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

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3
votes
2answers
121 views

Global stable and unstable manifolds of quadratic 2D differential system

Show that $x^* = (1, 2)$ is a fixed point of the system $x_1' = 2 + 3x_1 − 2x_2 − x_1^2 + 2x_1x_2 − x_2^2$ $x_2' = 3 + 4x_1 − 3x_2 − x_1^2 + 2x_1x_2 − x_2^2$ Determine $W^s(x)$ and $W^u(x)$, the ...
2
votes
2answers
364 views

Finding a Lyapunov function for the differential system $x_1'=-8x_1^3-x_2$, $x_2'=-4x_2-4x_1^3$

I've got the following system of equations: $$ x_1'=-8x_1^3-x_2 \qquad x_2'=-4x_2-4x_1^3 $$ I'm trying to check, if the equilibrium point in $(0,0)$ is stable or not. I am supposed to find so called ...
1
vote
1answer
72 views

Slowing down a nonlinear differential system to compute its asymptotics

How do we solve following system of differential equations. $$x'(t)=- \frac{x}{2}+\frac{x}{2}[\lambda y-\frac{1}{2}(1+\lambda)x+1-x-y]$$ $$y'(t)= \frac{x}{2}[-\lambda y+\frac{1}{2}x]$$ $$x(0)= \...
2
votes
1answer
124 views

Limits of integration of solution $u$ of $-u''=f$ with $u(0)=u(1)=0$

Suppose I have to solve the following boundary value problem: $$-u''=f\quad\text{and}\quad u(0)=0,\ u(1)=0.\qquad \mbox{Where $f \colon [0,1] \to \mathbb{R}$.}$$ In the solution to the exercise, ...
0
votes
0answers
20 views

Tangent curve of a vector field

Given vector field $X=f(x,y)\frac{\partial}{\partial x}+g(x,y)\frac{\partial}{\partial y} $. We have $\frac{f}{g}(x,y)$ constant along the lines through origin. Hence $\frac{f}{g}(x,y)=k(y/x)$. How ...
0
votes
1answer
31 views

Robertson's system of ODEs

I have just one problem concerning this system of ODE's. The plot shows us the concentrations of A, B and C. Doesn't the mass of the system change? When looking at the graph the green curve goes up ...
0
votes
1answer
34 views

Solving an Exact Differential Equation

I have the following d.e. that is exact. $\cos \theta dr - (r\sin\theta - e^\theta)d\theta = 0$ which is the same as: $\cos \theta dr + (-r\sin\theta + e^\theta)d\theta = 0$ I am trying to solve ...
0
votes
3answers
42 views

Converting a differential equation

Consider an ODE $\frac{dy}{dx}=h(x,y)$ such that $h(rx,ry)=h(x,y)$, which implies $h(x,y)=k\left(\frac{x}{y}\right)$. (Why?) Show that this ODE can be changed to a separable ODE for $u=u(x)$, if ...
0
votes
1answer
30 views

Solving a simple Ordinary, first order differential equation

I have been having a problem with this simple equation. It is asking me this: Find all values of $k$ for which the function $y=\sin(kt)$ satisfies the differential equation $y′′+ 7y = 0$. I have ...
0
votes
0answers
60 views

Recommendations for PDE Problem Book

Currently, I began self-studying partial differential equations as a prerequisite for quantum mechanics and differential geometry. My background covers multivariable calculus, ODE, and basic linear ...
1
vote
1answer
100 views

Closure Under Addition And Scalar Multiplication

I'm stuck on my homework on "Definition of A Vector Space" in my Differential Equations and Linear Algebra class. The section of problems I'm on ask you to determine whether or not each problem is ...
0
votes
3answers
270 views

Using implicit differentiation on trigonometric fraction and logarithm

I am supposed to solve the following using implicit differentiation. $$\sin\left(\frac{x}{y}\right)+\ln(y)=xy$$ This is what I have so far: $$\cos\left(\frac{x}{y}\right)\frac{d}{dx}\left(\frac{x}{...
0
votes
2answers
27 views

Need help solving first order nonlinear ODE

I am having problems with this ODE: (2lny - ln(y)^2)dy = ydx - xdy Doesn't seem to be separable and I don't know what kind change of variables could be done
0
votes
0answers
28 views

Is this map defined from an ODE a contraction?

