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

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11
votes
3answers
648 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
264 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 ...
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 ...
26
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 ...
1
vote
1answer
964 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
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
380 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 ...
3
votes
1answer
328 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) = ...
2
votes
3answers
668 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 ...
15
votes
1answer
15k 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 ...
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 ...
1
vote
2answers
424 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} + ...
65
votes
4answers
74k 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 ...
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
696 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 ...
1
vote
2answers
113 views

Does $e^{tx} + x = t - 1$ yield a solution/s to $\frac{dx}{dt} = \frac{e^{-tx} - x}{e^{-tx} + t}$?

Problem: Use Existence Theorem to determine if $x(t)$ implicitly defined by $$e^{tx(t)} + x(t) = t - 1 \tag{*}$$ yield a solution/s to $$\frac{dx}{dt} = \frac{e^{-tx} - x}{e^{-tx} + ...
1
vote
3answers
176 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 ...
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
66
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 ...
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. ...
15
votes
8answers
5k 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
6answers
261 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 ...
2
votes
4answers
577 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
157 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 ...
1
vote
1answer
271 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 ...
39
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 ...
9
votes
6answers
1k views

Differentials Definition

Please define differentials rigorously such that they give a consistency to their use in the following links. I have read Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? What is the practical ...
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?
7
votes
4answers
446 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 ...
4
votes
2answers
1k views

Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$.

Given the function $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} & x \neq 0 \\ 0 & x = 0 \end{array}\right. $$ show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$. So I have to show ...
3
votes
2answers
417 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.
4
votes
2answers
149 views

Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $ F (x,y,y')=y'^2(x)$ After solving the ...
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
202 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
71 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 ...
1
vote
1answer
77 views

Differential equation for Harmonic Motion

Particle undergoes simple harmonic motion. Initially Its displacement is $1$, velocity $1$ and acceleration is $-12$ Compute displacement and acceleration when the velocity is square root of $8$. ...
0
votes
1answer
605 views

Orthonormal vectors in Polar coordinates, show $\hat{e}_R=\frac{(x,y,z)}{r}$

Definitions Unit vector has length 1. Orthonormal vectors are orthogonal and unit vectors. RobJohn's suggestions for the basis in polar coordinates, here, satisfy the criteria but how can ...
11
votes
3answers
475 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
249 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
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
113 views

stability of a linear system

The linear system: $y''(t)+4y'(t)=4(\lambda -1)y(t)+z(t)$ $z'(t)=(\lambda -3)z(t)$ Determine the stability of the system as a function of the parameter $\lambda\in\mathbb{R}$. ...
1
vote
0answers
353 views

Green function Sturm Liouville equation problem …

I need help with this new one, I don't even know how to start... $$Ly=-y''-y$$ $$y(0)=y(1)=0$$ I started doing this exercise and came to the conclusion that the result is given to us by $$G(x, ...
1
vote
2answers
850 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
2answers
201 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 ...
0
votes
1answer
65 views

How to solve implicit differential equation?

Suppose we go from the equation and go backwards: $$y=c\,e^x+e^{2x+c}$$where $c$ is any arbitrary constant. Now, $$y'=c\times(e^x)+(2e^c)\times(e^{2x}).$$ Solving for $c$: we get ...
0
votes
1answer
313 views

ODE $y''+ 9y = 6 \cos 3x$

I have this equation: $y''+ 9y = 6 \cos 3x$ $$ m^2 + 9m = 0\\ m(m + 9) = 0\\ m_1 = 0;\\ m_2 = -9;\\ y_h = c_1 + c_2 e^{-9x}\\ r(x) = 6\cos3x\\ y_p = K\cos3x + M\sin3x\\ y'_p ...
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). ...
45
votes
0answers
511 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 ...