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

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Differential equations, theoretical question about lowering the power of a basic differential equation.

In the text book it says we can solve:(The area of existence and uniquesness of the equation is $G= R \times R^n$)$$x^{(n)}=f(t), f \in C(R) \tag{1}$$ the following method: Integrating (1) $n$ times ...
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1answer
30 views

Steady states of a system

How can I find the steady states? I am aware that the condition is to equal 0 but I am not able to say how many steady states there are... $$\begin{cases} \dot x=x-y^2 \\ \dot y= -x+2y-z^2 \\ \dot z= ...
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1answer
83 views

The initial condition for a heat equation with stationary solution subtracted

I am presented with the following question for exam revision: Heat is supplied at a prescribed rate $Q(x) > 0$ (per unit volume) to an isotropic conducting rod that occupies the region $0≤x≤L$...
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0answers
49 views

nondimensionalisation for an ODE

I need help to understand 'nondimensionalisation' for ODEs better. I stacked how to deal with a constant input in a prey-predator model. I reproduced the following ODE system $$\frac{dx}{dt}=X_{0}-ax$$...
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2answers
160 views

Solving $\frac{dx}{dt} = A \frac{ (1-x)}{(t-t^2)} - \frac{(B*x -C*x^2)}{(t-t^2 )*(t-x)}$ (using wolfram/mathematica)

I would like to solve the following non-linear ordinary differential equation: $$\frac{dx}{dt} = A \frac{ (1-x)}{(t-t^2)} - \frac{(B*x -C*x^2)}{(t-t^2 )*(t-x)}$$ -I need an analytic solution. -I ...
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4answers
224 views

Solve the differential equation (if exact): $(5x + 4y)dx + (4x - 8y^3)dy = 0$

I need to test this to see if it is exact. My conclusion is the it is indeed exact because $My = 4 = Nx$. then I integrated $Mdx$ and $Ndy$. $$ \int Mdx = \int (5x+4y)\, dx = (5/2)x^2 + 4xy +f(y) $$ $...
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2answers
394 views

If the equation is exact, solve: $(2x-1)dx + (3y+7)dy = 0$

I'm trying to understand the concept of determining if a differential equation is exact. My professor told me that the general strategy is as follows: The term in front of $dx = M$ The term in front ...
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1answer
103 views

Need help with this proof, I don't understand it , could anyone clarify some of the details. System of linear Differential equations.

$$(*)X'=A(t)X - system$$ $$(*)PX(\alpha)+QX(\beta)=0.$$-border conditions, where P,Q constant square matrices $n \times n $. Let $Y(t)$ be the fundamental matrix for the system $(*)$ normed for$ t= \...
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1answer
49 views

Finding a general solution to a first order linear differential equation, using the integration factor method

I am trying to solve a very simple ODE using a integrating factor, I have a solution but its trivial and I'm unsure how to find the general solution. $$\tau \frac{dV}{dt} = E_0 - V$$ Where $E_0$ and $...
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1answer
52 views

How many particular solutions to a second order linear differential equation?

If I have the differential equation $$y''-y'-2y=3x+2,$$ I can make the guess that a particular solution is of the form $$y = Ax+B,$$ and determine the coefficients. Do I always have exactly one ...
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3answers
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Trying to understand the true meaning of integral and Derivative in calculus [duplicate]

I'm solving a physics question, and i just encountered some question i had no idea how to start, i just got the right answer and inside it it has something in math i never thought possible, I know ...
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1answer
18 views

Unit step response of $p(D) = D+kI$ without initial condition?

The full problem description is: $p(D)=D + kI$ where $D = \frac{d}{dt}$ and $I$ is the identity operator, $k$ is a constant number and $k \neq 0$. Find $x(t)$ such that $p(D) \, x = u(t)$ where $u(t) =...
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2answers
40 views

Meaning of the following, partial derivatives..

What is the meaning of $${\partial^kG \over \partial t^k} \in C$$ how is this function explained $G(t,s)$, does it mean that the k-th derivative of $G$ is continuous. I've done some studying on this ...
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1answer
35 views

Differential equation in inhomogeneous cosmological model

How to solve this differential equation: $$\left(\frac{dR(r,t)}{dt}\right)^2 = \frac{F(r)}{R(r)} + f(r). $$ This differential equation is occuring in Tolman Bondi Lamatre inhomogeneous cosmological ...
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1answer
52 views

Initial Value Problem using Laplace Transforms

Solve using Laplace Transform: $$y''(t)+2y'(t)+5y(t)=xf(t), \\ y(0)=1,y'(0)=1$$ where $x$ is a constant. Once the solution is found, evaluate the limit as $t \to\infty$. Progress: If I have ...
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2answers
94 views

Temperature Gradient and Fourier's Law

I'm having trouble understanding temperature gradient in the context of fourier's law of heat conduction. Namely, I do not understand how the change in temperature is approximately equal to what ...
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0answers
66 views

