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

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0answers
7 views

Evaluate Higher Order Homogenous Differential Equation $ (D^4 - D^3)y=0; y(0) = 1 = y'(0), y''(1) = 3e, y'''(1) = e $

I'm getting following values of constants: $C1 = 0$ $ C2 = 0$ $C3 = e$ $ C4 = 1$ But in my solution manual of the book, the constant values are coming like this: $C1 = 0$ $ C2 = 0$ $C3 = 1$ $ ...
3
votes
1answer
22 views

How to tell if you have specified sufficient initial data for a differential equation?

I recently learnt that the following 'wave equation' is not well-posed $$ \begin{cases} \partial_{tt}u=\triangle u, & (0,1)\times\mathbb R^d\\ u(0,x)=u(1,x)=0,&x\in\mathbb R^d \end{cases} $$ ...
1
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0answers
19 views

Finding a Relation Between Two Sequences Satisfying Recurrence Relations

Consider the following recurrence relation for $C_i(r)$s $$\begin{align} &C_0(r)=r-r_2 \\ &q(r+n)C_n(r)+\sum_{k=0}^{n-1}[(r+k)\alpha_{n-k}+\beta_{n-k}]C_k(r)=0, \qquad n\ge1 \end{align} \tag{...
1
vote
2answers
36 views

How to solve this using power series method $\left(x^2+2\right)y''\:+\:xy'\:-\:y=0$

$\left(x^2+2\right)y''\:+\:xy'\:-\:y=0$ What's next after this $\sum _{n=2}^{\infty }\:n\left(n-1\right)a_nx^n+2\:\sum _{n=0}^{\infty \:}\left(n+2\right)\left(n-1\right)a_{n+2}x^n+\sum _{n=2}^{\infty ...
1
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0answers
17 views

A Simpler way of Thinking About Integrability

I have recently been looking into the concept of integrability, and the more I look into it, the less sense it seems to make. There seems to be several kinds of integrability and their definitions ...
2
votes
2answers
2k views

Modeling a chemical reaction with differential equations

The problem says: Two chemicals $A$ and $B$ are combined to form a chemical $C$. The rate, or velocity, of the reaction is proportional to the product of the instantaneous amounts of $A$ and $B$ ...
1
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4answers
55 views

I can't derive the integrating factor of this first order ODE from the Dover textbook

I'm a junior mechanical engineering student. I can't derive the integrating factor of this first order ODE $(x^ 2 - y^2 - y) dx - (x^ 2 - y^2 - x) dy = O$ The textbook provides 5 integrating ...
3
votes
1answer
134 views

Differential equation based on chemical kinetics

In chemical kinetics, the law of mass action gives us reaction rates of the form $$r=k x^a y^b$$ where $r$ is the time derivative of either $x$ or $y$ times a constant $$r=-\frac{dx}{\beta dt}=-\...
4
votes
3answers
124 views

Should we re-define Sine?

Sine is usually defined as the ratio of the opposite side to an angle to the hypotenuse in a right angle triangle. Another common definition is based on the unit circle. However I think these ...
0
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1answer
27 views

Solutions to Sturm-Liouville equation continuous even with discontinuous coefficients?

In the physics paper here (should be open access), the author first studies a Schrödinger equation in the form of a Sturm-Liouville equation $$\frac{d}{dx}\frac{1}{m(x)}\frac{d}{dx}\phi(x) = -\...
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votes
1answer
20 views

Reduction of order of 2nd order ODE (without $y_0$ or $y_1$ provided) [on hold]

Please help to reduce $y''+y'=0$. There are no initial condition nor a solution to start from. From a similar question, I've tried a suggestion to let $W = y'$. But don't come to an agreeable ...
3
votes
1answer
42 views

Show that there aren't negative eigenvalues.

