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

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24 views

2nd order linear differential equation with non-constant coefficients

Considering the equation $2y''+(x+1)y'+3y$ where $X_0=2$. Find the general term in each solution. That is, the general term for Y1,Y2 where $y=A_0(Y_1)+A_1(Y_2)$ I've solved this as ...
3
votes
3answers
71 views

How do we find $u(x)$?

I want to know how to find $u(x)$ in the below question: $$u''(x)+{e^u}^{(x)} = 0\\ x \in[0,1]\\u(0) = u(1) = 0$$ Please explain briefly how this was done?? Thanks!!
1
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1answer
34 views

Solving a non-linear differential equation

Given the differential equation $$\ddot{y} + y + y^3 = 0$$ where $y = y(t)$, $\dot{y} = \frac{dy}{dt}$. By multiplying this equation by $\dot{y}$, assuming $\dot{y}>0$ and integrating, find ...
3
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2answers
455 views

How to find a conserved quantity in this differential equation.

Consider the system: $$\ddot x = x^3 -x$$ What is the method to follow to find a conserved quantity for this system? So far what I have is: $\dot x = y$ and $\dot y = x^3 - x$ and I can find the ...
1
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1answer
26 views

Stuck with weird results when solving $\frac{d^2y}{dt^2} + 4 \frac{dy}{dt} +4y = e^{-\alpha t}$.

Find the general solution of the differential equation $$\frac{d^2y}{dt^2} + 4 \frac{dy}{dt} +4y = e^{-\alpha t}$$ where $α$ is a constant and $α ≠ 2$ Normally questions which ask to ...
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0answers
163 views

Car traveling on a bumpy road (ODE)

The suspension system of a car traveling on a bumpy road has a stiffness of $k = 5\times 10^6$ N/m and the effective mass of the car on the suspension is $m = 750$ kg. The road bumps can be considered ...
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0answers
35 views

Scheme for mixed partial derivative from Taylor

I need to deduce a scheme(finite difference) for the partial derivative: $$\frac{\partial^3 u}{\partial t \partial x \partial x} $$ How can I deduct it from Taylor polynomial? Thanks for your help
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1answer
94 views

Solve Bratu problem using Python

I am going crazy trying to solve the Bratu problem using Python: $$y''(x)+ e^{y(x)} = 0, \quad \lambda = 1, \quad x \in(0,1),$$ $$y(0) = y(1) = 0$$ I have to solve this using the tridiagonal ...
0
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1answer
25 views

Is this equality for derivatives true?

I was trying to solve a problem and I came across the left term in the equality below (which is part of a differential equation), were $\mu$ is a constant: ...
3
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1answer
46 views

Can someone explain linearisation on nonlinear systems to me?

I want to find all critical points of the following nonlinear system: $$\def\b{\begin{pmatrix}}\def\e{\end{pmatrix}}$$ $$\b y_1' \\ y_2'\e = \b 5y_2 -15 \\y_2^2 - y_1 ^2\e$$ Then use linearisation ...
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1answer
15 views
1
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1answer
112 views

Questions about Lyapunov functions

I'm trying to find a Lyapunov function for the nonautonomous ODE $z'=g(z)$ with $z=(x,y)$ and $$ g(x,y):=(-2x-y^4,-y-x^2). $$ For the sake of analyzing the stability of the rest point $z=0$, one does ...
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0answers
33 views

Analytic Solution of Second Order Nonlinear odes

Any idea how to find analytic solution of the following ODE. $y''+0.1 y'+y^{5} = \sin (t)$ I will really appreciate your response! Shah
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1answer
47 views

$0$ is an stable equilibrium of $x' = Ax$ iff $A$ is semisimple, given that all of its eigenvalues have real part 0.

