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

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Let $(I_\eta, y_\eta)$ be maximal with $y_\eta(1) = \eta$ (IVP). Show for $0 < \eta < 1$ we have $y_\eta(t) < t^{\frac 4 3}$, $t \in I_\eta$.

Consider the differential equation $y' = X(t,y)$ with $X(t,y) = \frac 1 3 y^{\frac 1 4} + t^{\frac 1 3}$, defined on $\mathcal D_X = (0,\infty) \times (0,\infty)$. For $\eta > 0$ let $(I_\eta, ...
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24 views

What is the process of nondemensionalizing an equation?

Question: I need to scale time by $\frac{1}{I}$ and species by $P$ for the following equation $\frac{dS}{dt}=I(1-\frac{S}{P})-\frac{ES}{P}$ where P - Size of the source pool of species on the ...
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21 views

ODE for the normal distribution [on hold]

The normal density function $\phi(x)=\tfrac{1}{2\pi}e^{-\frac{x^2}2}$ can be described via the ODE $$\phi^\prime(x) = -x \phi(x)$$ under the condition $\int_{-\infty}^\infty \phi(x) = 1$. Is there ...
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18 views

Determining the equilibrium solution of a direction field for a first order ODE

Consider the equation $dy/dt = f(y)$ and suppose that $y_{1}$ is a critical point, that is, $f (y_1) = 0$. Show that the constant equilibrium solution $φ(t) = y_1$ is asymptotically stable if $f' ...
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1answer
12 views

First Order Differential Equation Problem Substitution or bernoulli

I am trying to solve the equation $$dy/dx + xy = y^4$$ using Bernoulli's method but it seems to fail since I end up with $$dv/dx -1/3(xv) = -1/3(v)^-8 $$ I am not sure what to do... Any help would ...
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10 views

Time taken to empty a hemispherical shaped tank

The tank has a radius of $2$m when initially filled and has an outlet of cross section $12$ cm2 Outlet flow as I calculated goes according to the law $V(t)=0.6\sqrt{2gh(t)}$. Having found out the ...
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1answer
36 views

Solve for $y' + Py = ae^{bt}$

How do I solve $y' + Py = ae^{bt}$? My attempt: $y' + Py = ae^{bt}\Rightarrow Py - ae^{bt} + 1.\frac{\mathrm{d} y}{\mathrm{d} y}=0$, where $M(t,y)=Py - ae^{bt}$ and $N(t,y)=1$. $M_{y}=P$, and ...
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1answer
20 views

ODE: Why do we change our variable here?

I was trying to solve a matrix equation $\dot x = Ax + Bu$ Rearranging yields $\dot x - Ax = Bu$ Let $I = e^{-At}$ our integrating factor so $d(xe^{-At})/dt = e^{-At}Bu$ Then $xe^{-At}$ = $x_0 ...
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1answer
11 views

Why does solving the spherical Bessel equation using Frobenius series produce two quadratic equations for the exponents at the singularity?

The spherical Bessel equation is: $$x^2y'' + 2xy' + (x^2 - \frac{5}{16})y = 0$$ If I seek a Frobenius series solution, I will have: \begin{align*} &\quad y = \sum_{n = 0}^{\infty} ...
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Show that the limit points of a system of differential equations are $p \in D$ and $\partial D$

Consider the following system of differential equations: $ \left\{\begin{matrix} \dot {x}=y-x+x^3\\ \dot{y}=-x \end{matrix}\right. $ By linearization, it's easy to see that $(0,0)$ is a ...
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1answer
36 views

Differential Equation with biology!

I am working on a growth model for bacteria as a function of a nutrient, and I am stuck. So the differential equation I am supposed to be solving is $\frac{dN}{\ DT} = k(C_0 -\alpha N(T)) N$ The ...
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2answers
50 views

Boundary conditions which yield exactly one solution of the differential equation $u'' + u = 0$

Consider the ordinary differential equation: $u'' + u = 0$. Give an example of boundary conditions which yield exactly one solution $u$. Progress The equation of solutions is $$A\cos x + B\sin x ...
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1answer
19 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 ...
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1answer
20 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
35 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
31 views

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

Dear Math expert, Please solve part c of the question. Thanks in advance for your support! I'm able to determine (a) Determine xh and (b) Determine xp. But I'm not able to understand the question ...
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0answers
17 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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34 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
28 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
19 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
25 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
31 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
17 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
16 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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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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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 ...
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Solution of 2nd order linear ODE with regular singular points, and complex exponents at singularity

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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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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10 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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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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19 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
31 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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30 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 ...
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1answer
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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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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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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 ...
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2answers
149 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
28 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) = ...
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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 ...