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

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0
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1answer
41 views

Differential equation of order 2

Solve the differential equation $$y''(x)-4y'(x)+8y(x)=10e^x\cos(x)$$ I am not able to solve this particular differential equation. Please help. I know the answer is: $$y(x) = ...
0
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2answers
53 views

First Order Differential Equation $y' \cos x +y=\sec x+\tan x$

I'm stuck on a seemingly straight forward problem as follows: $$\cos x \frac{dy}{dx}+y=\sec x+\tan x$$ I have rearranged the equation to be: $$\frac{dy}{dx}+\sec x \cdot y=(\sec x+\tan x)\sec x$$ ...
0
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0answers
26 views

Application of the Existence and uniqueness theorem

Given the initial value value problem $$y'=\frac{10xy^{0.4}}{3}$$ $$y(0)=-1$$ has a unique solution on some open interval that contains $x_{o}$. Find a solution and determine the largest open ...
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0answers
27 views

What is the geometrical interpretation of differential equation?

I was wondering what was the geometrical interpretation on differential equations, is there one ?
2
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2answers
31 views

To find an extremal of a given functional

I have to find extremal of following : $\int_0^1 [(y')^2 + 12 xy] dx$ with $y(0) = 0$ and $y(1) = 1$. I applied the Euler's equation $\frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial ...
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0answers
51 views

How to solve Energy Balance equation by numerical method

Good Day I am new to heat transfer technique please give me some suggestion on solving energy balance equation $$a \frac{\partial T_p}{\partial t}=\frac{\partial}{\partial x}\left(b\frac{\partial ...
2
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1answer
45 views

2nd order differential equation with missing y'

I have the following 2nd order differential equation: $$y'' + p(x) y =0, \tag{1}$$ where $p(x)$ involves only first order of $x$, for example, $p(x)=ax+b$. Any suggestion how to obtain or guess a ...
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0answers
27 views

book suggestion on manifolds

I've to learn differential equations on Manifolds. Can any one please suggest some books/lecture notes for differential equations on Manifolds ?
2
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1answer
49 views

Differential Equation $y'=\frac{1}{x^2-1}, y(0)=0$, am I missing something?

I'm encountering a singularity of sorts when working out this D.E. It seems like a very straight forward problem and, assuming I'm going wrong, I'm wondering where. The problem: Solve the initial ...
2
votes
2answers
32 views

Integrating Factor - Exact Equation problem.

I have stumbled with a problem I can't seem to solve. $$(x^2 - y ^2)dx - 5xy dy = 0$$ We know that $$u(x,y) = \frac{1}{(x M + y N)}$$ if the equation is HDE (Which it is..I believe). Excuse my ...
3
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2answers
34 views

Autonomous equation having $\frac{t^2}{1+t}$ as a solution

Find an autonomous equation having $\displaystyle\frac{t^2}{1+t}$ as a solution. So the desired function $f$ should depend only on $x$, if I'm not wrong in the form $x'=f(x)$, that means the goal ...
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1answer
40 views

Fourier sine series of $f = \cos x$

Let $f:(0,\pi) \to \mathbb{R}$ defined by $x \mapsto \cos x $ Show that the Fourier sine series of (odd extension) is given by $$\sum\limits_{n=2}^\infty \frac{2n(1+(-1)^n)}{\pi(n^2-1)}$$ So far, ...
2
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0answers
21 views

Stability of non-autonomous stochastic differential equation

I'm looking for a good reference or insight to under what conditions can I prove stability (or instability) for the following general n-dimensional non-autonomous stochastic differential equation: ...
2
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0answers
23 views

Determining linear independence of three simple functions for a third order ODE. (2.9-7)

This is a very similar post to one previous by me but I felt that not all questions were satisfactorily answered. But I am sincerely grateful to those who tried. I would like a sharp independent eye ...
3
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5answers
41 views

Separable diff. eqn: $(1+x^2)y' = x^2y^2, x > 0$

I have been given a step-by-step answer which I just cannot understand or follow. $\begin{eqnarray} &(1+x^2)y' &= x^2y^2 + y\cdot1 \\ \iff& \frac{1}{y^2} &= \frac{x^2}{1+x^2} ...
2
votes
1answer
28 views

To find solution of differential equation

Find the continous solutionof $$\frac{dy}{dx} +y = G (x),\qquad x \geq 0,\quad y (0) = 2 $$ where $$G (x) = \begin{cases} 3 & \text{when }x\in [0, \pi/2) \\ \cos x & \text{when $x\ge ...
0
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1answer
19 views

How to solve this first-order nonlinear ordinary differential equation?

