# Tagged Questions

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

105 views

### Coupled differential-integral equation

This is coming from a physics paper I'm reading. It's been a while since I've done much differential equation solving and the system here is a bit unorthodox in that I'm actually searching for the ...
36 views

### Exact solution of a third-order ODE

Does anyone knows some exact solution of this nonlinear equation? $f'''=af^2f'-bf'-C$ Thanks!
52 views

### Continuity of the period of solutions of a second order ODE, with respect to their initial conditions

There is a second order ODE $$\ddot{x} + b(x) \dot{x}^2 + c(x) = 0$$ with continuous, and locally lipschitz coefficients b, c : $\mathbb{R}\to\mathbb{R}$. Assume the ODE has 2 partial periodic ...
35 views

### Verify that $P(t)=P_0e^{kt}$, $t>0$ is a solution of $\frac{dP}{dt}=kP.$

A city has a growing population at a rate proportional to the current population, that is: $$\frac{dP}{dt}=kP.$$ Verify that $P(t)=P_0e^{kt}$, $t>0$ is a solution of the equation. There is more ...
76 views

### Leibniz rule; Solving differential equations

Could you help me with a question? I get stuck at ii), Define the function $$I(x):=\frac{1}{\pi} \int^\pi_0 \cos(x\sin\theta) d\theta$$ i) Via application of Leibniz rule (or otherwise) calculate ...
43 views

### Central force is planar in n-D

So in 3-D the differential equation $$\ddot{\bf{r}} = -\frac{f(r)}{m}\bf{r}$$ is shown to be planar by noting $$\bf{r} \times \dot{\bf{r}}$$ is constant. But isn't the differential equation planar (...
53 views

### How does the substitution $u=\sin y$ transform $\frac{dy}{dx} = \frac{1}{\cos y} + x \tan y$ into a linear differential equation? [closed]

How to transfer $$\frac{dy}{dx} = \frac{1}{\cos y} + x \tan y$$ into a linear a linear differential equation? The answer is $u = \sin y$, and I don't know why...
46 views

### differential equation with shifted argument

I am trying to solve, in general, the following differential equation : $y(x+a) = \frac{d^2}{dx^2}y(x)$ With $y :\mathbb{R} \to \mathbb{R}$ and $~a\in \mathbb{R}$ Do you know of any techniques ...
36 views

### Finding the least possible order of D.E given the particular solution.

Let $$y=x^2 \exp (3x) + \sin x$$ be solution of initial value problem with constant coefficients then what is the least possible order of differential equation if y solves the homogeneous linear ...
41 views

### General Solution of the ODE $\eta' - 2 \zeta \eta = K\left(\zeta^2+1\right)^2$

I am trying to find the function $\Phi(\zeta)$ which is determined by the ordinary differential equation $$\Phi'' - 2 \zeta \Phi' = K\left(\zeta^2+1\right)^2$$ where $K$ is a constant from my model, i ...
271 views

### Deriving the Airy functions from first principles

I have just started reading about the Airy functions and am stuck on a particular step of their derivation. But first here is some background information to give this question some meaning, more ...
31 views

### Diffusion equation involving dirac delta term

I've ran across the following diffusion equation: $$\frac{\partial c_i(r,t)}{\partial t}- a \nabla^2c_i(r,t)=b \delta[x-x_1(t)]$$ where $a$ and $b$ are constants related to the context, $\delta$ is ...
98 views

### How to get the coordinates of the center of the ellipse after approximation

I create an algorithm recognizing ellipses in images. I have five coordinates (points) possible ellipse. (8.8) (7.4) (6.3) (3.6) and (2.2) I use the formula of the conical section of the ...
20 views

### Optimization problem with differential equations as constraints

I have formulated an optimization problem which I have to solve for a project but i do not have enough math skills to solve it. The problem is an optimization problem, whose constraints include both ...
79 views

### Solve this differential equation.

How should I approach this problem? $$\dfrac{dy}{dx}=1+y^{2}$$ given that $y (2) = 0$.
49 views

