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

learn more… | top users | synonyms (1)

6
votes
2answers
104 views

Geometric series of an operator

In solving a first order linear differential equation $(1-D)y=x^2$ where $D\equiv \frac{d}{dx}$ the way I learnt was that we proceed as ...
0
votes
1answer
46 views

Find a linear combination of $u_n$'s satisfying $u(x,1) = \sin(2\pi x) -\sin(3\pi x)$

I have the following problem: $$u_n(x,y) = \sin(n\pi x)\sinh(n\pi y), \;\;\;n = 1, 2, 3, ...$$ Find a linear combination of the $u_n$'s that satisfies: $$u(x,1) = \sin(2\pi x) -\sin(3\pi x)$$ Any ...
1
vote
0answers
17 views

Using given conditions to determine a variable $\alpha$ from a differential equation

I am given that at 1 am a snow plough has travelled 2 miles, but it takes until 3 am to travel a further one mile. Also let $m$ be the distance travelled in miles by snow plough, $t$ be the time in ...
4
votes
3answers
69 views

Stability of a feedback system

Take the following feedback system: $\dot{x} = (\theta - k_1) x - k_2 x^3$ Now my book says: For $\theta > k_1$, the equilibrium $x = 0$ is unstable. I wonder why... Furthermore my book ...
0
votes
0answers
25 views

Solving the D.E:$\bigg(\dfrac{1}{t}+\dfrac{1}{t^2}-\dfrac{y}{t^2+y^2}\bigg)dt+\bigg(ye^y+\dfrac{t}{t^2+y^2} \bigg)dy=0$

By trying to solve: $$\bigg(\dfrac{1}{t}+\dfrac{1}{t^2}-\dfrac{y}{t^2+y^2}\bigg)dt+\bigg(ye^y+\dfrac{t}{t^2+y^2} \bigg)dy=0$$ And checking if the D.E. is exact, we partial derivate: ...
0
votes
1answer
58 views

Solving the Exact DE: $x\dfrac{dy}{dx}=2xe^x-y+6x^2$

Solving the Exact DE: $$x\dfrac{dy}{dx}=2xe^x-y+6x^2$$ I first re-wrote the ecuation, in order to have $M(x, y)$ and $N(x,y)$: $$(2xe^x-y+6x^2)dx-x dy$$ And, by having the partial derivative: ...
0
votes
1answer
120 views

How to solve a differential equation involving $\sin$ and $\cos$ functions?

I am trying to solve the following differential equation (1) : $ \frac{d}{dt} V(t)+ \frac{1}{C} V(t) = \frac{B}{C} \sin{(\omega t)} $. (where $B$,$C$ and $\omega$ are constants) I approached this ...
0
votes
0answers
44 views

initial value problem using picards theorem

I don't know how to do this problem and it would be a great one to know so I could do the similar problems like this one. How would I able to solve this ? thanks in advance.... Initial value problem ...
0
votes
1answer
35 views

differential equation..part of the integral in the answer

I am doing a differential equation and part of the integral in the answer I don't understand. Where did the e^​t go from the top when they were integrating ?
1
vote
1answer
29 views

Showing the range of this PDO is dense.

Consider the PDO; $Q:=-p(x)\frac{d^2}{dx^2}-p'(x)\frac{d}{dx}+q(x)+1$, an unbounded operator on $L^2(\mathbb{R})$ densely defined on compactly supported infinitely differentiable functions ...
0
votes
1answer
38 views

Converting 2nd order differential equation to state space

I've got two 2nd order differential equations that I need to convert to state space in order to express them as first order ODE's to model something in Matlab (using ODE45, for what it's worth). They ...
0
votes
0answers
25 views

Tangent solutions of $x' = f\left(\frac{x}{t}\right)$

Can someone help me in the following exercice? Lef $f:\mathbb{R}\longrightarrow \mathbb{R} $ be of $C^1$ class and $r\in\mathbb{R}$ such that $f(r) = r$. Show that If $f'(r) < 1$, ...
0
votes
1answer
55 views

Exact numerical solution to non-linear ODE

We're given the initial value problem below: \begin{align} y'(t)=4t\sqrt{y(t)}-\lambda (y(t)-(1+t^2)^2), \quad y(0) = a, \quad (a,\lambda)\in\mathbb{R}^2 \end{align} For which $\lambda$ and $a$ ...
1
vote
0answers
36 views

Figuring out impulse response

I need a little help with figuring out this problem. I understand most of it but the main part I don't understand is: The signal $h''(t)+2*h'(t)+2*h(t)$ is of finite duration. In the problem we are ...
0
votes
0answers
71 views

Solving a weird Diff equation…

Good day people I am modelling a water bottle rocket. Using the conservation of mass : $$-{\rho}vA + \frac{d}{dt}∫dM = 0 \tag{1}$$ Since the mass, O2 pressure, O2 volume and velocity change over ...
2
votes
0answers
88 views

Does the solution to this ODE have a closed form?