Obtain the solution to the DE $$\dfrac{dy}{dx} = \dfrac{1+y^2}{1+x^2}$$ (A) $\dfrac{Cx}{1-Cx}$ (B) $\dfrac{Cx}{1+Cx}$ (C) $\dfrac{C-x}{1-Cx}$ (D) $\dfrac{1-Cx}{x+C}$ (E) ...
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2answers
23 views

Differential equation with $(\frac{dx}{dt})^2$

I have had very little training in differential equations, so this might be a stupid question. Is it okay to solve something like $t = x^2({dx\over dt} )^2$, by doing: $t(dt)^2=t^2(dx)^2 \rightarrow ...
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0answers
40 views

High dimensional Differential Evolution

I want to minimize a cost function with Differential Evolution (DE) algorithm and I have 55 unknown parameters as an input for DE algorithm. Therefore, the DE should search in high-dimensional space ...
0
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1answer
29 views

string displacement function by d'Alembert's formula

Consider an infinite string stretched taut on $x$ axis from $-\infty$ to $\infty$ . Let the string be drawn aside into a curve $y=f(x)$ and released, and assume that its subsequent motion is described ...
0
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1answer
21 views

First Order Differential Equations - length of the arc joining two points on it

What curve lying above the x axis has the property that the length of the arc joining any two points on it is proportional to the area under that arc?
2
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2answers
29 views

Determining linear independence of three simple functions for a third order ODE. (2.9-6)

I would like a sharp independent eye other than my own to review my work here. I have a few questions I would like answered. Did I actually answer/solve all parts of this problem? Determinants of ...
3
votes
3answers
58 views

Finding a particular solution to a differential equation [closed]

what is the particular solution for the following differential equation? $$D^3 (D^2+D+1)(D^2+1)(D^2-3D+2)y=x^3+\cos\left(\frac{\sqrt{3}}2x \right)+xe^{2x}+\cos(x)$$ I tried Undetermined Coefficients ...
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2answers
53 views

Singular solution to ODE and singularities

I'm a bit confused about those two, i'll try to explain. I get an ODE like $$ \frac {y \cdot dy }{\sqrt{y^2+1} } + \frac {x \cdot dx}{ \sqrt {x^2 +1}} = 0 $$ What I'm not sure about : 1) at y=0 the ...
1
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1answer
19 views

Solve the pde $(x^2-y^2-yz)p+ (x^2-y^2-zx)q=z(x-y)$

I'm solving this by Lagrange's method. Lagrange's auxiliary equation is: $\frac{dx}{x^2-y^2-yz}=\frac {dy}{x^2-y^2-zx}= \frac{dz}{z(x-y)}$ From the first two ratio and the last ratio: ...
3
votes
1answer
34 views

ODE Separable Equation

Let $y = Φ(x)$ be a solution to $y' = y(5-y)(8-y)$ subject to $y(0) = 7$. Determine $\lim_{x \to ∞} Φ(x)$. Workings: I'm thinking I have to solve the differential equation. $y' = y(5-y)(8-y) dy$ ...
0
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1answer
25 views

Verify that each given function is a solution of the given partial differential equation

$\alpha^2u_{xx}$ = $u_t$; $u_1(x,t)= e^{-\alpha^2t}\sin x$ I took the derivate of u1 and then took the second derivate and plugged it into $\alpha^2u_{xx}$ = $u_t$ as $u_{xx}$ but it is looking ...
0
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1answer
17 views

Solving the Helmholtz equation

If I wanted to figure out how to do a simulation with the Helmholtz equation, how would I do it? Or, what kinds of techniques would I have to learn in order to figure it out? Background: 1st year ...
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3answers
69 views

Is there another way of solving this differential equation?

$y'+y\tan x=\dfrac{1}{\cos x}$ I found the integrating factor to be $e^{-\ln\cos x}$, but then I get into having to integrate $\dfrac{-\ln\cos x}{\cos x}$ , which seems to be quite a mess. Is there ...
0
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1answer
28 views

First order differential equation problem

Suppose we have $$ \frac{dy}{dx} +f(x)y = r(x) $$ and it has two solutions $y_1(x)$ and $y_2(x)$ then how to prove that solution of differential equation $$ \frac{dy}{dx} +f(x)y = 2r(x) $$ Will be ...
1
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2answers
41 views

What's wrong with my solution for the following differential equation?

I have the following DE: $$y'+2xy-xy^4=0$$ It's a Bernoulli equation, so I converted it to: $v'-6xv=-3x$ and the integrating factor being $e^{-3x^2}$ after doing the necessary steps, I find myself ...
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2answers
27 views

First order DE, need help

I am trying to solve this equation by inspection: $$(xy-y)dx+(x^2-2x+y)dy=0$$ Hints would be very helpful.. Thanks
1
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0answers
58 views

Solving Laplace equation in polar coordinates

I have some assignments to do and I don't even know where to start. The notes in the course aren't too good, so I didn't understand too much from them. Given $$ \Omega = \{(x, y) \in \mathbb{R}^2 , ...
1
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2answers
68 views

