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

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0
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1answer
34 views

Second order, homogeneous, linear boundary value problem

I could not solve differential equation. $$x^{"}- 3x^{'} -9 e^{6t} x = 0 , x(0)=0 , x(1)=1 $$ that $e^{6t} = \exp(6t)$ please help me.
0
votes
1answer
21 views

integrals and differential equations [on hold]

proof that $x \in \mathbb{R_*^+}$ $\int_{0}^{+\infty} \dfrac{e^{-xt}}{1+t^2}dt=\int_{0}^{+\infty} \dfrac{\sin t}{x+t}dt$ (you can Use :differential equations between two functions)
2
votes
1answer
26 views

Repairing solutions in ODE

Recently I encounter something interesting that I hope to hear from your opinions: Suppose we are given a ODE $\frac{dy}{dx}=y$, with no initial condition. Naively, we divide both sides by $y$ and ...
1
vote
0answers
46 views

Finding the critical values of a response curve

I have the motion of a forced spring: $$x'' + \kappa x' + x - x^3 = \varGamma \cos(\omega t) \ \ \cdots \ \ (1)$$ and I am investigating the stability of its solutions with forcing period $T = ...
1
vote
1answer
50 views

Showing that a solution to an ODE is bounded without solving the ODE

Consider the differential equation: $2y'-y^2=-\alpha^2$ where $\alpha>0$ ($\alpha$ is a constant). Ons solution to this equation is $y(x)=\alpha$. Without solving the ODE, show that any bounded ...
0
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0answers
10 views

Integration by parts applied to weak form of boundary value proble

In my finite element textbook the proof for strong and weak form equivalence is determined as such: $$\int_0^1w_{,x}u_{,x}dx = \int_0^1wfdx + w(0)h$$ Integrating by parts and making use of the fact ...
2
votes
6answers
184 views

How to solve $y''' = y$

I'm trying to solve the following differential equation $ y''' = y$ and given conditions: $ y(1) = 3$, $y'(1) = 2$ and $y''(1) = 1 $ I began by making it: ...
3
votes
0answers
50 views
+100

How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $ M ^ n $ be a Riemannian manifold, $ f: M \rightarrow M $ be ...
0
votes
1answer
37 views

Solve the homogeneous differential equation $y^2+x^2y'=xyy'$

$y^2 + x^2y'=xyy'$ Here's what I did: $y^2=(xy-x^2)y'$ $\frac{y^2}{xy-x^2}=y'$ $\frac{\frac{y^2}{x^2}}{\frac{y}{x}-1}$=y' $v=\frac yx$ $\frac{v^2}{v-1}=v+x\frac{dv}{dx}$ ...
0
votes
0answers
13 views

Elementary differential equations, difference equation

Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance.
2
votes
1answer
33 views

Solving $y''-4y=x^2 e^{2x}$.

I want to solve the differential equation $$y''-4y = x^2e^{2x}$$ Clearly $y_1 = e^{2x}$ and $y_2 = e^{-2x}$ are linearly independent solutions of the homogeneous equation. I would propose $y = ...
1
vote
3answers
25 views

I can't figure out this simplification in a differential equation

I was watching PatrickJMT's video on first-order differential equations and while I think I should see what he's doing on the left side here from line one to line two, I just can't. I ran it past my ...
0
votes
2answers
37 views

Let $A$ be a single $p\times p$ Jordan block. Find general solution to $\dfrac{dx}{dt} = Ax$

Let $A$ be a single $p\times p$ Jordan block. Find the general solution to $\,\dfrac{dx}{dt} = Ax$. What should I approach first? Please help!
0
votes
1answer
29 views

Infimum and supremum of $\int_{0}^{1} e^{x(t)}(x'(t))^{2} \; dt$

Find infimum and supremum of $$\phi[x]=\int_{0}^{1} e^{x(t)}(x'(t))^{2} \; dt$$ where $x \in C^{1}[0,1]$ and $x(0)=0$ and $x(1)=\log 4$. It's easy to show that $\sup \phi[x]=\infty$, but what about ...
0
votes
0answers
16 views

Topological conjugacy

Prove that any two linear systems with the same eigenvalues $ \pm i \beta$, $\beta \neq 0$, are conjugate. What happens if the systems have eigenvalues $ \pm i \beta$ and $ \pm i \gamma$ with $\beta ...
1
vote
1answer
25 views

Bounding a linear functional in $L_2[0, 1]$

For each f in $L_2[0, 1]$ let $\phi(t)$ be the solution of $y' + ay = f$ that satisfies $\phi(0) = 0$, where a is a constant. Define $l: L_2[0,1] \to \mathbb{C}$ by $l(f) = \int_0^1 \phi(t) dt.$ ...
0
votes
1answer
24 views

On a substitution to solve a high order differential equation (exercise).