### How to solve this fourth-order singular nonlinear ODE?

Hi I would like to hear your suggestions on solving the following fourth-order singular nonlinear ODE regarding $u=u(x)$ $\alpha u'''' + u'u''' + (u'')^2 = 0$ where prime denotes derivative w.r.t. x ...
48 views

### Find on which $z=x+iy\in\mathbb{C}$ the function $f(z)=(\overline{z}+1)^3 - 3\overline{z}$ is differentiable

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses some difficulty. If it's any indication of difficulty, the exercise is ...
78 views

### solving $y' - yy'x^2-x=0$

How can i solve this? $$y' - yy'x^2-x=0$$ I only got to the homogeneous solution wich I found is (I just divided by $y'$) $$y=\frac{1}{x^2}$$ But I don't know how to get the particular solution, ...
21 views

### sustitution $x=z-h$ and $y=w-k$ reduce the differential equation. $\frac{dy}{dx}=f(\frac{ax+by+c}{dx+ey+f})$ a homogeneous equation

help with this excercises, I´m sorry for the english :) If $ab \neq bd$ then, we can choose the appropriate constant $h$ and $k$ adequately so that the sustitutions $x=z-h$ and $y=w-k$ reduce the ...
75 views

18 views

### SIR model parameter notation

I am reading on SIR models and I found this article In the article it has three groups as one without vaccination, one with only whole cell(wP) vaccination, and one with only acelluar(aP) vaccination....
28 views

### Solving Second order differential equation

I have a second order ordinary differential equation of the form : $a \frac{d^2p(x)}{dx^2} + x \frac{dp(x)}{dx} + p(x) = 0$ Can anyone tell me how I can solve it? Thanks in advance .
41 views

### General integral of a linear system of ODEs

I want to find the solution of \begin{cases} x'= -5x-y+e^t \\ y'= 2x-3y \end{cases} $$A = \begin{bmatrix} -5 & -1 \\ 2 & -3 \end{bmatrix}$$ I calculated the exponential matrix $e^{tA}$ ...
43 views

### Is it posible that a first order diferential equation doesn't “fit” in any known method?

I've been doing excercises for first order differential equations, there are like 7 methods so far and I wonder, what if one day i find a differential equation which doens't fit in any method? I'm not ...
58 views

### Solve a differential equation using variable separation

I am trying to solve the following differential equation: $y'=xy^{2/3}$ with the initial condition: $y(0) = 0$ Here is my progress so far: We separate the fariables: $\int y'y^{-2/3}dy = \int x dx$ ...
100 views

### How to solve this 1st order ODE

$$\frac{dy}{dx} = \frac{2x^2 + 3y^2 - 7}{3x^2 + 4y^2 + 8}$$ This does not satisfy the exactness and I can't find any integrating factor to transform it. I can't make it homogeneous too. Thanks in ...
27 views

### Method to check if solutions of given differential equation are identically zero or bounded.

I am given a differential equation $y'+ 2y = 0$. Then which of the following options is correct one? A) every solution is identically zero. B) all solutions are unbounded. C) every solution tends ...
36 views

### Help with a linearly dependent proof on differential equations

Show that any two solutions $y_1$ and $y_2$ of the equation $y' + p(x)y = 0$ are linearly dependent. How do I prove this question?
40 views

### A question in differential equations

Show that if $y_1$ and $y_2$ are linearly independent on $\alpha < x < \beta$ and $y$ is any function such that $y \neq 0$ on $\alpha < x < \beta$, then $yy_1$ and $yy_2$ are also linearly ...
42 views

### Laplace Transform with initial value

Use the Laplace transform to solve the following initial value problem: $$y'' + y = 2t$$ with $y(\pi/4) = \pi / 2$ and $y'(\pi/4) = 2 - \sqrt{2}$. I understand this type of problems ...
35 views