Eigenvalue problem $y''+py=0$, $y(-2)=0$, $y(2)=0$

The problem states to find the non-negative solutions to the eigenvalue problem given by $y''+py=0$ where p is a parameter which may be varied. Solving this differential equation for the general ...
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2answers
48 views

Emulating a parabola in my game for a jump

I am currently having some trouble understanding how to plot a parabola with the x and y coordinates.In my game a player needs to jump from point a to point b and the jump would look something like ...
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1answer
27 views

differential equation, partial derivatives: Why is the following true?

$$x_1'=f_1(t,x_1,...,x_n) \\ x_2'=f_2(t,x_1,...,x_n) \\ ....... \\x_n'=f_n(t,x_1,...,x_n)$$ This a system of equations, now the text book says let's differentiate the first equation of the system ...
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1answer
31 views

I am trying to maximize the following constrained optimization and I need help.

$$ \arg \max\limits_{C,D} \quad tr\{C^{-1}D\} + \log(det(C)) - \log(det(D)) \\ \mbox{sub. to} \quad tr\{C\} \le k \\ \quad \quad D > 0 $$ I did the following. Rewrite the above ...
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2answers
46 views

How to solve this non-linear differential equation

I want to solve the following differential equation: $$y''=e^{x}(y')^2$$ then I substitute $y''=u'u$,$y'=u$ so I got: $u'=e^{x}u$ But then I don't know how to solve this, may be separate variables ...
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1answer
41 views

Reverse engineering a differential equation from singular points

I've been struggling to find a way to reverse engineer a differential equation based on knowing it's singular points. In this case, I'd like to create a flow on $[-1,1] \times [-1,1]$, which has ...
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236 views

Physical meaning of linear ODE $xy''+2y' + \lambda^2 x y = 0$

As reported by Wikipedia - Sinc function, $y(x)=\lambda \operatorname{sinc}(\lambda x)$ is a solution of the linear ordinary differential equation $$x \frac{d^2 y}{d x^2} + 2 \frac{d y}{d x} + \lambda^...
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1answer
248 views

Solve by separation of variables: $\frac{dx}{dy}y\ln|x| = \big(\frac{y+1}{x}\big)^2$

I need to solve the problem above using separation of variables. I got as far as the below but it seems too complex to be right. Am I wrong somewhere? Because I think my final answer needs to simplify ...
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2answers
33 views

Cauchy problem: $y '(x) + (\tanh x) y(x) = \sinh x$, $y(0) = 1$.

I am using the formula for the inhomogeneous first order differential equation but I don't know how to apply the Cauchy problem to it. Like I still have the $c$ constant with this inhomogeneous first ...
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1answer
204 views

Solve the differential equation using separation of variables: $\frac{dy}{dx} = e^{3x+2y}$

I need to solve using the method of separation of variables. I got to this point but I'm not sure if I'm on the right track: $$\frac{-1}{2e^{2y}} + C = \frac{e^{3x}}{3} + C$$ It looks like now I ...
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1answer
44 views

Please help to solve this differential equation

I am not able to find a proper solution of the following differential equation: $$y''(x) + \frac{b}{y(x)} = a$$ where a, b are constants I need to have $y(x)$ as a function of $x$. Any help regarding ...
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2answers
1k views

Changing variable in a second derivative

I want to convert the differentiation variable in a second derivative, but it's a bit more complicated than the case of the first derivative. For context, the variable ETA is a dimensionless density ...
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1answer
27 views

Ways to further simply recursive relation

I was working on a power series solution in my ODE class and I had found that my $a_n$ seemed to be defined as $$a_n=\frac{a_o}{(n^{2}(n-1)^{2}…(n-(n+2))^{2}}$$ but I am having trouble understanding ...
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2answers
201 views

Quick way to solve $yy''-(y')^2=y^4$.

This is a simple yet ugly ODE (arising from Euler-Lagrange equations): $$yy''-(y')^2=y^4$$ What method could I use to quickly solve it? I began to notice that by dividing by $y^2$ I can write it as ...
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0answers
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Solving Linear ODEs over the space of distributions

I am encountering in my work linear differential equations with coefficients that involve things like Heaviside functions and delta Dirac functions. I know how to find things like Green's functions or ...
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1answer
36 views

Boundary conditions of ODE are derivatives, constant falls out

I am trying to solve an ordinary differential equation, and I have the fundamental solution, but do not know how to get the constants. The solution is $ T(r)=Ar^2+C_1\ln(r)+C_2$, I have the ...
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1answer
42 views

Laplace equation in a circle - where is my mistake

We want to solve $r^2u''_{rr}+ru'_r+u''_{\theta \theta}=0$ where $0\leq \theta <2\pi$ and $0\leq r \leq 2$, given that $u(2,\theta)=\cos(2\theta)$. I managed to work out a simple solution, but ...
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3answers
76 views

Why $\omega$ in $x = \cos(\omega t + \alpha)$ , $\omega$ isn't considered an arbitrary constant?