Let $f:\mathbb{R}^+\to \mathbb{R}$ be a $T$ periodic function, i.e. $f(t+T) = f(t)$ for all $t\in \mathbb{R}^+$. Let $\sigma\in \mathbb{R}^+$. Define a weighted $\mathcal{l}_2$ norm, $\left\| \cdot \...
0
votes
0answers
32 views

Find the kernel

Find the kernel of each of the following transformations on C^∞: a. The third derivative D^3, that is, D^3(f)=f′′′ b. The linear operator D-2I, that is, (D-2I)(f)=f′−2f c. The linear operator D^2=(...
1
vote
1answer
27 views

Autonomous System: Finding all orbits connected to a fixed point

Given the autonomous system $$ \begin{pmatrix} \dot{x}\\ \dot{y} \end{pmatrix} = \begin{pmatrix} y\\ |x| \end{pmatrix} $$ I found that all orbits in the left halfplane are circular $$ \frac{d}{dt}\| (...
0
votes
0answers
21 views

normal forms of vector fields reference

Does anyone can give reference or a good book for the technique of calculating normal forms of vector fields under the context of ODE system i.e for equations of the type: $$ \dot{X}=F(X) $$ where $F$...
0
votes
1answer
62 views

Nonlinear differential equation with singularity: numerical solution?

I want to numerically solve the following problem over some interval of time: \begin{eqnarray*} \dot{\theta}_1 & = & \theta_2\\ \dot{\theta}_2 & = & \frac{g - l\theta_2^2\cos\theta_1}{...
2
votes
2answers
88 views

Euler-Lagrange Equation has no solution?

I've been asked to compute the Euler-Lagrange equation and second variation of the functional $$I[y]=\int_{a}^{b}(y'^2+y^4)dx$$ with boundary conditions $y(a)=\alpha$, $y(b)=\beta$. It's easy to see ...
0
votes
0answers
17 views

Writing explicit scheme - numerical methods for PDEs

I've got a problem writing a code for the explicit scheme. In the end i'd like to plot a temperature destribution for a given case. Let's say i have a material of a lenght L with given boundary and ...
2
votes
2answers
44 views

Inhomogeneous system of differential equations

To my grief this week I had a very incoherent class about differential equations and I find myself unable to solve Problem: $$\begin{cases} x'(t)=2x(t)-y(t)+4t \\ y'(t)=x(t)+e^{-t} \end{cases} $$ ...
0
votes
1answer
97 views

Using fourth order Runge-Kutta to solve a second order ode in MATLAB

For a ball of mass $m$, radius $r$, falling freely through air of density $p$, the force of air resistance is proportional to the density of the air, the square of the ball's speed, $v$, and the area ...
0
votes
1answer
21 views

How do I interpret these equations of motion in the presence of air resistance?

For a ball of mass $m$, radius $r$, falling freely through air of density $p$, the force of air resistance is proportional to the density of the air, the square of the ball's speed, $v$, and the area ...
0
votes
1answer
66 views

How I can solve this differential equation use implicit function theorem? [closed]

I have to find the solution of the next differential equation: $$x'(t)=\frac{-\frac{\partial F(t,x(t))}{\partial t}}{\frac{\partial F(t,x(t))}{\partial x}}$$ I need help I get all mixed up with this ...
2
votes
1answer
47 views

Harmonic Oscillators: Differential Equations

The book being used for this course is Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris W. Hirsch. The question is as follows. Suppose there are two masses $m_1$ ...
4
votes
1answer
310 views

Bounds on norm of matrix exponentials

Given $A = n\times n$ matrix with the real parts of its eigenvalues are contained in $[\alpha, \beta]$ where $-\infty < \alpha \leq \beta <\infty$. For any $\epsilon > 0$ and any norm $||.||$ ...
0
votes
0answers
29 views

Solving strange differential equations?

