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

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0answers
8 views

Can someone verify my derivation of a differential equation involving elliptic integrals, please?

I'm trying to determine the relationship between the major and minor radii ($a$ and $b$, respectively) of an ellipse of constant perimeter and variable eccentricity, and I've been thinking that ...
0
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0answers
12 views

Solution of a Partial Differential Equation

Problem statement Solve $\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial y}=y$ using the change of variables $\left\{\begin{matrix} u=ax^2+y \\ v=x \end{matrix}\right.$ for a suitable ...
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0answers
22 views

How do I solve the differential equation $r(t)^2 + r^{'}(t)^2 = 1$, where $r$ is a smooth real-valued function?

How do I solve the differential equation $r(t)^2 + r^{'}(t)^2 = 1$, where $r: \mathbb R \rightarrow \mathbb R$ is a smooth real-valued function ? In Calculus I've seen linear (higher-order) ...
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3answers
24 views

Identify the Differential Equations from the given problem [on hold]

Dear Math expert, Please solve the above problem. Thanks in advance for your support!
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0answers
15 views

Differential equations. Task. [on hold]

$$f: \mathbb{R}^2 \to \mathbb{R} d_{(x,y)} f =(4x^3y+3x^2y^2)dx + (x^4 + 2x^3y)dy $$ in every point $(x,y) \in \mathbb{R}^2$ Determine: 1) $ \frac{df}{dx}(1,-2)$ 2) $\frac{df}{dh}(2,-3) , h = ...
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0answers
21 views

How to solve $\int_{x}^{x+a} f_X(u) du=e^{-2\lambda_1 x} \int_{x-a}^{x} f_X(u) du$

How to solve equation of the type \begin{align*} \int_{x}^{x+a} f(u) du=e^{-\lambda x} \int_{x-a}^{x} f(u) du \end{align*} we want to solve for $f(x)$ where $\lambda,a$ are some constants. Things I ...
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1answer
24 views

If $u : \Bbb R \to \Bbb R$ satisfies $u' + 2\pi x u = 0$, why does $\hat{u}$ (the Fourier transform) also satisfy this?

I'm trying to understand why if a function $u : \Bbb R \to \Bbb R$ satisfies the differential equation $u' + 2\pi x u = 0$, then so does the Fourier transform. The properties I have that I can use ...
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1answer
16 views

Refreshing solving second order ODE

I have a boundary value problem for the following differential equation $$\frac{d^2 v}{d \chi^2} = q^2 \left( v - C \right), \; 0<\chi<S \; and \;\; v(0)=v(S)=0 $$ where $q$ and $C$ are certain ...
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1answer
23 views

Solution of nonhomogenious differential equations

Kindly help me regarding below math problem. How can I prove? Show that if $y_1(x)$ is a solution of $$y'' + ay' + by = f_1(x)$$ and if $y_2(x)$ is a solution of $$y'' + ay' + by = f_2(x)$$ ...
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0answers
28 views

Solution techniques for f'(x)=f(g(x))

I stumbled over this seemingly natural question and was surprised, that I couldn't find a satisfying answer. Differential equations of the type $f'(x)=g(f(x))$ are studied for all kind of classes of ...
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0answers
15 views

Asymptotic solutions to generalized Airy equation

I am interested in asymtotic solutions, for $x \gg 0$ and $x \ll 0$ of the following differential equation: $\frac{d^ny}{dx^n} + yx = 0$ Here $n$ is an integer $\ge 2$. For the particular case of ...
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1answer
34 views

Ordinary differential equations of order zero?

Is $x+y+2=0$ a differential equation without derivatives of order $n$, $n>0$? Could it be called a differential equation (for unknown $y(x)$) of order $0$? If not, can we define differential ...
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2answers
14 views

Guess maximal solution of ODE ($y^{'} = X(t,y) = \frac 1 3 y^{1/4} + t^{1/3}$) on the form $y(t) = at^p$.

Suppose I have the following ODE: $y^{'} = X(t,y) = \frac 1 3 y^{1/4} + t^{1/3}$ defined on $D_X = (0, \infty) \times (0,\infty)$. I want to guess a maximal solution of the form $y(t) = at^p$ for ...
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0answers
31 views

Cauchy-Euler problem [on hold]

I cannot solve this Cauchy-Euler problem. $$x^2y''-xy'+2y=2x$$
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1answer
16 views

Solving second order nonlinear ODE given boundary condition at infinity

I am trying to solve the following differential equation $$\frac{d^2 u}{dx^2} = - \frac{d V}{du} \; \; , \;\; where \;\; \; V = \frac{1}{2}u^2 - \frac{1}{4}u^4 $$ And the given boundary conditions are ...
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0answers
13 views

Lipschitz continuous function [on hold]

let $y:\mathbb R \rightarrow \mathbb R$ be differentiable and satisfy the ODE $dy/dx=f(y)$ ; $y(0)=y(1)=0$ where $f$ is a lipschitz continuous function then what are the properties of $y$ that it ...
2
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0answers
37 views

Is my solution for the form of the steady state solution of the temperature of a rod at a regular singular point correct?