We know the SHM differential equation is of second-order $$\dfrac{d^2 x}{dt^2} = -{\omega}^2 x$$ . So, the solution of this equation must contain two arbitrary constants. And also we know that $x = \...
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1answer
26 views

Masspoint - curve - differential equation.

On what curve in a vertical plane must a masspoint of mass m move, so that, with gravitational acceleration, it falls down with a constant vertical velocity $v_z=v_0=const.$? The initial values at $t=...
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1answer
26 views

Expressing multiplication as percentage changes.

I have seen many times that for example when we have a formula: $$A=\frac{B\cdot C}{D}$$ where $A, B, C, D$ are some variables; we can express it in the following way as 'growth rates':$%\Delta%$ $$\...
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1answer
148 views

General solution of Transport equation

General solution of Transport equation (homogeneous): Method of Characteristics $$u_t+cu_x=0 (\star)$$ We know that if $f: \mathbb{R} \to \mathbb{R}$ is differentiable then $u(x,t)=f(x-ct)$ is a ...
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2answers
370 views

Solve $x\frac{dy}{dx}=y(1+\ln y-\ln x)$

$$x\frac{dy}{dx}=y(1+\ln y-\ln x)$$ I realize that it can be rearranged to make it clear that it's a homogeneous first order differential: $$\frac{dy}{dx}=\frac yx(1+\ln \frac yx)$$ Using the ...
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1answer
162 views

Using an inverse operator to find a particular solution to a differential equation.

I am just learning about inverse operators in solving a differential equation, but I don't understand exactly how they work. For example, find a particular solution to $$4y''-3y'+9y=5x^2$$ using ...
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1answer
25 views

Finding the solution to a differential equation using an integratring factor

I need to find the solution to the following differential equation: $$y'+a(x)y=R(x)$$ With $a(x)$ and $R(x)$ smooth functions. I can re-order this to the following: $$1.dy+(a(x)y-R(x))dx=0$$ $Q:=1$ ...
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1answer
60 views

separable or non separable? alternative way to solve the equation?

I've got a problem with this diff. equation: $h'(2h-z)(\frac{1}{2} -z)= 2h (h-2)$ I tried hard to somehow transform the first equation in order to reach a separable equation but failed to get a ...
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Heat equation proving smoothness

I have a question regarding a PDE course: Let $T$ be the strongly continuous semigroup which belongs to the heat equation, thus with generator $A$ is the Laplacian. Suppose we have $g \in C^{\infty}...
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1answer
67 views

Why is the solution to $y' = y^n$ always in polynomial form EXCEPT when $n = 1$?

Could someone explain (intuition-wise) why the differential equation $$y' = y^n$$ for $n \in \mathbb{N}$ seems to always some kind of polynomial solution (or a ratio of polynomials, etc.) except ...
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286 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 ...
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If a Point Admits an Integral Curve on an Interval then a Neighborhood Does too On the Same Interval

$\newcommand{\R}{\mathbf R}$ Let $V:U\to \R^n$ be a (continuous) vector field on an open set $U$ of $\R^n$. Suppose we have a point $\mathbf p\in U$ and an open interval $I\subseteq \R$ such that ...
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1answer
18 views

Zero-points of solutions to differential equations

Consider a simple differential equation: $$\displaystyle\frac{dy}{dx}=\displaystyle\frac{y}{x}\text{.}$$ By considering it as a separable equation, the solutions are $$y=Cx\text{,}$$ where $C$ is a ...
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1answer
55 views

Mathematica - Searching a clever way to do a computation

I am solving this differential equation using Mathematica $$x ~y''(x)+\alpha ~ y'(x)+\frac{1-3\mu (x)}{(1-x) x}=0$$ where $\mu(x)$ is a polynomial: $$\mu (x)=a+b\left(\frac{1}{1-x}\right)+c\left(\frac{...
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3answers
46 views

Homogenous equation to higher order ODE

Hello I have a quick question in regard to general form of the solution to $$y^{(4)}-2y^{(3)}+y''=0$$ I had thought to find this solution we would consider $r^{4}-2r^{3}+r^{2}=0$ which factors as $...
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3answers
39 views

Solving easy first-order linear differential question.

Question Solve $y'=2x(1+x^2-y)$. My attempt Rearranging gives $y'+2xy=2x(1+x^2)$. Thus, the integrating factor is $e^{\int2x\,dx}=e^{x^2}$ and multiplying the equation throughout by this gives $e^{...
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1answer
72 views

A difficult 2nd order differential equation

I've been doing some revision in dynamics, and I reduced a problem to solving the differential equation (stripping away the constants) given by $$\ddot x = x^2$$ How can I find the general solution ...