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

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Cauchy problem with parameter

Find the solution $u_{\lambda}(t)$ of the following Cauchy problem: \begin{cases} u'(t) = 1 + \lambda \sin u(t) \\ u(0) = 0 \end{cases} as the changes of $\lambda \in ]0, 1[$ and evaluate (for each ...
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117 views

Analytical solution to a nonlinear ODE

How might I analytically solve the following differential equation? $$yy'' = y' + y^3$$ I've tried certain substitutions ($y = ux$ etc.) but none of them work.
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1answer
45 views

An exercise in Treves related to Cauchy-Riemann operator

This is part of the exercise 5.10 from the book "basic linear partial differential equations" by Treves: " Let $P(z)$ be a polynomial in one variable, with complex coefficients. Describe all ...
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1answer
115 views

Showing that a sequence of Picard iterates converges

I have a sequence of functions: $$y_{n}(x) = 1 + \int \limits_0^x 1 + t^2 + y_{n-1}^2(t)\,\mathrm dt$$ With $y_0 = 1$. I'm trying to show that this converges in a box $-1 \le x \le 1$ and $-10 \le ...
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2answers
124 views

How to prove: $f(x)$ is differentiable on $(0,+\infty)$

The function $f(x)$ is defined on $(0,+\infty)$. We know $f'(1)$ exists and we have that $$\forall x,y \in(0,+\infty), \quad f(xy)=yf(x)+xf(y)$$ How to prove:$f(x)$ is differentiable on ...
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2answers
47 views

derivative after changing variable

I have just studied a lesson about derivative of a function but I still confuse in the following case. Suppose that I have a function: $$ f(x) = 2x^2 + 3x + 1$$ and I want to calculate ...
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1answer
284 views

Solving Differential Equations theoretically and using matlab

i am trying to solve the initial value and elliptic boundary value problems below. but now i need some help solving them using matlab. for the elliptic problem, any method is ok, but for the initial ...
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3answers
471 views

Differential Equation: Am I missing a trick?

I am trying to solve the following calculus problem: Show the function $$\displaystyle y(x)=\int_{0}^x \sin(x-t)f(t)dt$$ solves the differential equation $$y''+y=f(x)$$ I have put ...
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121 views

Analytical solutions of Thomas Fermi equation

The Thomas Fermi model of atoms and nuclei is used in many applications of atomic and nuclear physics. The ODE related to this model is: $$\frac{d^2}{dx^2}\phi(x)=x^{-\frac{1}{2}}\phi(x)^{3/2}$$ with ...
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1answer
33 views

How to solve this system of the 1st order equations?

This is a problem from the book: $$x_1' = x_2\\ x_2' = -x_1\\ x_1(0) = 2\\ x_0(0) = 0$$ The problem says transform the system of the 1st order differential equations into a single differential ...
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2answers
55 views

What does d f(t,x) = 0 mean?

A differential equation that can be written in the form $d\phi(t, x) = 0$ for some continuous and differentiable function $\phi(t, x)$ is called exact. What does $d\phi(t, x) = 0$ mean?
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56 views

Eigenvalue equation under neumann boundary condition

Let $c(x)\in L^{\infty}(0,a)$, a>0. We consider the problem $-u''+c(x)u=\lambda u(x),\quad x\in (0,a),$ under the boundary conditions $u'(0)=u'(a)=0$. Show that $u>0 $ in $[0,a]$ and that ...
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35 views

Solving Differential Equations to satisfy a condition

I have two differential equations $$\frac{dX}{dt} = -\frac{d\cdot N(0)\cdot X}{m+X}$$ and $$\frac{dY}{dt} = \frac{d\cdot N(0)\cdot X}{m+X}$$ with initial conditions $X(0) = X_0$ and $Y(0) = 0$. ...
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2answers
43 views

Using the substitution $u=x^3$, find the general solution of $xy''+y'+9x^5=0$.

Using the substitution $u=x^3$, find the general solution of $xy''+y'+9x^5=0$. I have no idea about above question? Could somebody suggest me a solution or resource for such problems?
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0answers
22 views

Is there any relation between positive definite operator and an operator that satisfies maximum principle?

Suppose $L$ is a self adjoint differential operator which satisfies maximum principle. Maximum principle: Assume that $u(x)$ satisfies $u(0)\geq 0$ and $u(1)\geq 0$. Now $L$ is said to satisfy ...
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3answers
61 views

Definition of homogeneous ODE

In my lecture notes, it gives this following definition of a homogeneous ODE: A differential equation is called homogeneous if it can be written in the form $x′=f(\frac{x}{t})$ Then in one of ...
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2answers
61 views

Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
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1answer
108 views

numerical update rule for discretized hawkes excitation process

So I think I am just misunderstanding some simple notation or something and would appreciate some help. I am trying to replicate this model in an agent based model, but I cannot seem to figure out the ...
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1answer
35 views

2nd Order Differential Equation - general solution

I have a 2nd order differential equation here. The main thing I need help for is working out what the general solution will take the form of. The equation I have is: $$ ...
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0answers
31 views

Derivation of an integral identity from the kdv equation.