Consider the following two initial value problems: Problem 1: $\frac{dy}{dx}=\sqrt{\frac{1}{2\cos x}-\frac{y^2}{4}}, \ \ y(0)=-\sqrt{2}$ Problem 2: $\frac{dy}{dx}=\sqrt{\frac{1}{2\cos ...
1
vote
0answers
53 views

How can solve this differential equation (third equation )?

How can I solve this differential equation? $$ \frac{dy}{dx}=\sqrt{\frac{A}{y}+\frac{B}{y^2}+\frac{C}{y^4}+\frac{D}{y^5}+\frac{1}{(\frac{1}{y}+\frac{3}{y^2})^2}} $$ where $A,B,C,D$ are constants.
0
votes
0answers
24 views

Approximation Methods for DEs

This is related to Euler's equation, but I'm being posed with the question: Show that $y(t) = \int e ^ {-u^2}$ satisfies the initial value problem $dy/dt = e^{-t^2}$, $y(0) = 0$. Can anyone clue me ...
1
vote
1answer
28 views

Problem related to DE

Solve and comment on the solution behaviour at $|x|$ approaches infinity (bounded or unbounded as $x$ approaches $\pm\infty$. $$\frac{1}{1+x^2} + \sin y + y'\left(x\cos y + ...
0
votes
1answer
30 views

Finding the inverse of a complicated function

I need to find $\rho=B^{-1}(t)$, where $$B(t) = \int_{0}^{t}\frac{1}{k-\frac{Bs+m_0}{As+w_0}}ds = \frac{A}{(kA-B)^2}\{(kA-B)t+[m_0-kw_0+(kA-B)\frac{w_0}{A}]ln\big|\frac{(kA-B)t}{kw_0-m_0}+1\big|\}.$$ ...
1
vote
0answers
19 views

Show that $\det\Phi(t)$ is the unique solution of the scalar equation

Show that $$\det\Phi(t)=\det\Phi(t)\exp\left[\int_{t_0}^{t}\sum_{j=1}^{n}a_{jj}(s)ds\right]$$ is the unique solution of the scalar equation $$y'=\left(\sum_{k=1}^{n}a_{kk}(t)\right)y$$ satisfying ...
0
votes
0answers
65 views

Dependence On Initial Conditions and Parameters.

I'm having a hard time getting started with this problem. I'm not even sure if this can be done by computing some derivatives or what not or if I need to use a proof for this solution.
1
vote
2answers
410 views

Implicit form of general equation

Find, in implicit form, the general solution of the differential equation: $$\frac{dy}{dx}= \frac{2y^4e^{2x}}{3\left(e^{2x}+7\right)^2}$$ I am struggling to make any sense of this. What I have ...
6
votes
1answer
101 views

Assistance Solving A Second Order Nonlinear ODE (Converted into a First Order)

I am trying to find the solution to $y''=y+y^2$ I noticed that if I multiplied by $y'$ on both sides and integrated, the result would be $\frac{1}{2}(y')^2=\frac{1}{2}y^2+\frac{1}{3}y^3+c$ I have ...
1
vote
1answer
54 views

solve the differential equation by integrating directly

I am trying to solve a differential equation and I don't know how to solve it when it comes to integrating directly. I'd like to know how to do this so I can start doing other problems. Thanks in ...
1
vote
1answer
57 views

Ordinary Differential Equation - Boundary Conditions Question

The following problem has brought up some misunderstandings for me - Find the eigenvalues λ, and eigenfunctions u(x), associated with the following homogeneous ODE problem: $$ {u}''\left ( x \right ...
0
votes
1answer
30 views

Solutions space of sturm Liouville when $p>0$ is one dimensional?

Let $y_1(x)$ be a non trivial solution of the ODE $(p(x)y')'+q(x)y=\lambda y$, with $y(a)=y(b)=0$, and $p(x)>0$ in all the interval $[a,b]$. Prove/disprove: every solution $y_2$ is of the form ...
4
votes
3answers
373 views

Theory of the Mathieu Operator

How important is the theory of the Mathieu operator in mathematics/applied mathematics? What are the major mathematical concepts required to study it? The Mathieu operator is an ordinary periodic ...
0
votes
1answer
41 views

Compound Growth

This is along the same lines as the population growth problem but a little more involved. I'm really lost on how to go about it. The question is: Suppose that each time a savings bank compounds ...
0
votes
2answers
78 views

A Complex Variable ODE

suppose $f$ is a holomorphic function on some domain $D$ satisfying $f'(z)=af(z) $ for some >constant a. show that $f(z)=Ce^{az}$, for some constant $C$
1
vote
2answers
54 views

Solve $y'(t) = \dfrac{1}{1+ty}$

Does the following reasoning make sense? \begin{gather} \dfrac{dy}{dt} = \dfrac{1}{1+ty},\\ 1+ty \; dy = dt, \\ \int1+ty \;dy = \int dt,\\ y+t\dfrac{y^2}{2} = t+C. \end{gather}
4
votes
6answers
1k views

New & interesting uses of Differential equations for undergraduates?

I'm teaching an elementary DE's module to some engineering students. Now, every book out there, and every set of online notes, trots out two things: DE's are super-important, vital, can't live ...
0
votes
1answer
84 views

Have trouble making a phase line plot with Maple.

