$

I am completely stuck on the following problem: Consider the linear system: $x'(t)=A(t)x(t)$ where $A(t)$ is an $n$ by $n$ matrix. Assume that $\lim_{t\to \infty}A(t)=B$. Suppose that each eigenvalue ...
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If $f'(x)=f(x)+\int_{0}^{1}f(x)\,dx$ and $f(0) = 1,\,$ then what is the value of $\, \int_0^1 f(x)\,dx=$?

If $\displaystyle f'(x)=f(x)+\int_{0}^{1}f(x)\,dx\,$ and $\,f(0) = 1.$ Then what is value of $\displaystyle \int f(x)\,dx\,?$ $\bf{My\; Try.}$ Let $\displaystyle \int_{0}^{1}f(x)\,dx = A\;,$ Then ...
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Changing form of differential equation

How would you change the differential equation: $3xy^{''}+y^{'}+12y=0$ into a form where the coefficient of the first term is 1? leaving just $y^{''}$ I should probably know how to do this, but I'm ...
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1answer
26 views

Limit of a continuously differentiable function that statisfies

Let $x(t)$ be a continuously differentiable for all $t>0$, and such that: $$\lim_{t\to \infty}[x'(t)+x(t)]=\alpha$$ I need to show that $\lim_{t\to \infty}x(t)=\alpha$ My goal is to show that ...
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1answer
28 views

Proving that the solution to the IVP exists given a condition on the right hand side

Consider the following IVP: $x'=f(t,x)$ $\ $and $\ $ $x(0)=x_0$ where $x\in \mathbb{R^n}$ and $t\in \mathbb{R}$. Suppose that for all $(t,x)\in\mathbb{R^{n+1}}$: $|f(t,x)|\leq b(t) |x|^2$. In order ...
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Interval of solutions of this differential equation

For the following differential equation initial value problem, $y' = \frac {-t + (t^2 + 4y)^{1/2}} {2}$, $y(2) = -1$ the interval of t for which the solution $y_1 = 1 - t$ is valid is $t \ge 2$ and ...
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21 views

Limit of Solution of an ODE

Consider the following differential equation: $y^{'}(t)=g(y)$ where $g$ is a continuous function from $\mathbb{R^n}$ to $\mathbb{R^n}$. Assume that $y(t)$ is a solution to the previous ODE. Suppose ...
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1answer
19 views

Projectile Motion using cos and sin theta???

Golfball is struck to clear a tree 20m away and 6m high at an angle of elevation of 40degrees. Find the speed of the ball when it leaves the ground. I've created my displacement equation with i and j ...
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34 views

How to find inverse laplace transform

$$ F(s) = \dfrac{6s+9}{s^2-10s+29} $$ How do you solve the inverse Laplace transform of this above equation?
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1answer
19 views

Why existence of Lyapunov function implies Lyapunov stability at the equilibrium point

Why existence of Lyapunov function (locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative semidefinite) implies Lyapunov stability (i.e for any ...
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1answer
28 views

How to solve Laplace initial value problem

$$ y''+36y = f(t) $$ $$ f(t) = \begin{cases} 1, & \text{0 ≤ t < 8} \\ 0, & \text{8 ≤ t < ∞} \end{cases} $$ $$ y(0) = 0 $$ $$ y'(0) = 1 $$
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On a cauchy problem (differential equation system) .

I am approaching the theory of these kind of problems but I am missing an example. I am tasked to solve: $$X'= \left( \begin{array}{ccc} 1 & 4 \\ 1 & 1 \\ \end{array} \right)X + \left( ...