Whilst my knowledge of differential equations is somewhat limited, I was under the impression that the following was a valid equation to be solved yet it is unrecognised by wolfram alpha and I have no ...
0
votes
0answers
24 views

How to show the order equality of modified Bessel function?

The modified Bessel function of order n of the first kind is given by $$I_n(x)=\sum_{m=0}^{\infty}\frac{(\frac{1}{2}x)^{2m+n}}{m!\Gamma(m+n+1)}$$ where $\Gamma$ is defined by an improper integral, $...
1
vote
1answer
178 views

System of differential equation with variable coefficent

How to solve this system of differential equations $x'(t)=\frac{a+s}{(1-t)d}x(t)-\frac{b}{(1-t)d}y(t)$ and $y'(t)=\frac{a}{(1-t)d}x(t)-\frac{(s+b+(1-t)c)}{(1-t)d}y(t)$ where a,b,c,d and s are ...
1
vote
0answers
66 views

Population growth model word problem

The question: There are two non competing populations with birth rates, death rates, and starting populations as follows. $B1(t)=\frac{1}{t+1}; D1(t)=\frac{1}{10}; P1=2; B2(t)=\frac{2}{t+1}; D2(t)=\...
0
votes
0answers
18 views

Show a solution to a differential equation

I have to show that $f(t)$ is a solution to $f'''+(\alpha +\beta )f''+(1+\alpha \beta +g_2)f'+g_1f=z'''+(\alpha +\beta )z''+(1+\alpha \beta )z'$ I have deduced that $u(t)=x'''+x''(\alpha +\beta )+x'...
2
votes
1answer
36 views

Ordinary Differential equation-integrating factor [closed]

Show that the differential equation $(3y^2-x)+2y(y^2-3)y'=0$ admits an integrating factor which is a function of $(x+y^2)$. Hence solve the equation.
0
votes
1answer
50 views

Solving the matrix differential equation $\dot \Delta P(t) = (A + P(t)C^{T}R^{-1}C)\Delta P(t) + \Delta P(t)(A^{T} + C^{T}R^{-1}CP(t))$

Here $P, \Delta P \in \mathbb{R}^{N X N}$ The initial condition $\Delta P(0)$ is given and the dynamics of $P(t)$ is known. $ A,C,Q,R$ are constant matrices of compatible dimensions. Since it is a ...
4
votes
2answers
245 views

$(\partial_{tt}-\nabla^2+\partial_t)f=g,\quad (\partial_t-\nabla^2+b)g=\partial_t f$

Hi I am looking for complete solutions for $f(r,t),g(r,t)$ given in the coupled linear partial differential equations below: $$ (\partial_{tt}-a\nabla^2+b\partial_t)f(r,t)=bg(r,t) $$ $$ (\partial_t-c \...
0
votes
1answer
42 views

How to show this Ricatti equation can be changed into a linear equation?

Problem: The equation $$ \frac{dy}{dx} = A(x) y^2 + B(x) y + C(x) $$ is called a Ricatti equation. Suppose a partical solution is given as $y_1 (x)$. Show that the substitution $$ y = y_1 + \frac{1}{v}...
12
votes
2answers
15k views

Explanation and Proof of the fourth order Runge-Kutta method

Runge-Kutte 4th order method is a numerical technique used to solve ordinary differential equation of the form $dy/dx=f(x,y), y(0)=y_0$ It gives $y_{i+1}$ in the form $y_{i+1} = y_i+(a_1k_1+a_2k_2+...
3
votes
3answers
114 views

Given $f(x+y)=f(x)f(y), f'(0)=11,f(3)=3$, what is $f'(3)$?

The question is this: Given \begin{align} f(x+y)&=f(x)f(y)\\ f'(0)&=11\\ f(3)&=3 \end{align} What is $f'(3)$? And my solution: On differentiating the equation $f(x+y)=f(x)f(y)$ ...
0
votes
2answers
43 views

Is it possible to find general solutions for $n$-th order Euler-Cauchy ODE?

