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

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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 $6 - Ce^{-x}$ ...
7
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1answer
431 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 ...
7
votes
3answers
557 views

A simple question about sine and cosine

I have been thinking about all of the different ways that I have encountered sine and cosine in my studies. There are no courses on trigonometry at my school, so perhaps that's why I feel like ...
7
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2answers
129 views

Integrating Factor - Not getting the answer given?

$y' - 4y = t$ My integrating factor is $e^{-4t}$ $\int e^{-4t}y'$ - $\int 4e^{-4t}y$ = $\int te^{-4t}$ $\int (e^{-4t}y)'$ = $\int te^{-4t}$ $e^{-4t}y$ = $-4te^{-4t}$ - $e^{-4t}$ I end up with ...
7
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3answers
705 views

Solving separable differential equation

Seems straight-forward but I've been unable to get it right. Here are my steps: $$y'(x) = \sqrt{-2y(x) + 28},\hspace{20 pt} y(-4)=-4$$ $$\int {1 \over \sqrt{28-2y} }\hspace{2 pt}\text{d}y = \int ...
7
votes
1answer
147 views

My first partial differential equation attempt

have I solved this correctly? My textbook is asking for the relation between $ \alpha $ and $ \beta $: $$ \frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{x^2}} $$ Textbook's proposed ...
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1answer
182 views

Reduction to standard form.

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
7
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2answers
184 views

Can't seem to solve this differential equation

Disclaimer: This IS homework. So I will outline the steps I've taken an where I'm stuck. I have the following DE: $$ xy' = y + x\cos^2\left(\frac{y}{x}\right) $$ I then rule out the possible ...
7
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1answer
416 views

Is there a basis-independent proof of Abel's identity?

Abel's identity states that if $X(t)$ and $A(t)$ are $n\times n$ matrix-valued functions such that $X'(t)=A(t)X(t)$, then $\frac{d}{dt}(\det X(t)) = \mathrm{tr}\,A(t) \cdot \det X(t)$. The ...
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1answer
76 views

Using Poincaré-Bendixson to prove that there is a periodic solution

I want to use the Poincaré-Bendixson theorem to show that there exists a nontrivial (and periodic) solution to $$z'' + [\log (z^2 +4(z')^2)]z' + z = 0.$$ Therefore I substituted $u = z'$ to get $$u' ...
7
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1answer
190 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
479 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 ...
7
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3answers
122 views

Integral formulation for the solution of $xy'' + y' = y$

Let's say that $y$ satisfies the following ODE: $$xy'' + y' = y$$ I want to formulate $y$ as a contour integral. I know that the final result I should get is: $$y(x)=\frac{1}{2i\pi} ...
7
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1answer
138 views

Reference request: a differential equation arising in geometry

$$ \frac{d\beta}{d\alpha} = \frac {\sin\beta}{\sin\alpha} $$ In what contexts (if any) is this equation known to occur?
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1answer
180 views

What's the name of this chaotic system? (Cool pics included.)

I found this playing with a 2D-ODE-system plotter I'm writing. Surely, since it's so simple, it's been found and extensively studied by someone. What's it called? I'd like to look it up and learn a ...
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2answers
356 views

Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer?

This is a slightly farcical question, for which I apologise. An undergraduate tutee of mine was faced with the following problem: Q. A particle of mass $m$ moving along a line is subject to a force ...
7
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1answer
286 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 & ...
7
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1answer
251 views

A higher-order differential equation involving absolute values and trigonometry

For a smooth function $f: (-\pi/2,\pi/2) \to \mathbb{R} $, if $\displaystyle\frac{|f''(x)|}{\sqrt{(1+f'(x))^3}} = \cos{x}$, and $f(0) = f'(0) = 0$, $f''(0) = 1$, ...
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2answers
383 views

Prove that a map is continuous

Let $r:[0,1]\to\mathbb R$ be a continuous function and let $u_\lambda$ be the unique solution of the Cauchy Problem: $$\begin{cases}u''(t)+\lambda r(t)u(t)=0,\quad\forall t\in [0,1],\\ ...
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1answer
32 views

Initial value problem

Solve the following initial value problem: $$\frac{d^2y}{dt^2}+2\frac{dy}{dt}+5y=0; y(0)=0 \text{ and } y'(0)=2 $$ I started off with the characteristic equation which is: $$ r^2+2r+5=0 $$ Using ...
7
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3answers
245 views

Distribution theory and differential equations.

