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

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11
votes
3answers
660 views

Infinite Series $\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$

How can we prove that? $$\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}=\frac{1}{3}e^{\frac{-x}{2}}x\left(e^{\frac{3x}{2}}-2\sin\left(\frac{\pi+3\sqrt{3}x}{6}\right)\right).$$ I think if we write the taylor ...
3
votes
2answers
270 views

Solving a differential equation?

I'm trying to analyze the transient state of a RC circuit. My book gives me the following differential equation: $$\frac{d(v(t))}{dt} + av(t) = c$$ for some constants $a$ and $c$. The book thens ...
29
votes
4answers
1k views

What am I doing when I separate the variables of a differential equation?

I see an equation like this: $$y\frac{\textrm{d}y}{\textrm{d}x} = e^x$$ and solve it by "separating variables" like this: $$y\textrm{d}y = e^x\textrm{d}x$$ $$\int y\textrm{d}y = \int e^x\textrm{d}x$...
7
votes
4answers
1k views

How to solve $y''' - y = 2\sin(x)$

$$y''' - y = 2\sin(x)$$ I'm doing differential equations and know pretty much all methods of solving them, but I haven't come across anything of a higher order than second yet. How do I go about ...
1
vote
3answers
1k views

differential equation : non-homogeneous solution, finding YP

hi i have a problem for this Differential Equations : $$ \frac{d^{3}y}{dx^3} - 9\frac{dy}{dx} = 10 - 4x $$ i know first we must solve the homogeneous equation: and my result is : $C_1 + C_2e^{3x} + ...
1
vote
1answer
993 views

Consider the following Sturm-Liouville problem

Consider the following Sturm-Liouville problem: $$X''+\lambda X=0,\quad X'(0) = 0,\, X(\pi) = 0,$$ where $X = X(x)$. Find all positive eigenvalues and corresponding eigenfunctions of the problem. Is ...
1
vote
1answer
391 views

differential equations, diagonalizable matrix

I have a question of differential equations of the form. $\textbf{x}'(t)=A*\textbf{x(t)}$, where x is an n-dimensional matrix, and A is an n*n real matrix. I have learned to solve this if a is ...
3
votes
3answers
3k views

How do you solve the Initial value probelm $dp/dt = 10p(1-p), p(0)=0.1$?

The problem is... $$ \frac{dp}{dt} = 10p(1-p),$$ $p(0)=0.1$. Solve and show that $p(t) \to 1$ as $t\to \infty.$ I know this is probably really simple, I was trying to go down the line of finding ...
69
votes
7answers
2k views

Does a non-trivial solution exist for $f'(x)=f(f(x))$?

Does $f'(x)=f(f(x))$ have any solutions other than $f(x)=0$? I have become convinced that it does (see below), but I don't know of any way to prove this. Is there a nice method for solving this kind ...
7
votes
3answers
1k views

Solving $-u''(x) = \delta(x)$

A question asks us to solve the differential equation $-u''(x) = \delta(x)$ with boundary conditions $u(-2) = 0$ and $u(3) = 0$ where $\delta(x)$ is the Dirac delta function. But inside ...
2
votes
3answers
690 views

Find the form of a second linear independent solution when the two roots of indicial equation are different by a integer

Consider the differential equation $$x^2y''+3(x-x^2)y'-3y=0$$ $(a)$ Find the recurrence equation and first three nonzero terms of the series solution in powers of $$ corresponding to the larger root ...
1
vote
2answers
427 views

Finding the general solution of a sixth degree differential equation

Find a differential equation whose solutions are $y_1 = e^{2x} + e^{-4x}\sin(3x)$ and $y_2 = e^{-2x} + 5e^{2x}$. Am I supposed to assume that $y_1$ and $y_2$ can take the forms: $y_1 = Ae^{2x} + e^{...
17
votes
1answer
16k views

Best Book For Differential Equations?

I know this is a subjective question, but I need some opinions on a very good book for learning differential equations. Ideally it should have a variety of problems with worked solutions and be ...
8
votes
6answers
302 views

Are the any non-trivial functions where $f(x)=f'(x)$ not of the form $Ae^x$

This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is ...
17
votes
4answers
1k views

The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions on a Rectangle

I am interested in solving the following biharmonic eigenvalue problem. $$\begin{array}{cccc} & \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y \le b \\ &...
18
votes
3answers
706 views

Find $f$ where $f'(x) = f(1+x)$

Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function such that $$f'(x) = f(1+x)$$ How can we find the general form of $f$? I thought of some differential equations, but not sure how ...
3
votes
1answer
331 views

Finding a solution basis

Find a real solution basis of $$y'=\left( \begin{matrix}-1&-2&0\\0&2&0\\-1&-3&2\\ \end{matrix} \right)y.$$ The characteristic equation of this matrix is $$P(t) = (1-t)(2-t)^2.$...
0
votes
4answers
2k views

