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

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2answers
237 views

how did he conclude that?integral

So the question is : Find all continuous functions such that $\displaystyle \int_{0}^{x} f(t) \, dt= ((f(x)^2)+C$. Now in the solution, it starts with this, clearly $f^2$ is differentiable at every ...
1
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2answers
4k views

Solving differential equation $x^2y''-xy'+y=0, x>0$ with non-constant coefficients using characteristic equation?

Whenever you deal with non-constant coefficients you usually use Laplace transform to solve a given differential equation, at least that's how how I learned it. But how would you solve the equation ...
1
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1answer
185 views

Multistep ODE Solvers

Write both a fourth order Adams Bashforth and Adams Moulton procedure to solve $$x'(t) = x(t)-y(t)-\exp(t);$$ $$y'(t) = x(t)+y(t)+2\exp(t)$$ with initial values $x(0) = -1, y(0) =- 1$ on the ...
1
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1answer
71 views

Solutions and attraction regions of following odes?

Assume a mapping $X: \mathbb{R} \to \mathbb{R}^d$. We know that the solution to ode $$ d X_t = (\mu - X_t) dt $$ is $X_t = (X_0-\mu) e^{- t} + \mu$, which indicates that $X_t$ converges to $\mu$ as ...
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1answer
613 views

Existence and Uniqueness Theorem

I had a question about how to do one of these problems. So here's the question: Given this equation $y'=\frac{-\cos(t)y(t)}{(t+2)(t-1)}+t$, find if the initial conditions $y(0)=10, y(2)=-1, y(-10)=5$ ...
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2answers
758 views

How to apply reduction of order to find a 2nd linearly independent solution?

I have some questions about writing a general solution, $y$, for $y''-y=0$ when $y_1 = e^x$ is a known solution. I do not understand the logic of the method of reduction of order. How do we apply ...
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2answers
161 views

Use two solutions to a high order linear homogeneous differential equation with constant coefficients to say something about the order of the DE

OK, this one utterly baffles me. I am given two solutions to an nth-order homogeneous differential equation with constant coefficients. Using the solutions, I am supposed to put a restriction on n ...
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3answers
274 views

General Solution for a given system of equations

Find the general solution of this system of equations: $$x' = \pmatrix{-1&0&0\\1&0&-1\\1&1&0}x$$ I got the eigenvalues to be: $\lambda = -1,\pm i$ The eigenvectors ...
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2answers
470 views

Using Octave to solve systems of two non-linear ODEs

How to solve following system of ordinary differential equations using Octave? $$\frac{dx}{dt} = - [x(t)]^2 - x(t)y(t)$$ $$\frac{dy}{dt} = - [y(t)]^2 - x(t)y(t)$$ Update: initial conditions: ...
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2answers
3k views

Difference between improper node and proper node for phase portrait

Can someone offer an explanation for the difference between these two? I see pictures of what seem to be examples of both, but it's hard for me to discern what a new portrait would be. Any help? ...
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0answers
48 views

Show that a function is solution to differential equation

I have a homogenous differential equation $a_0 y'' + a_1 y' + a_2 y = 0$ and a function $y(t) = t e^{\lambda_0 t}$. First I am assuming that $\lambda_0$ is a root in the characteristic polynomial. ...
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1answer
77 views

How to go about solving this question on differentials?

A ring of a planet has an inner radius of approximately 52,000 km (measured from the center of the planet) and a radial width of 19 km. Use differentials to estimate the area of the ring. (Round ...
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1answer
30 views

Poincare-Bendixson theorem contradiction help

Lets suppose p is asymptotically stable but not a singularity for the planar differential equation dx/dt=f(x), then for points x sufficiently closed to p we must have x(t) tends to p. so the limit set ...
0
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2answers
135 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } ...
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1answer
44 views

Derivative of a differential equation help??

