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

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6
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5answers
628 views

Finding a non constant solution to $ (x')^2+x^2=9 $

How do I find a non-constant solution this equation? I've tried to solve for $x$, but the final answer should be in the form of $x(t)=...$ $ (x')^2+x^2=9 $ I'm not sure where to start.
6
votes
2answers
326 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
761 views

Solving $-u''(x) = \delta(x)$

A question asks us to solve the differential equation $-u''(x) = \delta(x)$ with boundary conditions $u(-2) = 0$ and $u(3) = 0$ where $\delta(x)$ is the Dirac delta function. But inside ...
6
votes
5answers
188 views

Check $\;y=\dfrac{\sin x}{x}\;$ is solution of $\;xy'+y=\cos x\;$

How can I check that $\;y=\dfrac{\sin x}{x}\;$ is a solution of $\;xy'+y=\cos x\;$?
6
votes
3answers
425 views

Common term for differential equations and recurrence relations

Recently I have been working with recurrence relations (mostly linear), and systems of coupled recurrence relations. I have noticed a lot of common ground with differential equations. In a way, you ...
6
votes
3answers
3k views

Differential Equations without Analytical Solutions

In many talks, I have heard people say that the differential equation they are interested in has no analytical solution. Do they really mean that? That is: Can you prove a differential equation ...
6
votes
3answers
326 views

Why do we chose exponential function as a trial solution for second order linear differential equation with constant coefficient?

Why do we chose exponential function as a trial solution for second order linear differential equation with constant coefficient ? Can any other function be taken as a trial solution ?
6
votes
4answers
524 views

How to solve $y''' - y = 2\sin(x)$

$$y''' - y = 2\sin(x)$$ I'm doing differential equations and know pretty much all methods of solving them, but I haven't come across anything of a higher order than second yet. How do I go about ...
6
votes
2answers
2k views

Can a differential equation have non unique solutions?

There are theorems of existence and uniqueness of differential equations. I was wondering if it is possible that a differential equations has a solution but it is not unique.
6
votes
1answer
191 views

Why is the numerical solution of this equation unstable? Is this equation stiff?

I am trying to solve the following equation with an explicit fourth-order using the Runge-Kutta method: $$y' = t(y - t \sin t)$$ with initial conditions $y(0) = 1$ over the interval $[0, 10]$. The ...
6
votes
3answers
481 views

Solve $\frac{dx}{dt} = x^3 + x$ for $x$

This is a seemingly simple first order separable differential equation that I'm getting stuck on. This is what I have so far: $$\frac{dx}{dt} = x^3+x$$ goes to $$\frac{dx}{x(1+x^2)} = dt$$ Now ...
6
votes
3answers
108 views

Why don't the roots of this characteristic equation correspond to the given solution of this 2nd order ODE?

I am asked to solve $$ y'' + 9y = 6\mathrm{sin}(3x) $$ using the method of undetermined coefficients. The characteristic equation of this 2nd order ODE is $$ \lambda^2 + 9\lambda = 0 $$ and its ...
6
votes
3answers
246 views

What are other solutions to this differential equation, “similar” to $\sin x$ and $e^x$?

I've been studying electronics, where they make great use of the relationship between the sine and exponential functions ($e^{i \omega t} = \cos{\omega t} + i \sin \omega t)$. This relationship is ...
6
votes
3answers
3k views

Integrating with respect to different variables

I have started reading a book on differential equations and it says something like: $$\frac{dx}{x} = k \, dt$$ Integrating both sides gives $$\log x = kt + c$$ How is it that I can ...
6
votes
3answers
328 views

two identical point charges can't collide

I've convinced myself intuitively that if you place two massless classical particles with the same charge in $\mathbb{R}^n$, with arbitrary initial velocities and (distinct) positions, they will never ...
6
votes
2answers
194 views

Power series $x f''(x) + f'(x) + xf(x) = 0$

Find a power series with radius of convergence $R = \infty$ such that $f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$ satisfies $x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R$. How should ...
6
votes
3answers
683 views

Square root of differential operator

If $D_x$ is the differential operator. eg. $D_x x^3=3 x^2$. How can I find out what the operator $Q_x=(1+(k D_x)^2)^{(-1/2)}$ does to a (differentiable) function $f(x)$? ($k$ is a real number) For ...
6
votes
5answers
1k views

Modelling forces acting on a sail

I'd like to create a basic model of the forces acting on a sail (wind sail, like a tail ship) A couple of things I was thinking about: 1) can create a very simple model where wind is 'one' force ...
6
votes
4answers
479 views

Derivative of square of derivative?