I've been trying to solve this Sturm-Liouville theory problem. Show that the problem: $$\left\{\begin{matrix} y''+(x+\lambda)y = 0\\ y(0)=0\\y(1)=0\end{matrix}\right.$$ doesn't have ...
1
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0answers
18 views

PDE basic traffic flow problem

I am analyzing a basic example of traffic flow presented here http://people.uncw.edu/hermanr/pde1/PDEbook/PDE_Main.pdf and have a question to the last transition in the traffic flow equation ...
0
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1answer
18 views

Use the Wronskian to determine a first-order inhomogeneous differential equation for $y_2(x)$.

The function $y(x)$ satisfies the linear equation $y''+p(x)y'+q(x)=0$. The Wronskian $W(x)$ of two independant solutions, $y_1(x)$ and $y_2(x)$ is defined as $\begin{vmatrix}y_1 & y_2\\ y'...
1
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0answers
66 views

What is wrong with my formulas for a mathematical model of a double pendulum?

I wanted to create a computer simulation on Matlab, using a model for a pendulum from this study (A double pendulum model of tennis strokes. Rod Cross. Uni of Sydney, 2006) - Link I wanted to use the ...
4
votes
1answer
79 views

Quasilinear second order ODE

Consider a smooth $u\colon\mathbb{R}\rightarrow\mathbb{R}$ satisfying $$ u^{\prime\prime}+a\left(u^{\prime}\right)^{2}+bu=0\text{ on }\mathbb{R} $$ with $$ u^{\prime}\left(x\right),u^{\prime\prime}\...
2
votes
2answers
69 views

Chemical reaction modeled by a differential equation

I am badly stuck on the question... so asking some help :) Consider a chemical reaction in which compounds $A$ and $B$ combine to form a third compound $X$. The reaction can be written as $$A ...
1
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1answer
70 views

Inner Product Examples, what is the points?

Example: For $ -\pi<x<\pi$, $$x =-2 \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin(nx)$$ and $$x^3 =-2 \sum_{n=1}^{\infty} \left( \frac{\pi^2}{n}-\frac{6}{n^3} \right)(-1)^n \sin(nx)$$ by ...
0
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1answer
26 views

How to compute the slope for a 3 or multi-dimensional equation.

If I have an equation $Z=X^2+Y^2+3X+6Y+5$ and want to find the slope at the point $x=2$, $y=1$. How do we compute it? I know for a two dimensional equation we can compute it by differentiation of $Y$ ...
0
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3answers
56 views

Finding Explicit Form of Function Defined by Definite Integral

Let $$f(y) = \int_{-\infty}^{\infty} e^{-x^2} \cos (xy) \> dx$$ One can show that $$f'(y) = - \int_{-\infty}^{\infty}xe^{-x^2} \sin (xy) \> dx$$ I'm interested in making an ODE involving $...
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0answers
23 views

Fundamental Set of Frobenius solutions

Consider the equation $$x^2 (\alpha_0 + \alpha_1x + \alpha_2x^2)y'' + x(\beta_0 + \beta_1x + \beta_2x^2)y' + (\gamma_0 + \gamma_1x + \gamma_2x^2)y = 0$$ Define $$p_j (r) = \alpha_jr(r − 1) + \beta_jr ...
0
votes
2answers
115 views

Difficult engineering second order DE, any pointers?

I have the following engineering DE: $$rR''+R'+\alpha r(R^2_0-r^2)\lambda^2R=0$$ Where $R(r)$ is Real, $r \geq 0$, $\alpha >0$. Boundary conditions $R(R_0)=0$ and $\Big(\frac{dR}{dr}\Big)_{r=0}=...
0
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4answers
94 views

Solving Differential Equation -trouble-

Given the equations: $$\dfrac{dy}{dx} - \dfrac{1}{x}y = \dfrac{1}{x^3}y^2,$$ and $y(1) = 1$, I am supposed to solve for $y$. Eventually through my work, I find $$x^{\color{red}1}v = -\int \dfrac{1}{x^...
1
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0answers
19 views