$dy/dx = y \sin x-2\sin x$, $y(0) = 0$ — Initial Value Problem

$$\frac{dy}{dx} = y\sin x-2\sin x, \quad y(0) = 0.$$ Initial Value Problem Hint says: Find an integrating factor
8
votes
4answers
281 views

Verify $y=x^aZ_p\left(bx^c\right)$ is a solution to $y''+\left(\frac{1-2a}{x}\right)y'+\left[(bcx^{c-1})^2+\frac{a^2-p^2c^2}{x^2}\right]y=0$

In order for the question that I have to make any sense I must first include some background information as given in my textbook: The standard form of Bessel's differential equation is $$x^2y^{\...
2
votes
4answers
607 views

Is this a correct/good way to think interpret differentials for the beginning calculus student?

I was reading the answers to this question, and I came across the following answer which seems intuitive, but too good to be true: Typically, the $\frac{dy}{dx}$ notation is used to denote the ...
4
votes
1answer
162 views

Are Exponential and Trigonometric Functions the Only Non-Trivial Solutions to $F'(x)=F(x+a)$?

Are exponential & trigonometric functions the only non-trivial solutions to $F'(x)=F(x+a)$? $F(x)=0$ would be the trivial solution. Then, for $a=0$ (or $a=2\pi i$), we have $F(x)=e^x$, and for $...
1
vote
3answers
178 views

Deducing the exact solution of a ODE

In page 53 of Arieh Iserles's A first course in the numerical analysis of differential equations, he presents the following ODE: $(\vec{y})'=\Gamma\cdot\vec{y}$, $\vec{y}(0)=\vec{y_0}$ Using the ...
1
vote
2answers
883 views

Legendre Polynomials: proofs

Does any one know, how to compute any of those two things? The relationship between Legendre polynomials and Shifted Legendre Polynomials. $\displaystyle\int_{-1}^1P_n^2(x)dx=\dfrac{2}{(2n+1)}$ for $...
1
vote
1answer
276 views

About the Legendre differential equation

Consider the Legendre differential equation $$ (1-x^2) y'' - 2xy' + n(n+1)y = 0 $$ Then its solution is given by $$ y = c_1 P_n (x) + \text{an infinite series} $$ In fact $y = c_1 P_n (x) + c_2 Q_n (x)...
66
votes
4answers
75k views

Teenager solves Newton dynamics problem - where is the paper?

From Ottawa Citizen (and all over, really): An Indian-born teenager has won a research award for solving a mathematical problem first posed by Sir Isaac Newton more than 300 years ago that has ...
25
votes
2answers
1k views

please solve a 2013 th derivative question?

$ f(x) = 6x^7\sin^2(x^{1000}) e^{x^2} $ Find $ f^{(2013)}(0) $ A math forum friend suggest me to use big O symbol, however have no idea what that is, so how does that helping?
8
votes
2answers
3k views

Exponential of the differential operator

I am not sure whether this question is even well-posed. But today I learnt that $e^Df(x) = f(x+1)$ where $D$ is differential operator and $$e^D \triangleq \sum_{i=0}^{\infty} \frac{D^i}{i!}.$$ (ref. ...
13
votes
5answers
5k views

Functions that are their Own nth Derivatives for Real n

Consider (non-trivial) functions that are their own nth derivatives. For instance $\frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x$ $\frac{\mathrm{d}^2}{\mathrm{d}x^2} e^{-x} = e^{-x}$ $\frac{\mathrm{d}^...
15
votes
8answers
6k views

Proof for exact differential equations shortcut?

Today in my math class, we learned about exact differential equations. During class, our teacher first taught us the accepted way to solve exact equations, but then, told us of a shortcut that one of ...
7
votes
4answers
448 views

Show $f''+vf' +\alpha^2 f(1-f)=0$ has solutions satisfying $\lim_{x \to - \infty}f=0$ and $\lim_{x \to \infty}f=1$ given $v\leq -2\alpha < 0$

I posted this question before but I took a completely different approach here, that's why I reposted as my previous question was already very long and took a different approach from here. I am given ...
3
votes
2answers
441 views

Solving inhomogenous ODE

I have an inhomogenous ODE. The main issue here is variables are matrices. It is bit of matrix calculus. A solution would be highly appreciated interms of x . I guess we can use same methods for ...
6
votes
4answers
1k views

Finding a non constant solution to $ (x')^2+x^2=9 $

How do I find a non-constant solution this equation? I've tried to solve for $x$, but the final answer should be in the form of $x(t)=...$ $ (x')^2+x^2=9 $ I'm not sure where to start.
5
votes
1answer
2k views