The stationary KdV equation given by \begin{equation} 6u(x)u_{x}−u_{xxx}=0 \quad \quad \quad(1) \end{equation} It has a solution given by $$\bar{u}(x)=−2\mathrm{sech}^{2}(x)+\frac{2}{3} \quad \quad ...
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23 views

How is this formula to solve linear ODE obtained?

On my textbook, at the end, there is a very short paragraph about differential equations where the formula $$y=e^{-\int a(x) \mathrm{d}x}\bigg[\int b(x)\cdot e^{\int a(x) ...
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1answer
31 views

Confusion over definition of attractor

The definition of attractor says that "A compact invariant set $M$ will be called an attractor if it has an open neighbourhood $O$ such that every trajectory in $O$ remains in $O$ and converges to ...
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1answer
72 views

Analyzing the stability of equilibria

There's a model with a condition $r>\mu$: $$\begin{align} S'&=r(S+I)-\beta SI-\mu S \\ I'&=\beta SI-(\mu +\alpha)I \end{align}$$ I can easily see that the equilibria of the second ...
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2answers
161 views

How can I solve these pde's?

Three different problem I got: 1.. $xu_x+2x^2u_y-u=x^2e^x$ and $u(x,x^2+x)=xe^x+x^2$ 2.. $yu_{xx}+(x+y)u_{xy}+xu_{yy}=0, \quad x\neq y$ 3.. $(y+xu)u_x+(x+yu)u_y=u^2-1$ Couldnt even start. Could ...
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51 views

Write a second order differential equation for which y(t)= 2 cost- t is a solution and verify

I've been trying to write a second order differential equation for which $y(t) = 2\cos t - t$ is a solution (and verify the solution). I've tried taking derivatives and finding the relationship ...
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2answers
163 views

Solving 2nd degree ODE with Euler method in MATLAB

I am trying to solve the equation below; $$\ddot{x}= -x + sin(t)$$ by the initial conditions; $$x(0) = 0 \\ \dot{x}(0)= 0$$ my MATLAB code is as follows: ...
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1answer
70 views

Annihilator Method Help

I'm having trouble on some annihilator method problems for homework and was wondering if anyone would be able to point me in the right direction. The problem is: $$y''-8y'+20y=5xe^{4x} \sin(2x)$$ I ...
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1answer
39 views

Doubt on limit set of an ODE

Suppose, the initial value of a 1st order ODE lies in the limit point set for the ODE. Is it true that any solution for the ODE (value $\forall t$) also lies in the limit set ?
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73 views

Intuition or wisdom for stability and instability properties of locally linear system. Boyce, p513, Table 9.3.1

Our instructor requires us to memorize this table for our differential equations exam. So I wonder if anyone has some deeper intuition or observation to help with this? For example, I noticed ...
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329 views

For this 2 by 2 locally linear system, how to determine that this “indeterminate” critical point is a centre? Boyce, p516, Question 9.3.12

$12.$ (a) Determine all critical points of $\dfrac{dx}{dt}=(1+x)\sin y$ , $\dfrac{dy}{dt}=1−x−\cos y$ . (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of ...
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List of IVP known to have periodic solutions

I am looking for a list or review article describing differential equations and corresponding initial conditions which result in periodic solutions.
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25 views

Picard's uniqueness theorem

The differential equation $$\frac{dy}{dt} =ay; y(0)=0$$ satisfies the condition of Picard's uniqueness theorem. But, I find that it can have multiple solution. How lipschitz condition helping to get a ...
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0answers
63 views

To be Lyapunov stable solution and don't be stable solution asymptotically

Find the parameters $ a, b \in R $ of equation $$ y''' + ay'' + by' + y = 0 $$ to the function $ y = e^{-x} $ be Lyapunov stable solution and don't be stable solution asymptotically? Am I correct? ...
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1answer
35 views

Well-posed ODEv definition

An ODE is well-posed if the solutions vary continuously over the initial conditions. What is the intuitive meaning/requirement for this ?
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136 views

Strong Lyapunov Function

By showing that $V(x_1,x_2) = (x_1)^2 + (x_2)^2$ is a strong lyapunov function for the system: $x_1’ = -x_2$ $x_2’ = x_1 + (x_2)^3 - x_2$ determine a region of ''attraction'' for the origin. I ...
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1answer
36 views

2 x 2 Phase Portrait for 2 x 2 Linear System with Real Coefficients. Boyce, p395, Figure 7.5.4a

The given general solution for some linear system is $ x= c_{2} \mathbf{ x^{(1)} }(t) + c_{2} \mathbf{ x^{(2)} }(t) = c_{1}\ \left(\begin{array}{l} 1\\ \sqrt{2} \end{array}\right)\ e^{-t}+ ...
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36 views

Solve $xy'=\frac{y^{3} +3yx^{3}}{7y^2+x^2}$ by using $z=\frac{y}{x}$ substitution.