I have solved Linear first order differential equations, I know how to separate variables, for higher orders I know the the substitutions that apply to the bernoulli equation and homogeneous ...
0
votes
0answers
29 views

Show that the system has a solution. [closed]

\begin{align} f_1=x'=x-x^3-xy^2 \\ f_2=y'=y-y^3-yx^2 \end{align} I get a critical point (0,0) and the set $\{x^2+y^2=1\}$. $$\frac{\partial f}{\partial \mathbf x} = \begin{pmatrix} 1-3x^2-y^2 & ...
1
vote
1answer
31 views

Show that $y''+(y^2+2y'^2-1)y'+y=0$ has a periodic solution.

I made the following system $$x'=y$$ $$y'=-(x^2 + 2y^2-1)y-x$$ The only critical point is $(0,0)$. I can get eigenvalues $\lambda = \frac{1\pm \sqrt3 i }{2}$. Then what should I do? Poincare-Bendixon ...
0
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0answers
15 views

Fourier series and Legendre polynomials

I am currently dealing with a problem that is based on this question that I cannot answer and therefore I wanted to ask you for help on this simpler problem: Consider the Legendre differential ...
0
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0answers
46 views
+50

Control Function with solution and fixed initial data on time interval, critical point of a cost functional?

Let $u(t)$ be a solution of the ODE $u''(t)+tu'(t) + u(t) = f(t)$ on the time interval $[0,T]$, with fixed initial data $u(0)=u_0$, $u'(0) = u_1$ where $f(t)$ is a control function. Find $f(T), ...
0
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1answer
46 views

Can the depicted function be a solution of an ODE with locally Lipschitz autonomous vector field?

Problem: Can x(t) depicted be a solution of a scalar differential equation x(dot)=f with locally Lipschitz autonomous
1
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0answers
38 views

A question on First Order Differential Equations

So, I've recently begun to tutor friends in math. I've only tutored classes that I've taken [algebra-multivariable calculus], and last night I was tutoring a friend in calc II. He pulled out a take ...
0
votes
1answer
33 views

Maximal solution of differential equation

Let $K\subset X$ be a compact set and let $x_0\in K$. Suppose that the maximal solution $x(t)$
1
vote
2answers
98 views

Solving ODE containing matrices

We have an ODE $ \psi'(t)_{_{3 \times 3}}=\psi(t)_{3 \times 3}(A_{3 \times 3}+B_{3 \times 3}t)\tag 1$ Given Data in Question We have no quarentee that $\psi'(t),\psi(t)$ both have inverse A,B are ...
1
vote
0answers
17 views

How to estimate the local error and the global error for Runge-Kutta method

How to estimate the local error and the global error for Runge-Kutta method in practice? I have no idea. I recieved a nice answer on the question at other site
1
vote
0answers
22 views

Best approach to matrix representation of system of nonlinear ODEs

I have this system of ODEs: $$ \frac{dS}{dt}=\pi S-\beta S Z\\ \frac{dZ}{dt}=\alpha S Z - \delta Z $$ I am trying to rewrite them in the form : $$ \pmatrix{\dot{S}\\\dot{Z}}=\mbox{diag}(S,Z) ...
0
votes
1answer
32 views

Advection equation with source u/x

I am trying to solve following equation: $$ u_t + u_x + \frac{u}{x} = 0 $$ With initial condition: $$ u(x,0) = 0 $$ And with boundary condition given at x = 15: $$ u(15,t) = sin (wt) $$ I tried to ...
0
votes
1answer
9 views

Showing uniqueness of non-linear second order differential equation with initial values with some condition.

Assume $f \in C(\mathbb{R})$ and $g\in C^1(\mathbb{R})$. Show that IVP problem $$y''+f(y)y'+g(y)=0$$$$y(a)=b , y'(a)=c $$ has a unique solution. my strategy: if assume $y=x_1$ and $y'=x_2$ ...
0
votes
0answers
12 views

Quadratic stability linear time varying system

Consider the linear time-varying system $$ \dot{x} = A(t) x, $$ where $x \in \mathbb{R}^n$ and $A: [0,+\infty) \rightarrow \mathbb{R}^{n\times n}$ is continuous. It is known (see for instance, [1, ...
-3
votes
0answers
41 views

Differential equation $(x^2+6xy+2y^2)dx + 2x(x+y)dy$

To be solved the differential equation: $(x^2+6xy+2y^2)dx + 2x(x+y)dy = 0$ if $y(1)=-3$. ... Not directly integrable, so I start by setting $\dfrac{y}{x} = u \iff y = ux \iff dy = udx + xdu$. So the ...
0
votes
0answers
22 views

Variation of constants for system $x' = Ax + B(t)x$

I came across this in the proof of a theorem about the stability of the solution to $x' = Ax + B(t)x$, $x \in \mathbb{R}^n$ ( Verhulst's Nonlinear ODE's, chapter 6). The proof states that such a ...
0
votes
1answer
17 views

