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

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1answer
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 ...
1
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1answer
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!
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0answers
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 ...
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1answer
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 ...
4
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1answer
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 ...
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1answer
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 (...
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0answers
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...
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0answers
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 ...
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1answer
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 ...
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1answer
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 ...
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1answer
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 ...
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1answer
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 ...
3
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2answers
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 ...
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0answers
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 ...
2
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2answers
79 views

Solve this differential equation.

How should I approach this problem? $$\dfrac{dy}{dx}=1+y^{2}$$ given that $y (2) = 0$.
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1answer
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 ...
2
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1answer
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 ...
4
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3answers
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, ...
2
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1answer
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 ...
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0answers
75 views

Verify $y=x^{1/2}Z_{1/3}\left(2x^{3/2}\right)$ is a solution to $y^{\prime\prime}+9xy=0$

This question is a sequel to this previous question. As before, some background information is needed first as follows from my textbook: The standard form of Bessel's differential equation is $$x^...
2
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1answer
49 views

Flow of a vector field, equivalent characterizations

Given a vector field $X$ on the manifold $M$ its flow is, loosely speaking, a map $F= F(t,x)= F^t(x)$, in the variables $(t,x)\in I\times M$, such that the curve $t\mapsto F^t(x_0)$ is the unique ...
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0answers
29 views

Helping understanding equation

reading script I've found following equation: We have $$x'(t)=f(t,x(t),p) \\ x(t_{0})=x_{0} $$ where $p$ is parameter $$\frac{d}{dp}\frac{d}{dt}x(t,p0)=\frac{d}{dp} f(t,x(t,p_0),p_0)=\frac{d}{dx}...
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0answers
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....
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1answer
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 .
2
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1answer
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 $e^{tA}$ ...
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2answers
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 ...
0
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2answers
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: $\int y'y^{-2/3}dy = \int x dx$ ...
2
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2answers
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 ...
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1answer
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 ...
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2answers
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?
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1answer
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 ...
3
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1answer
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 ...
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1answer
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 ...
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1answer
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 ...
2
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1answer
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}...
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1answer
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 ...
1
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1answer
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 ...
1
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1answer
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 ...
0
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1answer
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 ...
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0answers
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 ...
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2answers
37 views

How to solve this matrix equation

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)...
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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 "...
3
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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.
0
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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 ...
1
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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}...
2
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1answer
44 views

Stability of homogeneous linear differential equation with variable coefficients

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), \...