The steady state temperature distribution of a rod given by: \begin{equation} \frac{\textrm{d}p(x)y'}{\textrm{d}x} - y = 0,\; 0 \leq x \leq 1,\; \text{and} \;y(0) = 0, \end{equation} ...
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0answers
14 views

Determine the equilibrium temp distribution for a 1D rod with the following sources and boundaries.

Q=0 du/dx(0) =0. u(L)=T So ,my attempt is that u(x) = Ax + B, so du/dx = A implies A=0 and so u(L) = 0 + B = T so the solution becomes u(x) = T. But I have a feeling it's not right or I'm ...
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0answers
9 views

region of xy-plane for which the differential equation unique solution [on hold]

Determine a region of xy-plane for which the differential equation $(y-x)y'=y+x$ would have a unique solution whose graph passes through a point $(x_0,y_0)$ in the region.
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0answers
22 views

first-order differential equation problem

Given that $y=\sin(x)$ is an expicit function of the first-order differential equation $\frac{dy}{dx}=\sqrt{1-y^2}$. Find an interval I of definition, the solution interval. So I got to the point ...
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1answer
18 views

Help with an introduction to differential equations?

I am taking linear methods this year and im trying to get some more review for differential equations. This is a problem that I ran across: a) Show that the constant function y(x) = 0, for all x, is ...
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0answers
18 views

Rodrigues formula Associated Laguerre polynomial

Could you find the rodriguez formula of $$L_n^{\beta }\left(x^2\right)$$ knowing that $$\frac{\left(e^x x^{-\beta }\right) \frac{\partial ^n\left(e^{-x} x^{\beta }\right)}{\partial ...
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2answers
28 views

is it possible to intergrate this function to get x(t) and y(t)?

say you have a function as below; $d^2V(t)/dt = -B^2V(t)$ B is a constant Initial conditions $V_x(0) = V$, $V_y(0) = 0$ I can't see how to integrate to get x(t) and y(t); I ended up with ...
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0answers
24 views

Solving nonlinear differential equation using boundary value at infinity

I want to solve the following differential equation subject to the condition that $f(0)=0$ and $\lim_{x\rightarrow\infty}f(x)=1$. Also $|f| < 1 $ always. Can anybody suggest me a concrete way ...
0
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1answer
17 views

getting a new differential equation from an old one.

Suppose I have the following logistic differential equation: $$f'(x) = f(x)(1-f(x)), f(0) = 1/2 $$ and suppose that $ x = 2y - a$ for some positve constant $a$. How do I write a differential ...
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0answers
14 views

Comparison theorem for parabolic partial differential equations

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $J\subseteq\mathbb{R}$ be an intervall $T\in(0,\infty)$ and $f\in C^0\left(\overline{\Omega}\times[0,T]\times J\right)$ be locally Lipschitz ...
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0answers
28 views

A few queries of the method of variation of parameters

I've been reviewing my knowledge on the technique of variation of parameters to solve differential equations and have a couple of queries that I'd like to clear up (particularly for 2nd order ...
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1answer
34 views

First-order nonlinear differential equation

How would I solve this differential equation for $y(x)$? $\frac{dy}{dx} = \frac{y-xy}{x-xy}$ $y -\ln(y) = x - \ln(x) + C$ I'm not sure what to do at this point. I looked it up on WolframAlpha and ...
3
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2answers
148 views

Limit of solution of differential equation without solving the equation.

Given $$x'(t)=A-B\left(x(t)\right)^2, \quad x(0)=0.$$ Is it possible to find $\lim\limits_{t\to\infty}x(t)$ without solving the differential equation? Assuming $\lim\limits_{t\to\infty}x'(t)=0$ gives ...
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2answers
22 views

general solution of second order linear de

Let 1, x and $x^2$ be solutions of second order linear non homogeneous differential equation $-1\lt x\lt 1$. Then find the general solution. I only know that general solution is sum of complementry ...
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2answers
28 views

Can a differential equation with real coefficients have solution with complex coefficients?