Finding general soon for Euler equation given a trial function

Use $y=x^r$ as a trial function to find the general solution to the Euler equation: $2x^2y''+3xy'-y=0$ ; $x>0$ I have no idea how to start this, as I am only able to work with second order ...
0
votes
0answers
52 views

The Runga-Kutta method with a adaptive step

I have some questions about this method. I use Richardson extrapolation for select a adaptive step [Solving Ordinary Differential Equations I - Nonstiff Problems 167-168p]. What mean $\varepsilon$ ...
0
votes
0answers
23 views

Uniqueness of a differential equation

Let $I_o=[t_0,t_0+T]\subset\mathbb R$, where $T>0$, $f\in C^0(I_0\times\mathbb R;\mathbb R)$ and satisfying Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
0
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0answers
20 views

vector space differential equations

Hi! I am working on some differential equations homework and we are up to the linear algebra part. This particular homework set on Vector space is due, but my teacher has not taught the material yet ...
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0answers
27 views

Showing a system is fully self adjoint for general unmixed boundary conditions

I have been asked to look at the following questions and I'm struggling to solve it. Let $Ly=a_2(x)y''(x)+a_1(x)y'(x)+a_0(x)y(x) , a<x<b$ such that $L^*=L$. i.e. $L$ is a self adjoint linear ...
1
vote
0answers
8 views

Simplifing a Cauchy product to find the recurrence relation when solving a differential equation using a power series solution.

I'm having trouble finding the recurrence relation of the following non linear differential equation: $y''(x)+p(x)y'(x)+y^2(x)=0$ with $y(0)=1$ and $y'(0)=0$ where ...
0
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0answers
11 views

solving three first order differential equations simultaneously with varying coefficient

I need to solve 3 first order differential equations simultaneously. I can solve this equation when [A] is constant. But in this case, as I will explain, [A] is function of z. By omitting the uz, I ...
0
votes
0answers
33 views

What is the equation family of the projectile-motion-with-air-resistance eqn?

The general form of the equations of projectile motion with air resistance are (from here) $s_y(t) = -\frac{mg}{k}t + \frac{m}{k}(v_{yo} + \frac{mg}{k})(1 - e^{-\frac{k}{m}t})$ and $s_x(t) = ...
0
votes
0answers
33 views

Invariants of a nonlinear ODE

Given a nonlinear ODE and a simple constraint $x \leq c$ for some constant $c$, how can we describe the largest set (or an approximation thereof) such that if the initial value of the solution of the ...
0
votes
0answers
12 views

How could one go about constructing this relatively simple contagious diffusion-reaction model?

How could one go about constructing a contagious diffusion-reaction model showing the relationship between disease (e.g. Ebola) and number of available healthcare workers in an unevenly distributed ...
1
vote
0answers
14 views

Is there a relationship between the integrand in Green's Theorem and the test for finding an integrating factor for a differential form?

Green's Theorem has the formula $$ \int_C Mdx+Ndy=\int\int_D\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy $$ There is also a well known test for finding an integrating ...
1
vote
2answers
23 views

What is the correct answer to this diffferential equation?

[Question] When solving the differential equation: $$\frac{\mathrm dy}{\mathrm dx} = \sqrt{(y+1)}$$ I've found two ways to express $y(x)$: implicitly: $2\sqrt{(y + 1)} = x + C$ or directly: $y = ...
0
votes
0answers
26 views

Showing a second order DE has characteristic equation

Verify that $y''-2py'+p^2y=0$ has characteristic equation $(m-p)^2=0$ and has solution $y=e^{px}$ I began by trying to solve $r^2-2p+p^2=0$ but I'm kind of stuck where to go. Any help would be ...
0
votes
0answers
25 views

I still could not figure out

IT is our homework problem but I have already submit it. Today, I asked professor, but I still could follow what he said clearly. $\frac{dX}{dt} = \mu(x)$ and $X(0;x) = x$, where $x,X\in R^n$ For ...
0
votes
0answers
30 views

How do I solve this calculs problem [closed]

a) Find the general solution of $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 0.$$ b) Solve $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 8\cos 2t + 6\sin 2t.$$ with $y(0) = 4$, $y'(0) = 0 $ How ...
1
vote
3answers
32 views

The limit of a solution of the logistic equation as time tends to infinity

$$ \frac{dP}{dt} = 3P(4 - P),\quad P(0) = 2.$$ What value does $P$ approach as $t$ gets large, ie. as $t \to\infty$. How do I solve this? Is the idea to this question to first rearrange the equation ...
1
vote
0answers
42 views

How to solve a system of two differential equations describing the concentration in a leaky tank?

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
0
votes
0answers
21 views

Differential equation Worded Problem [duplicate]

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...