Consider $n$-th order Euler-Cauchy equation: \begin{equation} a_nt^n\frac{d^nx}{dt^n}+a_{n-1}t^{n-1}\frac{d^{n-1}x}{dt^{n-1}}+\cdots+a_{1}t\frac{dx}{dt}+a_0x=0,\quad a_0\ne0,a_{1},\cdots a_{n}\in\...
0
votes
1answer
81 views

Autonomous or non-autonomous control system?

The control problem concerns the field of economics but my problem is purely about mathematics. I hope my question fits for MathematicsSE. I have a following system in which $c$ is a control and $k$ ...
-4
votes
2answers
104 views

How can solve$ y''-y=0$? [closed]

I know the answer $y=C_1e^x+C_2e^{-x}$ But how can I get the answer? Please help me to solve $y''-y=0$
1
vote
0answers
65 views

System of differential equations - disagreement with paper

I have the following system of differential equations: $$A'-\frac{m}{r}A=(\epsilon-1)B \tag{1}$$ and $$B'+\frac{m+1}{r}B=-(\epsilon+1)A \tag{2}$$ where $A$ and $B$ are functions of $r$ and $A'$ and $...
0
votes
1answer
34 views

Equilibrium of an ordinary differential equation [closed]

I have been trying to show that the following ODE has either one or three non-zero equilibria, depending on the value of $r$, but with no real progress. $$\frac {dx}{dt}=rx(1-\frac{x}{K})-\frac{bx^2}{...
0
votes
0answers
30 views

Particular solution for $y''' (t) + 6y'' (t) + 14y' (t) = 4e^{t}$?

To find the general solution for $y''' (t) + 6y'' (t) + 14y' (t) = 4e^{t}$ the homogeneous solution can be found first, see: But is there a reason of choosing a particular solution in the form ...
3
votes
1answer
30 views

$f(\textbf{x}) = {a\over{|\textbf{x}|^{n-2}}} + b,\text{ }\textbf{x} \neq 0$

If $f(\textbf{x}) = g(r)$, $r = |\textbf{x}|$, and $n \ge 3$, show that$$\nabla^2 f = {{\partial^2 f}\over{\partial x_1^2}} + \dots + {{\partial^2f}\over{\partial x_n^2}} = {{n-1}\over{r}}g'(r) + g''(...
0
votes
1answer
72 views

Existence and uniqueness non autonomous ODE.

For the first order, non autonomous ODE $ \frac{dx}{dt} = \frac{x}{t} $ I have been asked to find the general solution, and then conclude on the existence and uniqueness of solutions satisfying the ...
2
votes
1answer
26 views

Why is this a solution to the ODE system

Given a system of differential equations $\dot{x}(t)=Ax(t)+Bu$ where $x,B \in \mathbb R^n$, $A \in Mat_{n\times n}(\mathbb R)$ and $u \in \mathbb R$, we want to find a solution. I was given a ...
2
votes
0answers
41 views

How to solve this differential equation with Fourier Transform?

Consider the differential equation $$\dfrac{\partial w}{\partial t} = -\alpha \dfrac{\partial w}{\partial x} + D \dfrac{\partial ^2 w}{\partial x^2}$$ together with the boundary conditions that $$\...
0
votes
1answer
134 views

Solving forced undamped vibration using Laplace transforms

I'm heaving trouble solving the following undamped forced vibration problem using Laplace transforms: $$\ddot{q}(t) + \omega_n^2 q(t) = \cos(\omega t).$$ I will show what I have done so far, and I'd ...
0
votes
1answer
34 views

characteristic equation - complex coefficients

I'm looking at a differential equation given by: $$a_0\frac{d^ny}{dt^n}+a_1\frac{d^{n-1}y}{dt^{n-1}}+\cdots+a_{n-1}\frac{dy}{dt}+a_ny$$ Where the constants are allowed to be complex. When I look in ...
0
votes
2answers
346 views

Differential Equations: Newton's Second Law and Hooke's Law

First off I wasn't sure if I am just stating that I am posting this question here because it appears in my differential equations course. (Although it seems rather like a physics question). Question ...