Solving ODE with irregular singular point

I want to solve the following ODE $$x''(z)+ \frac{\frac{d}{dz} \left(\frac{f(z)}{z^2}\right)}{\frac{f(z)}{z^2}}x'(z)+\frac{\omega^2}{(f(z))^2}x(z)=0$$ where $$f(z) = 1- 4 \left(\frac{z}{z_*}\right)^...
1
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1answer
49 views

General solution of a nonlinear differential equation

Nonlinear differential equation gone beyond my field of expertise but I'd like to know the details of a problem and to do that I should know the general solution of the following nonlinear ...
11
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0answers
110 views

Solving $f'(x) = f(f(x))$ [duplicate]

Is there any solution to the differential equation $f'(x) = f(f(x))$? I couldn't find any information on this kind of DE
0
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1answer
46 views

solution of the ODE $u du =ydx+xdy$

In this case $u=u(x,y)$. When I saw this I just went on to taking iindefinite integral both sides yielding $ u^2=4xy+K $. Yet, the book I am using now got $udu=d(xy)$, which yields $ u^2=2xy+K$. I'm I ...
3
votes
2answers
98 views

Solution of $f(x)^2\dfrac{d^2}{dx^2}f(x)=x$

I am stuck in finding the solution of this apparently simple differential equation: $$f(x)^2\dfrac{d^2}{dx^2}f(x)=x$$ with$f(0)=a$ and $f(0)'=b$ Using Maple the solution seems to be a combination of ...
0
votes
3answers
60 views

How to integrate the following: $\int{\frac{2y'y}{y^2+1}dx}$

I have encountered the following problem: $\int{\frac{2y'y}{y^2+1}dx}$ According to wolfram the solution is: $log(y^2 + 1)$ How was this solution derived and which rules were used?
6
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4answers
2k views

How do you solve the following separable differential equation: y'y = y + 1?

I just started learning about differential equations and encountered following equation: $$ y'y = y +1 $$ Wolfram alpha provided the following explanation: here But I'm not sure how the integration ...
1
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1answer
37 views

Solving ODE for practice

I'm doing self study and I can't solve this equation: $$ax + \ln y = y + b$$ Where I'm supposed to eliminate the arbitrary constants. The given answer is $(y - y^2)(y'') = (y')^2$ But my workings ...
0
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0answers
27 views

Singularities in a PDE

This is more of a general question rather than anything specific but I was just wondering if someone could point me toward resources which discuss singularities in a PDE rather than in an ODE (by ...
1
vote
2answers
82 views

Nonlinear 2nd order ODE

I have been looking at numerical solutions to the following nonlinear Bessel-type ODE: $$ xy'' + 2 y' = y^2 - k^2, $$ where k is a constant. In general, $y = \pm k$ is an asymptotic solution, and as $...
0
votes
1answer
39 views

Easier solution to first order non-linear differential equation?

Im am dealing with this differential equation: $$m\frac{dv}{dt}=mg-kv^2$$ where $m,g,k$ are constants. I am able to solve this by treating this as a separable differential equation, but that method ...
3
votes
2answers
59 views

Sum of square of function

If $f'(x) = g(x)$ and $g'(x) = - f(x)$ for all real $x$ and $f(5) =2 =f'(5)$ then we have to find $f^2$$(10) + g^2(10)$ I tried but got stuck
0
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1answer
49 views

Is going from $V_{\text{L}} = L \frac{di_{\text{L}}}{dt}$ to $\frac{ V_{\text{L}} } {i_L} = L \frac{d}{dt}$ allowed?

