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

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3
votes
2answers
96 views

Should we re-define Sine? [on hold]

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
votes
1answer
25 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) = -\...
1
vote
2answers
39 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 ...
-1
votes
1answer
19 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
vote
0answers
10 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
votes
1answer
17 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
vote
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
59 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
vote
1answer
66 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
votes
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
votes
3answers
55 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 $...
-1
votes
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
112 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
votes
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
vote
0answers
18 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
vote
1answer
48 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
votes
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
votes
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
59 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?
7
votes
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
vote
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
votes
0answers
23 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
81 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
votes
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
votes
0answers
31 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
vote
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
30 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
votes
0answers
12 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
vote
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
votes
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
votes
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
votes
2answers
27 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
votes
1answer
19 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 ...
-2
votes
4answers
54 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 ...
1
vote
2answers
383 views

Problem related with boundary value problem and eigenvalue, eigenfunctions

I was looking at previous year exam papers and was stuck on the following problem: For the boundary value problem, $\,\,y''+\lambda y=0; y(0)=0,y(1)=0, \,\,\exists$ an eigenvalue $\lambda$ ...
1
vote
1answer
31 views

First principle of differentiation needs to approximate a sufficiently small integral as area?

$$y(t+\Delta t) = e^{- \int_{t}^{t+\Delta t}H(t')dt'}y(t)$$ is the solution to the differential equation $$\frac{dy}{dt} = -H(t)y$$, $H(t)$ and $y$ are scalar. However, in showing that $$y(t+\Delta t)...
0
votes
1answer
38 views

How small need it be to approximate integral as one area of product of initial value times length.

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ How small need $\Delta t$ be to approximate $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$$ as $$a(t)\Delta t$$ [ Just one ...
2
votes
1answer
71 views

Integrating factor

Can anyone give me some hints as to how to solve the following question? I have to show that the equation below has an integrating factor of the form $t^2\theta^c$ where $c$ is an integer. $\...
0
votes
1answer
32 views

How can I use this initial condition for the heat equation

How can I use the following initial condition for a partial differential equation describing heat diffusion? $$f(x) = \begin{cases} 0, & 0<x<0.45 \\ 1, & 0.45<x<0.55 \\ 0, & 0....
1
vote
0answers
44 views

Singularities in an Equation

these might sound like extremely trivial questions but since my background is more in probability and statistics I'm not too sure what to do, or even what to read up on to understand what to do. I ...
-4
votes
0answers
37 views

Help me solve this differential equation

I have came across where the complementary function has complex roots. Please help me solve this equation Also please use this method (operator d method ) as I know only this method I am uploading ...