How does distribution theory plays role in solving differential equations? This question might seem to be very general. I will try to explain, please bear with me. I understand, distributions make it ...
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2answers
57 views

Show that any solution of second order differential equation has atmost a countable number of zeroes $?$

Question : Considered the second order differential equation $y''(t) + a(t) y'(t) + b(t) y(t) = 0$. then any solution of second order differential equation has atmost a countable number of zeroes ...
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1answer
166 views

$\nabla \cdot f + w \cdot f = 0$

Let $w(x,y,z)$ be a fixed vector field on $\mathbb{R}^3$. What are the solutions of the equation $$ \nabla \cdot f + w \cdot f = 0 \, ? $$ Note that if $w = \nabla \phi $, then the above equation is ...
7
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3answers
240 views

Differential equations solveable independently of coordinate system?

Looking from a physics viewpoint ODEs tend to look very differently when setting up the problem in different coordinate systems. For instance the Laplacian in spherical coordinates involves way more ...
7
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4answers
381 views

A Handwaving Proof of a Specific Existence and Uniqueness Theorem

My problem is as follows: Given the second order homogeneous linear differential equation with constant coefficients $$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+c\,y(x)=0,$$ is there a good heuristic ...
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3answers
535 views

How to find the general solution of $xy''-(2x+1)y'+x^2y=0$ when we know the general solution of $y''+2y'+xy=0$?

Given that the general solution of $y''+2y'+xy=0$ is $y=C_1\int_0^\infty ...
7
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1answer
404 views

ODE Laplace Transforms: what impulse brings an oscillating system to rest?

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
7
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1answer
98 views

Solve $y' + \frac1y + \frac1x =0$ Differential Equation

Do you have any suggestions for how to sole this differential equation? $y'+\frac1x + \frac1y =0$ ? :) I tried solving this by changing variable in the form of $v=x^\alpha*y^\beta$ but it didn't ...
7
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2answers
86 views

Why is Existence and Uniqueness for Navier-Stokes Easier in 2-D than in 3-D?

I know that existence and uniqueness for incompressible viscous flow in the 2-D case has already been established$^1$, and that doing the same for the 3-D case has yet to be shown. Not only that, but ...
7
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1answer
177 views

how to solve this differential equation $\frac{dy}{dx} = \frac{1+xy}{x(1-xy)}$ by substitution?

I've tried with this differential equation $\displaystyle \frac{dy}{dx} = \frac{1+xy}{x(1-xy)}$ , put $u=xy$ then $\displaystyle\frac{du}{dx}=x\frac{dy}{dx}+y$ So, It will be after editing ...
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1answer
116 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 ...
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1answer
370 views

$\frac{dS}{d\rho}$ Factor arising

To get details see: equations 29,30,31,34,44,50,51 We have known some solitary wave solutions, given by(equations 1 to 5) $$ \phi_1=p_1\cos \tau \tag{1}$$ $$\phi_2=\frac16 ...
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1answer
2k views

A Differential Equation with Trigonometric Coefficients

Suppose we have the following second-order differential equation: $\cos^2(x)y'' -\sin(x)y' + y = 0$ How do we determine its general solution? I couldn't even guess a particular solution; all my ...
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2answers
337 views

Deriving the addition formula for the lemniscate functions from a total differential equation

The lemniscate of Bernoulli $C$ is a plane curve defined as follows. Let $a > 0$ be a real number. Let $F_1 = (a, 0)$ and $F_2 = (-a, 0)$ be two points of $\mathbb{R}^2$. Let $C = \{P \in ...
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1answer
206 views

$\int_3^5{\frac{x^2}{1+x^2}dx}$ by differentiation under the integral

I'm trying an easy problem to get my bearings using the method here. The integral is $$\int_3^5{\frac{x^2}{1+x^2}dx}$$. I would like to proceed, if possible to solve by defining: $$F(y) = ...
7
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3answers
308 views

Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem

A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume". At that time for me to write ...
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1answer
228 views

Checking the stability of an equilibrium point

I have the linearization of a non-linear system about an equilibrium point as follows $$ \dot x = (-A+M)x, $$ where $x\in\mathbb{R}^3$, $A$ is a positive definite matrix and $M$ has its eigenvalues ...
7
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3answers
327 views

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in polar coordinates $$ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial U} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 U} {\partial \theta^2} = 0$$ ...
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1answer
84 views

Solve second order differential equations

There are two differential equations that I could not solve. Can someone please help me solve them? $$ (x^2+y^2)y′′-y(y^{′})^3+xy′-y=0 $$ and $$ xy^2y′′+2y^2y′-4xy(y^{′})^2+2x^2(y^{′})^3=0, $$ ...
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1answer
72 views

Stochastic Calculus Question

I'm new here and was hoping someone could help me answer this question. I'm reading a paper and I'm a bit confused on how they go from 1 equation to the next. They say: Let \begin{align} x(t) = {} ...
7
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1answer
103 views

$\cos(x+y)\frac{\partial z}{\partial x}+\sin(x+y)\frac{\partial z}{\partial y}=z+\frac{1}{z}$

Having difficulty in solving the following partial differential equation: $$\cos(x+y)\frac{\partial z}{\partial x}+\sin(x+y)\frac{\partial z}{\partial y}=z+\frac{1}{z}.$$ Will it be easier if we ...
7
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1answer
140 views

Limit points of the differential system $\dot {x}=y-x+x^3$, $\dot{y}=-x$

Consider the following system of differential equations: $$\dot {x}=y-x+x^3,\qquad \dot{y}=-x.$$ By linearization, it's easy to see that $(0,0)$ is a (nonlinear) sink. Show that there exists an ...
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1answer
75 views

global behavior of an ODE system

Consider the ODE system $$ (x',y')=f(x,y) $$ where $$ f(x,y)=((1-x/2-y/2)x,(-1/4+x/2)y) $$ in the first (open) quadrant. It is not hard to show that $z_0=(1/2,3/2)$, which is a equilibrium of the ...
7
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1answer
184 views

Solutions for $ \frac{dy}{dx}=y $?

Al-right, this may be a very basic question but I'm confused about this. We all know that one differential equation can only have one solution. Consider: $$ \frac{dy}{dx}=y $$ The solution is: $$ ...
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1answer
259 views

First-term approximation for singular perturbation of ODE (with two turning points)

I'm reading "Introduction to Perturbation Methods" by Mark Holmes, and I came across an exercise that I don't know how to approach. As I decided to independently read this book, I have no ...
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3answers
220 views

Riccati differential equation

I have to solve the following equation : $$ \frac{dx}{dt}(t)=-q x^2(t) +1 $$ with $x(0)=1$ and $q>0$. At first I consider the two cases: $q=1$, then I take the change of variable $ x= ...
7
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1answer
133 views

Periodic solution of differential equation y′=f(y)

Let $f∈C^∞(ℝ^2,ℝ^2)$ and $∀x∈ℝ^2$ $f(kx)=k^2f(x)$ for $k∈ℝ$ Show that any periodic solution of $y′=f(y)$ is constant. My attempt : Let $\lambda \in \mathbb{R}$. Let $g$ a periodic solution ...
7
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1answer
115 views

Solution to differential equation $f^{(n)}-(n+1)f^{(n-1)}-(n+1)nf^{(n-2)}-\dotsc-(n+1)!f=g$

Let $n$ be a given positive integer and $g$ be a continuous function. We are looking for a function $f \in C^n(\mathbb{R})$ such that $$f^{(n)}-(n+1)f^{(n-1)}-(n+1)nf^{(n-2)}-\dotsc-(n+1)!f=g.$$ It ...
7
votes
1answer
297 views

How to solve a differential equation associated with square wheels?

I'm looking for a general solution for $f(t)$ given an unrelated function $g(t)$ in $$f(t)^2 - 2g(t)f(t)\sin(t) - 2f'(t) + g(t)^2 - 2g(t)\cos(t) + 1 = 0$$ Is it possible to solve without knowing ...
7
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1answer
2k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...