Draw the phase line, and sketch several graphs of solutions in the $ty$-plane Is the directions for my homework. I have tried the Phase Portraits for Autonomous Systems under tasks, but I don't ...
0
votes
1answer
37 views

Equations differential systems.

Can you help me to solve this equation. How could solve it? $$x'_1=-2x_1-x_2$$ $$x'_2=4x_1-2x_2-2t^{-3}$$
0
votes
0answers
38 views

Complex Non-liner First order ODE problem

Good day people I am modelling a "water bottle rocket" using basic Continuum Mechanics. I have found a equation describing the acceleration of the rocket. I need to integrate this function to find ...
1
vote
1answer
36 views

Question on convergence to fixed points

Let an autonomous dynamical system (system of ODEs) be given by $$\frac{dx}{dt}=f(x)$$ where $f(x): \mathbb{R}^n \to \mathbb{R}^n$ and also $f$ is $C^2(\mathbb{R}^n,\mathbb{R}^n)$. We also assume ...
1
vote
1answer
13 views

Assuming F(x=) # of people living within X radius of walmart is F' nonpositive or nonnegative

So F(x)=# of people living within a radius of x miles from walmart First question - What does F(3) represent - my answer - 3 people within 3 miles from walmart second question - What does quanity ...
2
votes
0answers
36 views

Reduction of Order - Confusion

Find the solution of the following initial value problem: $y'' -y'\tan(t)+2y =0$ $y(0) = 1$, $y'(0) = \sqrt[]{3}$ I've attempted this problem a couple of times and keep coming up with an incorrect ...
3
votes
0answers
83 views

Solve ODE by Fourier transform, and versus by Laplace transform?

Regarding solving ODE by Fourier transform, I read a nice reply by O.L.. After applying Fourier transform to an ODE to obtain an algebraic equation, the reply showed that some terms involving the ...
0
votes
2answers
79 views

How does expanding by Taylor's theorem work here?

The problem I am trying to figure out a step in the proof of this book (p. 245), where it goes like this: \begin{equation}\tag{a}\label{eq:equal} F_i(x^0; t + dt) = F_i(x; dt)\end{equation} ...
0
votes
2answers
49 views

a differential equation equation related to fourier series

I am really struggling with this one. Any help is welcome! For equation $f''(z) + p(z) f'(z) + q(z) f(z) = 0$, where $p(z)$ and $q(z)$ are fixed polynomials. Given $f(0)=f_0$, $f'(0)=f_1$, prove that ...
0
votes
1answer
30 views

Is there a closed form solution for the motion of a particle with friction?

I am trying to find a solution to Newton's equation of motion $ \boldsymbol{F} = m \boldsymbol{\ddot{r}} $ assuming a constant force $ \boldsymbol{F} $ but accounting for kinetic friction which is a ...
1
vote
1answer
38 views

determine if given differential equation is linear

let us consider following differential equation and problem statement let us consider first of all $dy/dx$ $xdy=-(y^2-1)*dx$ or $dy/dx=-(y^2-1)/x$ and $dx/dy=-x/(y^2-1)$ they are reciprocal ...
3
votes
0answers
47 views

Show that this initial-value problem has a unique solution

I am trying to show that the following initial-value problem $$\frac{dx}{dt} = - x + tx^{1/2}; \quad x(2) = 2$$ has a unique solution on $I = [2,3]$. By letting $f(t,x) = - x + tx^{1/2}$ and $(t_0 ...
1
vote
1answer
47 views

Singular system of differential equations

Consider the following second order liner singular system $K x''( t ) = A x'( t )$,where $K$ is a singular matrix,$A$ is any matrix,$x(0)=\left<2, 1\right>,x'(0)=\left<0, 1\right>$. how to ...
0
votes
0answers
38 views

MatCont continuation data for use on other plotting softwares

I have been using MatCont for generating continuation figures for my model ODEs. Dissatisfied with the quality of figures on MATLAB, I want to use gnuplot for plotting of this continuation data. In ...
0
votes
0answers
13 views

Finding new state via Euler integration

I have a state; call it X = [x,y, $\theta$]. It's a pose. I know my linear velocity (v) and my angular velocity ($\omega$) (or at least decent approximations of them). I've been asked to find the new ...
0
votes
1answer
36 views

Stability of a Specific Hill's Equation

Consider the Hill's Equation $u''+a(t)u=0$ where $a(t)=a(t+T)$ for all $t$. Show that if $a(t)<0$ for all $t$, then the solution satisfying the inital condition $u(0)=u'(0)=1$ is unbounded as $t ...
0
votes
0answers
29 views

How to solve first order differential equation

it is been a while since I solved differential equations. Hope you can help me with this one: $w(t)=\frac{u'}{u}$. What is the type of this equation? Thank you!
1
vote
2answers
86 views

Population Growth - Ecology

This is a population growth/decay problem but it is just tripping me up. The question is: Suppose, in addition to births and deaths (with constant rates $b$ and $d$ respectively), that there is an ...