I was trying to solve this differential equation: $$2yy'' + 3y'^2 = 4y^2 $$ And I found this way to solver it: http://eqworld.ipmnet.ru/en/solutions/ode/ode0344.pdf but I don't understand why $w'_y ...
6
votes
3answers
218 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 ...
6
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1answer
170 views

Finite dimensional spaces

What are the finite-dimensional spaces $W$ of differentiable functions with this property: If $f$ is in $W$, then $\frac{df}{dx}$ is in $W$.
6
votes
2answers
189 views

Estimating rate of blow up of an ODE

Suppose I have a differential equation $x'=f(x)$ and $f(x)>0$ grows super-linearly. I.e., $\lim_{|x| \rightarrow \infty} |f(x)|/|x| \rightarrow \infty$. Several related questions: (1) Can I ...
6
votes
1answer
836 views

Infinitely many solutions to an ordinary differential equation

Show that the first order differential equation $y'(x)=\sqrt{|y(x)|}$ with intial value $y(1/2)= 1/16$ has infinitely many solutions on the interval [−1, 1]. My thought were to show that this ...
6
votes
3answers
300 views

Help understanding $e^{it}=\cos t+i\sin t$ by way of matrices and vector fields

I was brushing up on my complex arithmetic in preparation for a class in ODE's this semester and I found myself looking at Exercise 2.7.5 in Introduction to Complex Analysis for Engineers by Michael ...
6
votes
2answers
835 views

Why is this constant of integration taken as $\log A$ instead of just $C$?

Suppose we solve $$\frac{dy}{dx} = \frac{1 + y}{2 + x} .$$ Which can be written as the following and integrating both sides w.r.t. $y$ and $x$: $$\int\frac{1}{1 + y}dy = \int\frac{1}{2 +x}dx ,$$ we ...
6
votes
1answer
4k views

What's the difference between explicit and implicit Runge-Kutta methods?

I have been working on numerical analysis, just as a hobby. I am only aware of the basic fourth order Runge-Kutta method in order to solve problems. When I was digging deep into it, I found there are ...
6
votes
2answers
140 views

Solving a matrix as a differential equation

If I have a matrix $$A = \begin{bmatrix}5 & 4 & -6\\-2 & -1 & 2\\2 & 0 & -3\end{bmatrix}$$ how do I solve $x'=Ax$ as a differential equation? My text book explains this in a ...
6
votes
2answers
2k views

Exponential of the differential operator

I am not sure whether this question is even well-posed. But today I learnt that $e^Df(x) = f(x+1)$ where $D$ is differential operator and $$e^D \triangleq \sum_{i=0}^{\infty} \frac{D^i}{i!}.$$ (ref. ...
6
votes
1answer
390 views

Sloppy notation for differential equations

Why does one often use the following notation for differential equation: $$ y'=f(t)y$$ (this is just a particular example) ? What bothers me with this notation, which I have encountered in ...
6
votes
2answers
251 views

Inverse Laplace Transform help

Is the information below correct? Find the inverse Laplace transform of $$ F(s) = \frac{s}{s^2 + 4s + 13}$$ Soln: a) Complete the squares to simplify our denominator $$ s^2 + 4s + 13 = (s+2)^2 + 9 ...
6
votes
2answers
122 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 ...
6
votes
1answer
62 views

Differential inequality implies inequality for points at distance pi.

Given a function $f$ with $f+f''\ge 0$, show that $f(x)+f(x+\pi) \ge 0$ for all $x$. Note that for sine and cosine both inequalities become equations. It seems reasonable to look at $f+f''=g$, but ...
6
votes
2answers
295 views

Transition time in a Lotka-Volterra system

I am working with a set of real-valued ordinary differential equations based on the Lotka-Volterra competition equations: $$\begin{align} \dot{a_1} & = a_1 \left( 1 - a_1 - 2 a_2 \right) \\ ...
6
votes
2answers
173 views

Initial Value Problem with Repeated Eigenvalues

Given the matrix $$ A=\left(\begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & 1\end{array}\right) $$ For $X'= AX.\quad$ $X\left(0\right)=\left(\begin{array}{r}1 \\ 0 \\ ...
6
votes
2answers
2k views