$0$ is an stable equilibrium of $x' = Ax$ iff $A$ is semisimple, given that all of its eigenvalues have real part 0. I'm kind of confused here: I had understood that if all of the eigenvalues of $A$ ...
1
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1answer
36 views

Solution to Differential Equation $\left( 1-2\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)+2 \lambda h(x,y,z)=0$

I'm trying to solve the following Differential Equation: $\left( 1-2\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)+2 \lambda h(x,y,z)=0$ The unknown function is $w(x,y,z)$. The ...
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0answers
82 views

RLC Circuit and 2nd order linear DE

An RLC circuit consists of a voltage source in series with a resistor, a capacitor, and an inductor. An inductor is a coil of wire. When the current passing through the coil changes, a magnetic field ...
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0answers
31 views

Explanation of the Leibniz formula

I am reading the book Solving Ordinary Differential Equations I - Nonstiff Problems (1987) by Hairer et al. My question is from Section II, chapter 2 (Order conditions for RK methods), equation 2.4. ...
0
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1answer
34 views

What is wrong with my general solution and how to find $f(x)$

Given that $$y=\frac{1}{w}$$ Here is my working: $$\frac{d^2w}{dx^2}+2\frac{dw}{dx}+5w=-5x^2-4x-2$$ Auxillary Equation: $$a^2+2a+5=0$$ $$a=-1+2i,-1-2i$$ C.F $$w=e^{-x}(C \cos 2x+ E ...
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0answers
40 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
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1answer
19 views

how to find a function $f(x)$ such that $\lim_{x\to \infty} (\frac{y}{ f(x)})=1$

$$\frac{d^2w}{dx^2}+2\frac{dw}{dx}+5w=-5x^2-4x-2$$ Given that $$y=\frac{1}{w}$$ Where The particular solution (I found) is : $$w=e^{-x}(C \cos 2x+ E \sin 2x)-x^2$$ The general ...
0
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1answer
22 views

changing forms of constant of integration

In solving O.D.E in my book sometimes he changes the constant of integration in the form for example C=Sin(A) where C & A are constants obtain the general solution in an explicit form but how ...
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0answers
71 views

Change of Variables in Cauchy-Euler equations

So I'm working on the change of variables in the Cauchy-Euler equation. And I understand everything except one step. It's the same one step skipped in the answer to the very related question here. I ...
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2answers
27 views

Query about differential equations involving projectiles.

A projectile is launched at a speed $U$ at an angle $θ$ to the horizontal from $(x,y) = (0,0)$. Thereafter the projectile moves so that the second derivative of $X$ is $0$ and the second derivative of ...
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1answer
48 views

Please help me understand the concept of variable, and differentiation of variables.

`I am in the first year of college and know mathematical analysis in a very rigorous context, from high school/ math olympiads Imo's etc. But the concept of $df$ seems totally weird and ...
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1answer
45 views

Solving a non linear second order differential equation [closed]

How do I go about solving a nonlinear ordinary equation that is of second order? Such as $$y'' + ay^3 = 0$$ where $a$ is a constant.
2
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1answer
48 views

How to get rid of $(\frac{dw}{dx})^2$ term in a differential equation

My try: $$y=w^{-1}$$ $$y'=-w^{-2} \frac{dw}{dx}$$ $$y''=\frac{2}{w^3} \frac{dw}{dx} - \frac{1}{w^2} \frac {d^2w}{dx^2}$$ Substituting these to the first expression : ...
0
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1answer
37 views

How do I solve $yy''=y'-5y$ given that $y(1)=1$ and $y'(1)=-1$?

How do I solve $yy''=y'-5y$ given that $y(1)=1$ and $y'(1)=-1$? Do I have to integrate both sides of ODE? $$yy''=y'-5y$$ $$y''=\frac{y'}{y}-5$$ $$\int{y''dx}=\int{(\frac{y'}{y}-5)dx}$$ ...
0
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1answer
28 views

Differential equation: general solution for formula

I have following formula and I need the general solution: $$ \frac{d^{2}\theta}{d\xi ^{2}}-\mu ^{2}\cdot \theta =0 $$ EDIT Following solution was given: $$ \theta(\xi )=C_{1}\cdot exp(\mu \xi ...
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1answer
23 views

If $y$ is the solution of $\left\{y'=-y+\sqrt{t},y(0)=y_0>0\right\}$, then $\lim_{t\to\infty}\frac{y(t)}{\sqrt{t}}=1$

The homogeneous equation $$y'=-y$$ has the solution $$y_h(t)=ce^{-t}\;\;\;\;\;(c,t\in\mathbb{R})$$ In order to find a particular solution we can take the approach $$y_p(t)\stackrel{!}{=}c(t)e^{-t}$$ ...
0
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1answer
52 views

Overdamped and critically damped

Consider $y''+2by'+w^2y=0$. Show that as the limit of $b\to w$, the overdamped solution is equal to the critically damped solution. The roots are $D=-b\pm\sqrt{b^2-w^2}$. Thus, if $b^2>w^2$ ...
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0answers
56 views

