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

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-1
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0answers
19 views

Difference Equations and their applications

What are some interesting applications to difference equations? I've learned about first and second order difference equations, first order systems, different kinds of equilibrium solutions (locally ...
0
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2answers
16 views

Using method of undetermined coefficients

Here's the equation I'm working with: y''+y'-2y = 3e^(x)+4x I want to use the method of undetermined coefficients for this equation. The logical choice for our guess would be y = Ae^(x)+Bx+C. ...
1
vote
0answers
33 views

1-forms and zero simple

Let $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and $D\subset U$ such that $\varphi$ restricted to $\partial D=\gamma$ be distinct zero and we define $i(\varphi ...
1
vote
0answers
19 views

The Adjoint Equation

I have a simple question about the adjoint equation for second order linear differential equation. Given an equation of the form $$P(t)y'' + Q(t)y' + R(t)y = 0$$ Let $u(t)$ be an integrating factor ...
0
votes
1answer
12 views

Stochastic Differential equation, expectation and variance

The process is given by $$dU_t=-\gamma U_t\mathrm{d}t+\sigma\mathrm{d}X_t$$ where $U_0 = u$ and $\gamma, \sigma$ are constants. Can you help me out to solve the equation for $U_t$ and find the ...
-2
votes
0answers
15 views

A Bernoulli Differential Equations Problem [closed]

Please, can anyone help me in solving this Bernoulli's Differential Equation. $(xy+x)dx=((x^2)(y^2)+(x^2)+(y^2)+1)dy$
0
votes
0answers
11 views

Question on stable manifolds

If $x\in M$ is a hyperbolic fixed point of a diffeomorphism $\phi:M\to M$, then the stable manifold $$ W^s=\{y\mid \lim_{n\to\infty}\phi^n(y)=x\} $$ is the image of an injective immersion $$ ...
0
votes
0answers
19 views

Differential equation $tx'= \sqrt{t^2-x^2}$ [closed]

$tx'= \sqrt{t^2-x^2}, |x|\le|t|$ Is it sth like Clairaut equation?
0
votes
1answer
21 views

prove that $f(x,y) = x^2+y^2$ is continuous on rectangle R.

where $R = \{(x,y): |x|, |y| \leq \frac{1}{\sqrt 2} \}$ I am trying to use picard's theorem so I have to prove that f is continuous on R and that it's lipschitz continuous. How would I do this? I ...
1
vote
1answer
24 views

Determinant of solution of linear equation

Is there a direct way or method to know if the solution to a linear ODE is invertible? I mean, let $A(t)$ be a ($n$ times $n$) matrix and denote by $X(t)$ an unknown Matrix (of the same dimensions) ...
0
votes
2answers
40 views

Solve the initial value problem $u'(t)=u^2(t)+t,\;u(0)=1$

How can I solve the following initial value problem: $$\begin{cases}u'(t)=u^2(t)+t\\u(0)=1\end{cases}$$ This is a first-order nonlinear equation. The only method I know, to solve such an equation, is ...
0
votes
1answer
35 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
22 views

integrals and differential equations [closed]

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
votes
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
185 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
52 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
34 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
26 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
47 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
33 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, ...
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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
53 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 ...
0
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0answers
28 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
votes
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 ...