### Are weak (Sobolev) solutions to a linear ODE a classical ones?

Let $\Omega$ be an open subset of $\mathbb{R}$ and let $L$ be the differential operator $$Lf = \sum_{k=0}^{n-1} a_k f^{(k)} + f^{(n)},$$ where $a_k$ are reals. I would like to show that every ...
31 views

### Complete Vector field

I am reading "Geometry of Differential Forms". We want to show that on a smooth compact manifold, vector fields are complete. We claim that there is an interval $(-\epsilon ~ ~\epsilon)$ of time ...
28 views

### Find an expression for $\frac{dy}{dx}$ in terms of $x$ and $y$ and verify that $P$ is a stationary point.

A curve is defined by the equation $$2y+e^{2x}y^2=x^2+\frac{2}{e}$$ Find an expression for $\frac{dy}{dx}$ in terms of $x$ and $y$ \begin{align} 2y+e^{2x}y^2 & = x^2+\frac{2}{e} \\ 2\frac{dy}{dx}...
40 views

### If $\langle f(x),g(x)\rangle = 0$, and $f$ has periodic orbit, then $g(x)$ has equilibrium point

Let $x'=f(x), x'=g(x)$ be two ODE, with $f(x),g(x):\mathbb R^2\rightarrow\mathbb R^2$, such that $\langle f(x),g(x)\rangle =0$ for all $x\in\mathbb R^2$. If $f$ has a periodic orbit then $g$ has ...
37 views

### first orderr non linear ODE

I came along this first order non linear ODE, and cannot solve it. $$\frac{dv}{dt}=\frac{-b}{(vt)^2}+k$$ (where b and k are constants) The question asked to express v as a function of t. Thank you ...
65 views

### New Maths 9-1 GCSE for 2017 Sample Question

My teacher gave me some practice questions for my end of year exam which will be like the new GCSE and this question is very tackling to me. Could any with clear working solve the question and show me ...
29 views

### $c_1\cosh(x)+c_2\sinh(x)=A\cosh(x+y)$ always true?

My question: Can I rewrite $c_1\cosh(x)+c_2\sinh(x)$, which is a solution to a differential equation as $$A\cosh(x+x_0)$$ introducing the new constants of integration $A$ and $x_0$? How can I deal ...
18 views

### Definition of complete integrals - existence of envelope?

Let suppose a partial differential equation: $$\Phi(x,y,z,\partial_x z,\partial_y z)=0\qquad (1)$$ In some books I have found the following definition: Let $\Lambda$ and $\Omega$ be two open subsets ...
Consider the system of ODE in $\Bbb R^2$ $\dfrac{dY}{dt}=AY$ where $Y(0)=$ \begin{bmatrix} 0 \\ 1\end{bmatrix} $t>0$ where $A=$ \begin{bmatrix} -1 & 1 \\ 0 & -1\end{bmatrix} and $Y(t)... 1answer 40 views ### Collocation method for solving ODEs I am studying numerical methods for ODEs. At the moment I am trying to get the big-picture of collocation methods. As I understand it, there are two main things: Where to set the so called "... 2answers 87 views ### How to solve this differential equation$y(t) = y′(t)+ \frac{e^{2t}}{y'(t)}$I don't understand what a type of this equation and which method I could use for: $$y(t) = y′(t)+ \frac{e^{2t}}{y'(t)}$$ Please, help me. 1answer 39 views ### Solution to the linearly differential equation of order n. When we study the solution to the equation:$a_{n}y^{(n)}+a_{n-1}y^{(n-1)}+...+a_{1}y=0$where$a_{i}$is real coefficient$i\in \lbrace 1,...,n \rbrace$, we know that: if the characteristic ... 1answer 53 views ### Solution of$\nabla^2 f(x,y,z)=-f(x,y,z)$I am working on a problem and came across the following equation that I need to solve:$\nabla^2 f(x,y,z)=-k^2 f(x,y,z)$where the operator$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}...
I would like to know if a homogeneous linear differential equation, with variable coefficients which are periodic, is stable. So the differential equation can be written as,  \dot{y}(t)=A(t)y(t), \...