How to solve this ode with absolute value

Let $y(t)$ be continuously differentiable, $y(0)>0$, and $$|y'(t)|=|y(t)|, t\geq 0.$$ Then how to show that $y(t)=y(0)e^{t}$ or $y(t)=y(0)e^{-t}$. The main difficulty lies in that I could not ...
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1answer
49 views

Elliptic linear ODE

I have a rather short question: What does "elliptic" mean in the the context of linear ODE? Only found "elliptic" in the context of partial differential equations.
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1answer
2k views

Euler-Lagrange Equation example

I have been working on solving Euler-Lagrange Equation problems in differential equations, specifically in Calculus of Variations, but this one example has me stuck. I am probably making mistakes in ...
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0answers
21 views

Why does the set $\pi(C)\cap\pi(D)$ have $\gamma$-measure 1?

I have a question concerning the article Ergodic Theory and Linear Differential Equations by R.A. Johnson. My questions concerns the proof of Lemma 2.3 on page 27, namely the statement ...
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1answer
46 views

How to solve system of Differential Equations with 1 independent and 3 dependent variables

How can one solve this set of three differential equations in one independent variable "t" and three dependent variables A, B and F, which are functions of only t? $$ \frac{F(t) B''(t)+B'(t) ...
1
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1answer
49 views

Existence and uniqueness for the ODE $y''-y^{1/3}=0$

For the ode $ y'' - y^{1/3}=0 $, is there any way to check the existence and uniqueness of the solution? I know the Picard's Theorem, but it can only be used for the first order ode.
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1answer
33 views

Differential equation solution inconsistency

When solving the differential equation: $dy/dx = y^2$, with $y(0) = 1$ I've found $y = 1/(1-x)$ as the solution. The problem asks then for an explanation to why $x=3/2$ is an invalid point to ...
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1answer
40 views

Differentials word problem

The Questions Use differentials to find the approximate amount of copper in the four sides and bottom of a rectangular tank that is 6 feet long, 4 feet wide, and 3 feet deep inside, if the copper is ...
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0answers
240 views

Using MATLAB to solve a system of 2nd order non linear ODEs

I have 2 coupled non linear 2nd order ODEs which describe a particle's trajectory in space, subject to an initial horizontal and vertical velocity, and also to gravitational and aerodynamic forces. ...
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1answer
39 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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1answer
34 views

Help identifying differential equation extraneous solutions

For the following differential equation: dy/dx = (xˆ2)/y, with initial condition y(0)=10 I've found the solution: (½)(yˆ2) = (1/3)(xˆ3) + 50. However, I've found two answers for y(5): ...
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2answers
49 views

The volume of a spherical balloon (constant rate)

The volume of a spherical balloon is increasing at a rate of $3$ cubic inches per second. After you find the rate of change of the balloon's radius at the time when the radius is $8$ inches explain ...
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0answers
17 views

Positive Characteristic exponent, then $\lVert x(t)\rVert\to 0$?

Let $x(t)$ be any solution to the ODE $x'=a(t)x$. Say that $x(t)$ has strong characteristic exponent $\beta$ as $t\to\infty$ ($t\to-\infty$) if $\lim_{t\to\infty}\frac{1}{t}\ln\lVert ...
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3answers
96 views

Fixed points of: $\dot{x}=\sin(y) \qquad \dot{y}=\cos(x)$

How can you find the fixed points of this system: $\dot{x}=\sin(y)\\ \dot{y}=\cos(x)$ Normally I would suggest that you find the points when both functions are equal to 0.
0
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1answer
41 views

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 ...
8
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3answers
190 views

How to solve this differential equation: $x^2dy-y^2dx+xy^2(x-y)dy=0$

$$x^2dy-y^2dx+xy^2(x-y)dy=0$$ What I tried: $$\frac{x^2}{y^2} \frac{dy}{dx}+x(x-y)\frac{dy}{dx}=1\\$$ Let $h=-1/x, \; k=-1/y,\; dh=1/x^2 \, dx, \; dk=1/y^2 \,dy$ ...
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0answers
26 views

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: ...
0
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1answer
50 views

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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0answers
23 views

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 ...
0
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0answers
41 views

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 ?