Suggestion for a Lyapunov function

Consider the differential system $$ x'=x+y $$ $$y'=x-y+xy$$ What would be a Lyapunov function for this system at $(0,0)$? I have considered functions $V(x,y)=ax^{2n}+by^{2m}$ but none of ...
2
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0answers
31 views

suggestion for lyapunov function

Consider differential equation \begin{align}x'&=-t(x+y)\\ y'&=-y+x-y(y^2-6).\end{align} Can some one suggest a lyapunov function for it. I have examined $V(x,y)=x^2+y^2$ , ...
0
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1answer
20 views

Determing linear independence of three very simple functions. (2.9-23)

Why are the following three functions linearly independent if the first two can be multiplied to get the third? $$y_1 = x$$ $$y_2 = \frac{1}{x}$$ $$y_3 = 1$$ I am focusing on this for a course ...
1
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1answer
43 views

Legendre Series Recurrence Relation Divergence at $x=\pm1$, using Gauss test

How to show that the Legendre Series solution $y_{even}$ and $y_{odd}$, diverges as $x = \pm1 $. $y_{even} = \sum_{j=0,2,\ldots}^\infty a_jx^j$, where $a_{j+2}=\frac{j(j+1)-n(n+1)}{(j+1)(j+2)}a_j$. ...
0
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1answer
25 views

Differential Equation: Find the instability criterion

My attempt: Question (d) I took the derivative of the original differential equation, $$dI/dt = BI(N-I) -uI = g(I)$$ $$g'(I) = BN - 2BI - u$$ Set $$g'(I) = 0$$ Isolate $$I = Ro$$ $$I = Ro = ...
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0answers
38 views

Nature of the infinite differential sum operator?

Consider the operator $$ Hf = f + f' + f'' +\cdots = \sum_{i=0}^\infty \left[ \frac{d^i f}{dx^i}\right] $$ I am trying to determine what $ Hf $ is entirely in terms of $f$. I note the following ...
2
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1answer
28 views

Existence solution pendulum equation using the contraction principle

How can I prove that the differential equation $\ddot{x}(t)=-(g/\ell)\sin (x(t))$ must have a solution by using the contraction principle (by Banach). The numbers $g$ and $\ell$ are fixed constants ...
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0answers
30 views

The field mouse population satisfies the differential equation $dp/dt$ = 0 .5p - 450

I'm having trouble solving part b where it asks me to find the time of extinction if p(0) = $p_0$, where 0<$p_0$<900 I have done the work for this part b but I keep getting a negative number ...
0
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1answer
12 views

Laguerre Recursion Relation from two other recurrence relation

How to show this, $$xL_n'(x) = nL_n(x)-nL_{n-1}(x)$$ Laguerre recursion relation from these two recursion relations, $$L'_{n+1}(x)-L'_n(x)+L_n(x)=0\\(n+1)L_{n+1}(x)-(2n+1-x)L_n(x)+nL_{n-1}(x)=0$$ ...
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0answers
38 views

Is the calculation of Green's function correct?

I am not sure if all the calculations are correct.Could you check for me please ? ...
0
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1answer
28 views

To show solutions of a linear system lie on parabolas in phase space.

Given a linear system $\dot{x}=x$ $\dot{y}=2y$ To show solutions of a linear system lie on parabolas in phase space. Which solutions (if any) do not lie on parabolas? It is the second question ...
2
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1answer
38 views

Solving a differential equation $F-y'F_{y'}=C$, with $F(y,y')= \frac{1+2y'^2}{3y^3\sqrt{1+y'^2}}$

If $$F= F(y,y')= \frac{1+2y'^2}{3y^3\sqrt{1+y'^2}},$$ where $y=y(x)$ and $y'= y'(x)=\frac{dy}{dx}$, then how to solve the differential equation: $$F-y'F_{y'}=C, $$ that is: ...
3
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1answer
43 views

Second Order Differential Equation inhomogeneous

$$y''(s) - \frac{s^2}{c^2}y(s) - \frac{g}{s\cdot c^2}= 0$$ I am getting confused with what to do with the $$\frac{-g}{s\cdot c^2}$$ part
0
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2answers
46 views

Find the flow of a vector field

Question: Let $\mathbb{X}$ be the vector field given by $\mathbb{X}(x,y)=(x,y)$ Compute its flow $\Phi(x,y)$ Attempt: We have $\dot{x}(t)=x\therefore$$$\int_{x_0}^{x(t)}dx'=\int_{0}^{t}x(t')dt'$$ ...
1
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1answer
50 views

Lyapunov function and stability

Suposse we have $f:\mathbb{R}^2\longrightarrow\mathbb{R}^2$, $f(0)=0$, the system $z'=f(z)$. Let $z=(x,y)$ and $V$ a strict Lyapunov function, $V:U\longrightarrow R$, $U$ open, $V(0,0)=0$, ...
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0answers
27 views

Variable change for homogeneous equations

I got my brain for some days on this two differential equations with no luck at all, have tried different variable changes but it seem that I always get to a dead end, will be glad if someone could ...
1
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3answers
86 views

Good book for an introduction to differential equations for engineers

I will be leading a discussion class on differential equations for engineers this coming semester and I am wondering if anyone has a book that they could recommend. The book that will be used in the ...