Please can someone explain this to me in detail: if $y''+4y'+3y=14\cos(2x)$ and $z'''+4z''+3z'=-28\sin(2x)$ show that the $z=y+c$ where $c$ is a constant I know the second is the integral of the first ...
0
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1answer
305 views

Finding the interval where a solution is certain to exist for the equation $y' + (\tan t)y = \sin t$

Given the following problem: Determine (without solving the problem) an interval in which the solution is certain to exist for the initial value problem $y' + (\tan t)y = \sin t, \space y(2\pi) = ...
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1answer
74 views

Find the solution for a boundary value problem

Please, how can we find the solution of this second order boundary value problem $$-(e^{-2x}u')'-\ln(x^2+2)u= 2 e^ {-2x} - x \ln(x^2+2),\,\, x\in ]0,1[, u(0)=0,u(1)=1?$$ Or more generally, What's the ...
0
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1answer
37 views

Solve a differential equation and evaluate the solution at a particular value of independent variable

If $\frac{dy(x)}{dx}=(2-3i)y(x)$ where $i=\sqrt{-1}$, what is the value of $y(\pi)$?
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2answers
75 views

Differential Equations/IVP: $\frac{dy}{dt} = 4 - y^3$ and $y(-1)=2$

Transform the given initial value problem into an equivalent problem with the initial point at the origin. $$\cfrac {dy}{dt} = 4 - y^3 \\ y(-1)=2$$ I have no idea about how to solve it. Could you ...
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1answer
291 views

Wronskian-Differential Equations

The equations below are matrices: Consider the vectors $y^{(1)} (t)$=$\begin{pmatrix}t \\1 \end{pmatrix}$ and $y^{(2)}$ (t)=$\begin{pmatrix}t^2 \\2t \end{pmatrix}$ (a) Compute the Wronskian of ...
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1answer
79 views

Let $f$ be a field with only one singularity in the origin. Show that the phase diagram of the field $f$ has exactly three distinct orbits

Let $f:\mathbb R\to \mathbb R $ be a field with only one singularity in the origin. Show that the phase diagram of the field $f$ has exactly three distinct orbits which are the following: I need ...
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2answers
576 views

differential equations in SIR epidemic model and obtain Ro

I need to know why the differential equation system that expresses epidemic's model SIR in some texts appears: $$\frac{dS}{dt} =-\beta\frac{S}{N}I$$ $$\frac{dI}{dt}= \beta \frac{S}{N}I - \gamma I$$ ...
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2answers
261 views

Solve separable DE with integrating factor and homogeneous substitution

I just came out of test which asked to solve $$\frac{dy}{dx}=\frac{y}{x}$$ with $x,y>0$ in three ways: by separating the variables, using the substitution $y=vx$ and using an integrating factor. ...
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3answers
12k views

What exactly is steady-state solution?

In solving differential equation, one encounters with steady-state solution. My textbook says that steady-state solution is the limit of solutions of (ordinary) differential equations when $t ...
0
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2answers
420 views

Find a particular solution of the differential equation $-3y''-2y'+y=3xe^x$

Using the method of undetermined coefficients. Guess $(Ax+B)e^x$ Plug into diff eq: $-3[(Ax+B)e^x]'' - 2[(Ax+B)e^x]' + (Ax+B)e^x = 3xe^x$ Wolfram alpha simplifies this to: $A(x-2)=e^x(4B+3x)$. ...
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1answer
60 views

Initial values are lost (diff eq to Transfer function)?

I read eternal Julius O. Smith III and he says that $$x_{n-m} = z^{-m}X(z)$$ Particularly, difference relation $$y_{n} = y_{n-1} + x_{n}$$ is solved by by $$Y = z^{-1}Y + X = {X \over ...
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1answer
260 views

$\frac{dy}{dx} = 3y^{2/3}$ general solution?

What's the general solution of $\frac{dy}{dx} = 3y^{2/3}$ ? Im pretty sure this is a separable equation, but I'm not sure how to go forward? Just multiply by $dx$ and $\frac{1}{3y^{2/3}}$ well then I ...
21
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8answers
2k views

What's so special about sine? (Concerning $y'' = -y$)

In an attempt to actually grok sine, I came across the $y''= -y$ definition. This is incredibly cool, but it leads me to a whole new series of questions. Sine seems pretty prevalent ...
11
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1answer
477 views

Osgood condition

Let $h$ and $g$ be continuous, non-decreasing and concave functions in the interval $[0,\infty)$ with $h(0)=g(0)=0$ and $h(x)>0$ and $g(x)>0$ for $x>0$ such that both satisfy the Osgood ...
14
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3answers
1k views

Solving Differential Functional Equation $f(2x)=2f(x)f'(x)$

Find all functions satisfying $f(2x)=2f'(x)f(x)$ Under given condition, can't we find explicit solutions?
12
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4answers
370 views

Why solutions of $y''+(w^2+b(t))y=0$ behave like solutions of $y''+w^2y=0$ at infinity

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^\infty |b(t)| dt <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{t\to\infty} ...
15
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1answer
495 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2 = \pm \sqrt{-A \cos(x) - B \cos^2(x)+y'(x)-C},$$ where $C, A$ and $B $ are parameters. (The case that either ...
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1answer
621 views

What is, how do you use, and why do you use differentials? What are their practical uses?

