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

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1answer
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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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0answers
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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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0answers
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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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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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1answer
34 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 ...
2
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0answers
16 views

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
9 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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1answer
43 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
3k views

Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...
2
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0answers
32 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
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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0answers
52 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} ...
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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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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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0answers
32 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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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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0answers
16 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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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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1answer
18 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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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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1answer
24 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
30 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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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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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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4answers
110 views

Exponential of matrix

So, I'm wondering if there is an easy way (as in not calculating the eigenvalues, Jordan canonical form, change of basis matrix, etc.) to calculate this exponential $e^{At}$ with $$A=\begin{pmatrix} ...
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0answers
31 views

Cauchy-Euler problem [on hold]

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

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
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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
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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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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 ...
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0answers
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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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
28 views

Differential equation in Maple : No solution on $x = -1 .. 1, y = -1 .. 1$.

Backround: Yesterday in class we had a lab session (practical work ?) on ODE and I have a question. We plot the following contour (I am using maple) implicitplot(H(x, y) = 0, x = -1 .. 1, y = -1 .. ...
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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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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
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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0answers
28 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
18 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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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 ...
0
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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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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
22 views

Extracting differential equations [duplicate]

$$\frac{dx}{dy} = \frac{x(\alpha - \beta y)}{y(\delta x - \gamma)}$$ How do I extract two differential equations (y as a function of x and x as a function of y) from the equation above? I could ...
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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 ...
1
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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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3answers
168 views

Eigenvalue of some Sturm–Liouville problem

I have one simple question. How I suppose to show that $\lambda =0$ is an eigenvalue of some problem. Does it mean that I must have non-trivial solution for $\lambda=0 $? Thanks! UPD:I mean by ...
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1answer
39 views

Showing a system is fully self adjoint for general unmixed boundary conditions

I have been asked to look at the following questions and I'm struggling to solve it. Let $Ly=a_2(x)y''(x)+a_1(x)y'(x)+a_0(x)y(x) , a<x<b$ such that $L^*=L$. i.e. $L$ is a self adjoint linear ...
0
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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.