The Laplace transform of $\frac{d}{dt} f(t)$ would be sF(s), when f(0)=0, which is something you can find in a Laplace transform table. If there is a rule that prohibits mathematical operations from $...
0
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0answers
32 views

Heat Equation : Commutation of partial derivatives and summation

I'am having a problem when checking the validity of the solution i found for the heat equation: \begin{cases} U_{t}(x,t)=U_{xx}(x,t),\ {(x,t)\in (0,1)\times(0,+\infty)} \\ U(x,0) = x^2 - x\\U(0,t)=0\...
1
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1answer
25 views

Implicit method for ODE

I want to numerically solve the initial value problem of ordinary differential equation for function $u=u(t)$: $$ u'(t)=L(u). $$ I find an second-order implicit method: $$ u_{n+1}=u_n+\Delta t L(u_{n+...
0
votes
2answers
32 views

Matrix Differentiation of $-a^T X^T y$ on $a.$

In short; what is the correct differentiation of: $$S(a)=-a^TX^Ty$$ when differentiating: $$0=\frac{∂S}{∂a}= \;?$$ Long story is; I know that: $$J(a)=\underbrace{\:\:\:a^TX^TXa\:\:\:}_u\:\...
2
votes
0answers
29 views

'2nd order' Picard Iteration

I'm self-studying differential equations using MIT's publicly available materials. One of the problem set exercises deals with what I'm calling a second order Picard Iteration. To be explicit, we ...
0
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0answers
13 views

Are there any examples of higher order ireducible linear differential operators?

Given a monic, linear differential operator $L = D^n + f_{n-1}(x)D^{n-1} + \dots + f_1(x) + f_0(x)$, say $f_0, \dots, f_{n-1}$ analytic for simplicity's sake, we say that $L$ is irreducible if there ...
1
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1answer
382 views

is there are specific way to solve coupled first-order differential equations with coefficients varying?

suppose I have "n" coupled differential equation represented by the matrix, Y• = A Y , where Y• is the column matrix containing first derivatives, namely, y1•(t), y2•(t), ... yn&...
3
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1answer
26 views

Polar coordinates for vector field to find sticking flow

I am currently working on an impacting system which is basically just a spring damper and a circular enclosure. Because of the rotational symmetry of the problem I need the vector field in polar ...
0
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1answer
31 views

Differentiation of$ f^{-1}(x)$, where $f(x)=e^{x-1}+x^3-4x^{-3}+10$

if $f(x)=e^{x-1}+x^3-4x^{-3}+10$ then find $\frac{d(f^{-1}(x))}{dx}$ at $x=8$..... (here $f^{-1}(x)$ means inverse of $f(x)$) I was trying to solve this problem but was not able to find out the way ....
0
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2answers
28 views

Solve the following IVP with explicit solution

Given: $4 dx + 2 {cos(y)\over sin(y)} dy = 0, \qquad y(0) = {\pi\over 2}$ I've already test the exactness which is $0$ for the result of both derivatives. Then I found the potential function is ...
0
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1answer
20 views

How to find the total derivative of a function $f_a(y(t),x(t))$ subjected to parametric change with the parameter $a$

It is well known to find the total derivative of a function $f(x(t),y(t))$. I consider it as $Td_f$. What, if the function depends upon some parameter, say, $a$. Then, how to find the total derivative ...
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votes
4answers
55 views

Finding the polynomial [on hold]

Find a nontrivial polynomial function $p(x)$ such that $p(2x)=p'(x)p''(x)\not=0$
2
votes
1answer
177 views

Numeric solution of third order ODE

I need to solve the following third order (non-linear) ODE by numerical methods: \begin{equation}\tag{1} h^{3} \dfrac{d^3 h}{d x^3} = h-1. \end{equation} By assumption, the solution should approach $ ...
8
votes
4answers
2k views

How unique is $e$?

Is the property of a function being its own derivative unique to $e^x$, or are there other functions with this property? My working for $e$ is that for any $y=a^x$, $ln(y)=x\ln a$, so $\frac{dy}{dx}=\...
3
votes
3answers
61 views

Second-order non-linear ODE

$2tx'-x=lnx'$ I differentiated both sides with respect to x: $x'+2tx''=\frac {x''}{x'}$ Substituting $p=x'$, $p+2tp'=\frac{p'}{p}$ But I have no clue what can I do from here on. EDIT: $t$ is the ...