Question regarding usage of absolute value within natural log in solution of differential equation

The problem from the book. $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6 -y$ I understand the solution till this part. $\ln \vert 6 - y \vert = x + C$ The solution in the book is $6 - Ce^{-x}$ ...
6
votes
2answers
609 views

Show that the equation $y'=f(x,y)$ has a periodic solution

Given information: Let $f$ be a continuous function defined for all $(x,y) \in \mathbb{R}^2$. Let $f$ also satisfy a Lipschitz condition with respect to $y$. Let $f$ be periodic with respect to $x$ ...
6
votes
3answers
279 views

Solving $\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$

I recently came across this equation : $$\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$$where $f \in \mathcal{C}^1(\mathbb{R}, \mathbb{R})$. I've done the following, but I'm stuck at ...
6
votes
1answer
77 views

Nonlinear 1st order ODE

$$y'(x)=\frac{\cos (y(x))+y(x) \cos (x)}{x \sin (y(x))-\sin (x)}$$ Did I enter LaTeX correctly? I am self-learning the differential equation from a textbook and I need some help with above equation. ...
6
votes
1answer
144 views

How to solve the non-linear differential equation $y''=x-y^2$?

$y''(x)=x-y^2(x)$ I'm particularly interested in solutions when $x>0$. I've performed asymptotic analysis and reached the conclusion that solutions must behave as $\pm\sqrt{x}$ when $x\rightarrow ...
6
votes
1answer
152 views

Rank of a jet bundle of a vector bundle.

I am trying to understand the jet bundles but currently I am stuck on the following questions: Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) ...
6
votes
2answers
103 views

Geometric series of an operator

In solving a first order linear differential equation $(1-D)y=x^2$ where $D\equiv \frac{d}{dx}$ the way I learnt was that we proceed as ...
6
votes
2answers
413 views

Solving $(y+x^4y^2)dx+xdy=0$

How to solve (in terms of $y$) $(y+x^4y^2)dx+xdy=0$. I know I'm supposed to multiply by an integrating factor to turn this equation into an exact equation. In the previous exercise I proved that in ...
6
votes
1answer
207 views

The tangent half-angle substitution in differential equations

\begin{align} y & = \tan\frac\theta2 \\[8pt] \frac{1-y^2}{1+y^2} & = \cos\theta \\[8pt] \frac{2y}{1+y^2} & = \sin\theta \\[8pt] \frac{2\,dy}{1+y^2} & = d\theta \end{align} This, the ...
6
votes
1answer
224 views

numerical integration of equations of motion of large system of particles with lubrication forces

I have a large system of solid particles moving in the liquid. I use traditional Newtonian equations of motion for the particles. There are many different interaction forces between particles and the ...
6
votes
2answers
2k views

Finding integrating factor when IF will be a function of x and y

I'm not finding any resource or description or systemic methodology to find integrating factors when the integrating factor will be a function of both x and y. I'm on this problem, $$ ( y - xy^2 ) ...
6
votes
1answer
442 views

How do you solve $f'(x) = f(f(x))$?

A friend told me to solve the following differential equation: $$f'(x)=f(f(x))$$ I have no idea how to solve this! This doesn't seem to be an ordinary differential equation and I can't even solve ...
6
votes
1answer
109 views

Help needed in solving a system of DE

The system of DE is: $$\frac{dI}{db}=-\frac{b}{c}\frac{dJ}{db}-\frac{2ab+1}{2c}J$$ $$\frac{dJ}{db}=\frac{b}{c}\frac{dI}{db}-\frac{2ab-1}{2c}I$$ Assume that $a$ and $c$ are constants and both $I$ and ...
6
votes
2answers
284 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
3answers
175 views

What went wrong?

Intrigued by this question, one-dimensional inverse square laws, I started to try to find an answer and came up with what follows. However, I calculated the derivatives to double check myself, and ...
6
votes
1answer
271 views

$f(x)=\int_{0}^{+\infty} e^{-(t+\frac{1}{t})x}dt$ how to find $f(x)$?

$$f(x)=\int_0^{+\infty} e^{-(t+\frac{1}{t})x}\;dt$$ if while $ x>0 $ , $ f(x) $ has values I noticed some interesting relations for $f(x)$ as shown below: $$ \begin{align} t & ...