How do I solve $$xy'=\frac{y^{3} +3yx^{3}}{7y^2+x^2}$$ by using $$z=\frac{y}{x}$$ substitution? EDIT: The DE was wrong, it is now right, my apologies. I checked with wolframalpha which gave me ...
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1answer
98 views

Application of operator theory in ODE and PDE

I am looking for references of applications of operator theory (especially spectral theory) in ODE, PDE and possibly SDE. I have learnt operator theory in the general set up, but only know little ...
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5answers
107 views

How to think when solving $3\frac{\partial f}{\partial x}+5\frac{\partial f}{\partial y}=0$?

Solve this differential equation $$3\frac{\partial f}{\partial x}+5\frac{\partial f}{\partial y}=0$$ Usually, when we get these problems, they tell us what variable change is smart to do and we just ...
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40 views

Newton's differential equation

As we all know one of Issac Newton's many achievement was to use his theory of gravitation and his law of motion to determine the way the planets move. I am looking for a not too deep resource in ...
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2answers
37 views

System of linear first order DEs

Question: $$3\dot{x} + \dot{y} + 5x - y = 2e^{-t}+4e^{-3t}$$ $$\dot{x} + 4\dot{y} - 2x + 7y = -3e^{-t}+5e^{-3t}$$ Subject to: $$x(0)=y(0)=0$$ Attempt at a solution: I have gotten to: ...
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178 views

Stability of equilibrium in diff EQ symbiotic growth model

Consider the following system, which is designed to model a symbiotic relationship between two species: $$\dot{x}(t) = x(\epsilon_1 -\alpha_1 x + \beta_1 y)\\ \dot{y}(t) = y (\epsilon_2 + ...
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second order linear equation (nonhomogeneous)

Find the general solution of the nonhomogeneous differential equation $$y'' + 4y =\sin(\alpha t).$$ For all possible real $\alpha$ decide which values of $\alpha$ gives a limited amount of ...
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52 views

Where can I find phase portrait problems involving only general coefficients with solutions and the phase portrait graphed in the answer? [closed]

I tried Googling “phase portrait problem” and “phase plane problem” and “predator prey problem”. Example. In chapter 9.4 (p 521), Boyce solves and provides phase portraits for two examples of ...
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1answer
475 views

Critcal values of coefficient matrix with parameter where the phrase portrait changes - Boyce, p410, Question 7.6.19

The coefficient matrix contains a parameter $\alpha$. Determine the eigenvalues in terms of $\alpha$. Find the critical value(s) of $\alpha$ where the qualitative nature of the phase portrait for the ...
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2 by 2 Linear Homogenous System with Complex Eigenvalue. Boyce, p409, Question 7.6.4

I don't know how to align multiple equations, hence I post this screenshot. If someone can show me how, thanks. I think understand everything above the red line, but please inform me about any ...
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1answer
72 views

number of points of tangency of the zero divergence vector field and the equator of the sphere.

Let $V$ be vector field on the sphere $S^2$ and $\operatorname{div} V=0$. What is the minimum number tangency points of this vector field and the equator of the sphere?
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Consider The Differential Equation $\frac{dy}{dx} = 3y^{\frac{2}{3}}, y(0)=0$ [duplicate]

Consider: $$\frac{\mathrm{d}y}{\mathrm{d}x} = 3y^{2/3}, \quad y(0)=0$$ does exist a solution to this initial value problem? how many solutions if exist? If the answer for the first question is ...
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1answer
159 views

Reduction of DEs to Bessel equation

A question in my textbook asks me to write down the general solution to: $\frac{d}{dx}(x^2\frac{dR(x)}{dx}) + [k^2x^2 - n(n+1)]R(x) = 0$ in terms of Bessel functions. Now two similar questions ...
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75 views

Brachistochrone Problem to find out the path by which a bead travels in least time

The question is to find the shape of the curve down whcih a bead sliding from rest and accelerated by gravity will slip(without friction) from one point to another in the least time. So I proceeded ...