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

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2
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0answers
5 views

sufficient conditions for finite time of existence of integral curves of a vector field

Let $U\subset \mathbb{R}^2$ open, $\partial U\neq \varnothing$, $V\colon U\rightarrow \mathbb{R}^2$ smooth. Let $c\colon [0,t_{max})\rightarrow U$ be an integral curve of $V$, where $t_{max}$ is the ...
1
vote
0answers
8 views

Eigenvalue problem of a differential operator

It is known that $\sqrt z H^1_\nu(z)$ and $\sqrt z H^2_\nu(z)$ are two linearly independent solutions of the equation $$-\frac{d^2}{dz^2}\chi+\Big(\frac{\nu^2-\frac{1}{4}}{z^2}\Big)\chi=\chi\ \ \ \ \ ...
0
votes
0answers
12 views

How to solve this initial-boundary value problem for a PDE

Consider $$u_{tt}-a^2u_{xx}+u_t+a u_x=0,\quad 0<x<\infty,\quad t>0,(*)$$ where $u_t=\frac{\partial u}{\partial t}$ and etc. It is not so hard to use the method of characteristics to solve it ...
2
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0answers
28 views

Solve $ty''-y'-4t^3y=0$

I try to solve the time dependent differential equation $$ty''(t)-y'(t)-4t^3y(t)=0$$ Hint: Substitute $x=\sqrt{t}$. By using the given subsitution you can derive $$x^2y''(x^2)-y'(x^2)-4x^6y(x^2)=0$$ ...
1
vote
1answer
5 views

Determining a constant to a Bessel function solution to an ODE

I am using the generalized Bessel function to obtain a solution to an ODE. The derivative of the general solution is something like \begin{align} \frac{dT_{1}}{dx} = \frac{3}{4}Ax^{\frac{1}{4}} ...
2
votes
2answers
24 views

Help in solving $y' - \frac{y}{t} = b$

I think the integrating factor is $r(x)=e^{\int \frac{1}{t}dt} = e^{ln|t|}=t$. Multiplying the DE by the integrating factor to get $$ty' - \frac{y}= bt$. Here $x(t)=t$, so $(x(t)y)'=(ty)'=bt$. So ...
1
vote
1answer
11 views

How to use the initial condition in this DE: $x^2\frac{dy}{dx}=\frac{4x^2-x-2}{\left(x+1\right)\left(y+1\right)}$

Here is my proposed solution: \begin{align} x^2\frac{dy}{dx}&=\frac{4x^2-x-2}{\left(x+1\right)\left(y+1\right)}\tag{1}\\ \implies & ...
0
votes
0answers
22 views

Simplify Laplace equation in rectangle geometry

Consider Laplace's equation in a rectangle as shown in the following figure. The boundary conditions are shown in the figure. The problem is solved in the case of a1 =a2=1. Is there a way to ...
0
votes
1answer
49 views

This may be odd to some of you (soln to 1st ODE with NO constant of integration)

The question is to solve the following 1st ODE: $$\frac{dy}{dx} = 2 + \sqrt{y - 2x + 3}$$ If you use the following substitution $$u=\sqrt{y-2x+3}$$ to solve the problem, you should end up with a ...
0
votes
1answer
29 views

An ODE inequality

Suppose $Q$ is a positive smooth function of $t$ on time interval $[0,a]$, such that $$\frac{d}{dt}Q\leq 1+Q-Q^{1+b},$$ where $b$ is a positive constant. Is it true that $Q\leq ...
2
votes
2answers
59 views

Approximating solutions for the ODE $y'=\exp(y/x)$

I am currently trying to solve excercise 1-38 from Mathews and Walker. In this excercise I am asked to consider the differential equation: $$\frac{\mathrm{d}y}{\mathrm{d}x}=\exp(y/x)$$ for two ...
0
votes
0answers
12 views

Solution for an ODE given only at discrete points

The problem I have: For each $n \in \mathbb N$ I have $$\begin{align} x_0^n & \in \mathbb R \\ h_n & \in \mathbb R \\ x_k^n & = x_0^n + k \cdot h_n \text{ for } k \in \{0,1,\ldots n\} \\ ...
1
vote
1answer
32 views

Solving $y'(t)=\frac{1}{t^2+y^2(t)}$ [on hold]

Solve the following differential equation $$y'(t)=\frac{1}{t^2+y^2(t)}$$ I would appreciate some help with this problem. Thank you very much.
1
vote
0answers
15 views

Time period of periodic motion

Find time of one period in polar coordinates $( r, \theta) $ $ \dfrac {d \theta } {dt} = \dfrac{ \sin \psi } {r} $, $ \dfrac{dr}{dt} = \cos \psi ;$ obeying differential equation in a 2D plane $ ...
1
vote
1answer
41 views

Solving ODE $x' = \lambda x^2$

I am currently studying continuous dependence ODE theory, and there's one example given in our lecture notes, where I am confused how to solve it. The equation is: $\displaystyle x' = \lambda~x^2$ ...
2
votes
1answer
39 views

How to solve the ODE $2x\frac{dy}{dx}=C(1+(\frac{dy}{dx})^2)^2$?

I am struggling with this ODE I obtained when solving the Euler-Lagrange equation. Can any one help me with solving the ODE $$2x\frac{dy}{dx}=C(1+(\frac{dy}{dx})^2)^2$$ Thanks so much! It comes ...
0
votes
0answers
33 views

Please guide me what are the topics i need to study in maths from basic. [on hold]

I am not having good knowledge in maths.Please guide me what are the topics i.e (algebra,calculus,diff.eqn...)i need to study by step by step. please guide me.
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0answers
10 views

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, ...
2
votes
1answer
64 views

What is the process of nondimensionalizing 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 ...
0
votes
0answers
31 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 ...
0
votes
1answer
35 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' ...
0
votes
1answer
15 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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vote
0answers
14 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 ...
0
votes
2answers
59 views

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

How do I solve $y' + Py = ae^{bt}$? My attempt: $y' + Py = ae^{bt}\Rightarrow Py - ae^{bt} + 1.\frac{\mathrm{d} y}{\mathrm{d} t}=0$, where $M(t,y)=Py - ae^{bt}$ and $N(t,y)=1$. $M_{y}=P$, and ...
0
votes
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 ...
1
vote
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} ...
5
votes
0answers
31 views

Limit points of the differential system $\dot {x}=y-x+x^3$, $\dot{y}=-x$

Consider the following system of differential equations: $$\dot {x}=y-x+x^3,\qquad \dot{y}=-x.$$ By linearization, it's easy to see that $(0,0)$ is a (nonlinear) sink. Show that there exists an ...
0
votes
1answer
39 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 ...
1
vote
2answers
54 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 ...
0
votes
1answer
24 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
votes
1answer
27 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
49 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) ...
0
votes
3answers
34 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 ...
1
vote
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 = ...
2
votes
0answers
39 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 ...
0
votes
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 ...
0
votes
1answer
24 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 ...
2
votes
1answer
27 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 $$y'' + ay' + by = f_2(x)$$ ...
4
votes
0answers
36 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 ...
0
votes
0answers
18 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 ...
0
votes
1answer
36 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 ...
0
votes
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 ...
1
vote
0answers
33 views

Cauchy-Euler problem [on hold]

I cannot solve this Cauchy-Euler problem. $$x^2y''-xy'+2y=2x$$
0
votes
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 ...
-1
votes
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 ...
4
votes
0answers
75 views

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} ...
0
votes
0answers
15 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 ...
0
votes
0answers
11 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.
0
votes
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 ...
0
votes
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 ...