What is a differential? And how is it useful? What is its practical use? For example, in Electromagnetic Wave Theory as it pertains to diffraction gratings, we have an equation like this one: ...
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1answer
650 views

Eigenfunctions of the Laplacian

I am willing to offer a bounty for this one, so I will give you an exact idea of what I need: I am looking for solutions of $$\Delta \Psi(r,\theta)=k^2\Psi(r,\theta)$$ where $k\in \mathbb{R}$. Such ...
3
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3answers
375 views

Numerical Analysis References

Could anyone suggest any good (perhaps online ref papers) reference material on numerical analysis focusing on determining accuracy/estimated errors, rates/orders of convergence especially when ...
12
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1answer
451 views

Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
12
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3answers
639 views

When do the Freshman's dream product and quotient rules for differentiation hold?

This is motivated by looking at the calculus exams of some of my undergraduate students. A recurring mistake is assuming that the derivative of the product of functions is a product of derivatives and ...
9
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0answers
485 views

Complicated exercise on ODE

I have this exercise extracted from a examination of qualitative theory of ODE (in which we study the existence and uniqueness of solutions, and stability using the function of Lyapunov) I don't know ...
7
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1answer
184 views

Properties of the solutions to $x'=t-x^2$

Let $f_c$ be the solution to $$ \left\{ \begin{array}{c} x'=t-x^2 \\ x(0) =c \end{array} \right. $$ I'm trying to prove: If $c \geq 0$ then $f_c(t)$ is defined for all $t>0$ There is a ...
7
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1answer
105 views

dropping a particle into a vector field, part 2

Okay, so earlier I posted this question "dropping a particle into a vector field " as sort of a feeler question as i study line integrals in order to go into surface integrals and eventually ...
7
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4answers
952 views

Methods to solve differential equations

We are given the equation $$\frac{1}{f(x)} \cdot \frac{d\left(f(x)\right)}{dx} = x^3.$$ To solve it, "multiply by $dx$" and integrate: $\frac{x^4}{4} + C = \ln \left( f(x) \right)$ But $dx$ is not a ...
5
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0answers
104 views

Solve PDE by getting two ODEs

My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}_{\ge 0}\rightarrow \mathbb{C}$ $$ \partial_t f(x,t) = -\partial_x^2 f(x,t) + g(t)V(x)f(x,t).$$ I would consider this PDE to be solved if ...
4
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3answers
188 views

How $\frac{dx}{dy}=f(x)g(y) \Leftrightarrow \int \frac{dx}{f(x)} = \int g(y)dy$?

In my intro differential equations class we have often used the "equivalence" stated in title. It seems to me that somehow, the intermediate step $$ \frac{dx}{f(x)} = g(y)dy$$ is being used, in which ...
3
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2answers
111 views

Using the Jordan form Complex

Let $C$ be a complex $n \times n$ matrix with $\det C \neq 0$. Prove that there is a complex matrix $B$ such that $C = e^B$ Hint: use the Jordan form matrices for comlexas
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1answer
343 views

Cancelling differentials

I'll start with an example. In physics, $x(t)$ represents the $x$-position of a particle, and $v(t)$ its ($x$-)velocity. To determine the total displacement of a particle on the interval $[a, b]$, we ...
12
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4answers
1k views

Examples of nonlinear ordinary differential equations with elementary solutions.

I am looking for nice examples of nonlinear ordinary differential equations that have simple solutions in terms of elementary functions. (But are not trivial to find, like, for example, with ...
11
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4answers
351 views

How to prove that $\frac{d^n}{dx^n}(x^2-1)^n=0$ has $n$ real roots?

How do I prove that $$\frac{d^n}{dx^n}(x^2-1)^n=0$$ has $n$ real roots?
7
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2answers
351 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem. Let $f(x):\Omega\longrightarrow \mathbb R^n$ be a $C^2$ function where $\Omega$ is an open subset of ...
6
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2answers
281 views

Proving Nonhomogeneous ODE is Bounded

I am trying to prove the following: Let $x(t)$ be a solution of the IVP $$ \dot x=A(t)x+h(t), $$ where $A(t), h(t)$ continuous on $1\le t<\infty$. Further assume that $$ \int_1^\infty \| ...
6
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1answer
191 views

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...