Can a differential equation (with constant coefficients, linear or nonlinear) with real coefficients have solution(s) with complex coefficients? If so, are there any examples related to actual ...
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1answer
25 views

Show that this equation together with the boundary conditions $u(0) = 2, u(\pi) = 0$ has no solution

Consider the ordinary differential equation: $u'' + u = 0$. I have no idea how to solve this, no idea what so ever. Please help.
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1answer
34 views

General solution to diffeerential equation

Given the differential equation $$\frac{dy}{dt}=\frac{4t}{1+3y^2}$$ is this the general solution? $$y+y^3=2t^2+c$$ Can we continue to simplify it?
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1answer
20 views

Verifying transport equation solution

I have just started PDE's and I have the transport equation $u_t + au_x = 0$ which has the general solution $u(x,t) = f(x - at)$ In a book I'm reading it says this can be verified by substitution ...
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0answers
14 views

Second order linear ODE arising from Euclidean heat kernel

When solving for the Euclidean heat kernel $H(t,x,y) \in C^{\infty}((0,\infty) \times \mathbb{R}^n \times \mathbb{R}^n)$, one way to proceed is to look for a solution in the form $H(t,x,y) = ...
0
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1answer
47 views

How to solve: y'' + 9y = sin(3t)

I need to find the particular solution to the equation: $$y'' + 9y = \sin(3t)$$ I thought we were looking for a trigonometric forcing term on the form: $$y = a\cdot\cos(3t) + b\cdot\sin(3t)$$ But ...
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1answer
19 views

Help in solving linear differential equation.

The equation is: $(xy^4 + y)dx -xdy =0$ I brought the differential terms to the same side and then divided by $y^2$ to get this. $(xy^2)dy=d(y/x)$. I tried an alternate way to simplify it which ...
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0answers
24 views

Is it possible to show the uniqueness of formula for solution?

The motivation to this question can be found in: Show that any sequence $(u_{n})$ must tends to infinity as $n→∞$ My question is: Is it possible to show the uniqueness of the formula for the ...
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1answer
24 views

Value(s) of the parameter $a$ that give explicit formula's

For what value(s) of the parameter $a$ is it possible to find explicit formula's (without integrals) for the solutions to $$\frac{dy}{dt}= aty +4e^{-t^2}$$ The answer is $a=-2$. I don't know how to ...
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1answer
38 views

Reduce this third order ordinary differential equation to first order to use Runge Kutta

The ODE I'm working with is $$\dddot{x} + t^2\ddot{x} + 4x = 0$$ with $$x(0)=1, \dot{x}(0)=0, \ddot{x}=-1$$ I've written a very basic program in C++ to use the RK4 method to approximate a solution to ...
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0answers
29 views

Fokker-Planck equation - find probability density function

I have problem from my course, that I can't solve. If anyone can do it and explain, would be great. Find the probability density function $f(x,t)$, of $X_t$ where {$X_t$} is a solution of stochastic ...
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0answers
23 views

Substitution in a system of ordinary differential equations when terms of the same order derivative for different variables occur in the same equation

Let's say I have a differential equation such as: y'' - 2ty' + y = 0, y(0) = 2.1, y'(0) = 1.0 I can solve this (among other ways) by substitution and conversion ...
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1answer
27 views

Why aren't my Laplace transform and Undetermind Coefficients answers matching up?

I might be losing my mind this morning (I am, for sure), but I can't these two techniques to give me the same answer to a basic differential equations problem. The problem is $y''-8y'+27y=0$ with the ...
0
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0answers
32 views

Solution of a non-linear ODE system

I'd like to find an explicit solution for the following system of ordinary differential equations: \begin{cases} \frac{dx}{dt}=-x+\frac{ax}{1+x}+\frac{by}{1+y}+c\\ \\ ...
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0answers
14 views

Stick breaking point (discretized ODE)

I cannot find nontrivial solutions to the following problem. Let $x\in[0,1]$ and $y(x)$ be the deflection of the stick. Then this is described by the diff.eq.: $$\alpha^{-1} P y(x)+y(x)''=0 $$ where ...
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0answers
13 views

Does $-\Delta u\equiv u^p$ have non-positive radial solutions?

Let $p>1$ and $u:[0,R)\to\mathbb{R}$ be a radial solution of $$\left\{\begin{matrix}\displaystyle-u''-\frac{n-1}ru'&\equiv&u^p&&\text{on }(0,R)\\ u'&\equiv ...
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1answer
23 views

Decide the smooth function $r : \mathbb R \rightarrow \mathbb R$ of the equation $r(t)^2 + r'(t)^2 = 1$.

Suppose $r:\mathbb R \rightarrow \mathbb R$ is a smooth function and suppose $r(t)^2 + r'(t)^2 = 1$. I want to determine the function $r(t)$. I see that $r(t)^2 + r'(t)^2 = 1$, so I could take $r(t) ...
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1answer
46 views

Is there a unique solution of $\gamma(t)= f''(t)f(t) $ with $f(0) =0$ and $f'(0)=1$?

Consider, \begin{align*} \gamma(t) &= f''(t)f(t) \\ f(0) &=0 \quad f'(0)=1 \end{align*} where $f(t)$ is an unknown function and $\gamma(t)$ is a known function. Is there a unique solution ...
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0answers
12 views

diffusion equation [on hold]

I'm kinda lost with this problem. I don't know how to solve it. If somebody can help me I will be so thankfully. I'm so confuse.If somebody know a reference problem that would help a lot
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0answers
18 views