Finding Weak Solutions to ODEs

I'm wondering if anyone has a reference to a good set of notes on finding weak (distributional) solutions to ODEs, or has any tips or tricks. For example, $$ xy^\prime=0 $$ has a classical solution ...
4
votes
2answers
206 views

Help with special function differential equation

this is my first time to use this site. Please let me know if the equations are unreadable, latex isn't my first language. We've been covering Legendre, Bessel, and Confluent Hypergeometric ...
2
votes
1answer
73 views

Show that $y_1$ and $y_2$ are not Linearly Independent

Suppose that $y_1(x)$ and $y_2(x)$ are solutions of the differential equation $y''+py'+qy=0$ on $I$. How can I show that if $y_1$ and $y_2$ vanish at the same point then they are not linearly ...
11
votes
3answers
479 views

Why is it legitimate to solve the differential equation $\frac{dy}{dx}=\frac{y}{x}$ by taking $\int \frac{1}{y}\ dy=\int \frac{1}{x}\ dx$?

Answers to this question Homogeneous differential equation $\frac{dy}{dx} = \frac{y}{x}$ solution? assert that to find a solution to the differential equation $$\dfrac{dy}{dx} = \dfrac{y}{x}$$ we may ...
2
votes
2answers
257 views

Nonlinear Systems

I'm given the system of equations $x'(t) = -2y + x^2$, $y\,'(t) = 4x - 2y$, and I am supposed to find all steady state solutions. Once I find them, I need to determine what kind of steady states I ...
1
vote
2answers
208 views

Show that $y(t) = t$ and $g(t) = t \ln(t)$ are linearly independent

I need to show that $y(t) = t$ and $g(t) = t \ln(t)$ are linearly independent. I thought I could use the Wronskian as follows: $y'(t) = 1$ $g'(t) = 1 + \ln(t)$ So $W(y, g) = (t)(1 + \ln(t)) - t \...
1
vote
1answer
96 views

Uniqueness and Existence problem

I just need a bit of help with this question. If I know that $dg/dx = g^2$, and that $g(0) = g_0$, then I can solve: $$ dg/dx = g^2\\ \frac{1}{g^2} dg = dx \\ -\frac{1}{g} = x + \hat c \\ -g = \...
1
vote
2answers
108 views

find an approximate solution, up to the order of epsilon

The question is to find an approximate solution, up to the order of epsilon of following problem. $$y'' + y+\epsilon y^3 = 0$$ $$y(0) = a$$ $$y'(0) = 0$$ I tried to solve the given problem using ...
47
votes
7answers
1k views

What is the optimal path between $2$ fixed points around an invisible obstructing wall?

Every day you walk from point A to point B, which are $3$ miles apart. There is a $50$% chance each walk that there is an invisible wall somewhere strictly between the two points (never at A or B). ...
40
votes
2answers
3k views

Is it mathematically valid to separate variables in a differential equation?

I read the following statement in a book on Calculus, as part of my mathematics course: Technically this separation of $\frac{dy}{dx}$ is not mathematically valid. However, the resulting ...
48
votes
0answers
575 views

Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc… & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...
23
votes
2answers
450 views

Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$

Fix some even functions $f$ and $g$, differentiable, such that $f(0)=g(0)=0$ and $f'(0)=g'(0)=0$, and consider the autonomous differential system $$\left\{\ \begin{array}{lcr}x'&=&-y+f(x)\\ y'&...
3
votes
5answers
3k views

Could you recommend some classic textbooks on ordinary/partial differential equation?

I love R.Courant and F.John's Introduction to Calculus and Analysis because of its wide coverage, precise description and friendly written style. Are there any classic textbooks like it on ODE/PDE? ...
2
votes
1answer
552 views

Show $\nabla^2g=-f$ almost everywhere

let a continuous function $f(x,y,z)$ be absolutely continuous over every bounded region and let it be in $L^1$ that is $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} |{f(x',...
14
votes
3answers
762 views

When do the Freshman's dream product and quotient rules for differentiation hold?

This is motivated by looking at the calculus exams of some of my undergraduate students. A recurring mistake is assuming that the derivative of the product of functions is a product of derivatives and ...
18
votes
5answers
933 views

history of the double root solution of $ay''+by'+cy=0$

Motivation: It is a well-known fact that $ay''+by'+cy=0$ has solutions which are found from substituting the ansatz $y=e^{\lambda t}$ into the DEqn. It turns out that we replace the calculus problem $...
7
votes
2answers
20k views

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order ...
3
votes
1answer
321 views

Nonlinear Differential Equation question

I have a nonlinear Diffeq: $$\frac{d^2x}{dt^2}+\beta \frac{dx}{dt}+\epsilon \times e^{- \lambda x} = f(t) $$ where $f(t)$ is a function that is known, and $\beta$ and $\lambda$ are constants that ...