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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2answers
159 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 ...
1
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3answers
270 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 ...
1
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2answers
459 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: ...
1
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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? ...
0
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0answers
46 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. ...
0
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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 ...
0
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1answer
28 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
128 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{ , } ...
0
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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
297 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) = ...
0
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1answer
73 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
votes
1answer
36 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 $y(\pi)$?
0
votes
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 ...
0
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1answer
283 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 ...
0
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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 ...
0
votes
2answers
257 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. ...
0
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2answers
415 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: $A(x-2)=e^x(4B+3x)$. ...
-1
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1answer
59 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 ...
-3
votes
1answer
48 views

Find an integral factor of $3(y+1)dx-2(x-1)dy=0$ [closed]

What relationship exists between the $h$ and $p$ such that: $$(x-1)^h(y+1)^p$$ is an integral factor of $$3(y+1)dx-2(x-1)dy=0$$
-5
votes
1answer
259 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
votes
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
475 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
votes
3answers
993 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?
15
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1answer
488 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 ...
10
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1answer
619 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: ...
8
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1answer
644 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
votes
3answers
357 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
446 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 ...
11
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3answers
635 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 ...
7
votes
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
103 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
votes
4answers
937 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
votes
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
186 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
votes
2answers
109 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
1
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1answer
331 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
votes
4answers
343 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?
8
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0answers
469 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
votes
2answers
348 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
275 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
votes
2answers
312 views

Determinant called Grammian

Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if $$ \det\left( \begin{array}{ccccc} f_1 & f_2 & ...
6
votes
3answers
215 views

Solutions of homogeneous linear differential equations are a special case of structure theorem for f.g. modules over a PID

In this other post, Qiaochu Yuan comments that the solutions for the homegeneous linear differential equation with constant coefficients are a special case of the structure theorem for finitely ...
5
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1answer
188 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 ...
5
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2answers
325 views

The Green’s function of the boundary value problem

What is the Green’s function of the boundary value problem $$ \frac{\mathrm d^2 y}{\mathrm d x^2}-\frac{1}{x}\frac{\mathrm dy}{\mathrm dx}=1,\quad y(0)=y(1)=0, $$ this boundary problem is not self ...
4
votes
6answers
977 views

ODE introduction textbook

Unfortunately I have reached the maximum number of math classes I can take for my undergraduate degree. I still wish to study basic ODEs and basic number theory. What is a good textbook with an ...
3
votes
1answer
31 views

Uniqueness of solutions to linear recurrence relations

I understand that if I have a linear homogeneous recurrence relation of the form $q_n = c_1 q_{n-1} + c_2 q_{n-2} + \cdots + c_d q_{n-d}$, I can construct the characteristic polynomial $p(t) = t^d - ...
3
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1answer
108 views

Lipschitz continuity and differential equations

Does anyone have any ideas for this one? could use some help.
2
votes
2answers
372 views

How to solve this ODE?

For a certain problem, I am trying to solve the ODE $$\ddot{z}(t) - \omega^2 z(t) = f_0 \Big(e^{-i(\omega+\delta)t}+e^{-i(\omega-\delta)t}\Big)$$ where $\omega$ is a real and $\delta$ is close to ...
0
votes
1answer
102 views

Questions about the hyperbolic system of equations

$$u_t+A(x,t,u)u_x=b(x,t,u) \tag 1$$ $$u=(u_1, \dots, u_n), b=(b_1, \dots, b_n)$$ $$A=[a_{ij}], i,j = 1, \dots, n$$ $$$$ We set the question if there are characteristic directions at the path of which ...