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

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7
votes
4answers
736 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 ...
3
votes
2answers
206 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 ...
8
votes
3answers
519 views

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

How can we obtain following formula? $$\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}=\frac{1}{3}e^{\frac{-x}{2}}x\left(e^{\frac{3x}{2}}-2\sin\left(\frac{\pi+3\sqrt{3}x}{6}\right)\right).$$ I think if we ...
1
vote
1answer
806 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
958 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} + ...
3
votes
3answers
1k 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 ...
1
vote
1answer
249 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 ...
2
votes
3answers
378 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 ...
21
votes
4answers
929 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 ...
6
votes
3answers
810 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
370 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} + ...
61
votes
4answers
71k 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
3answers
629 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 ...
7
votes
1answer
6k 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 ...
2
votes
1answer
202 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) = ...
0
votes
4answers
1k 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
63
votes
7answers
1k 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 ...
14
votes
8answers
3k 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 ...
1
vote
3answers
130 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 ...
-2
votes
2answers
86 views

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$ [closed]

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$... if $x > 0$ and $cos(x)$ $> 0$
7
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 readviesr Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? What is the ...
22
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
423 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 ...
7
votes
2answers
2k 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. ...
3
votes
3answers
353 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 ...
3
votes
2answers
946 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 ...
6
votes
5answers
826 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
1answer
109 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 ...
4
votes
2answers
183 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 ...
1
vote
1answer
70 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
524 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 ...
1
vote
1answer
87 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
102 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
285 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
184 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
161 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
151 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 ...
21
votes
0answers
194 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 ...
3
votes
5answers
2k 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? ...
3
votes
1answer
287 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 ...
12
votes
1answer
700 views

Recursive solutions to linear ODE.

When finding the solutions to the simple ODE $$ y'- mxy= x^n \text{ ; } y(0) = 0$$ I found the following: Let $P_n$ be the particular solution for each integer exponent $n$. Then if we define ...
12
votes
5answers
3k views

Functions that are their Own nth Derivatives for Real n

Consider (non-trivial) functions that are their own nth derivatives. For instance $\frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x$ $\frac{\mathrm{d}^2}{\mathrm{d}x^2} e^{-x} = e^{-x}$ ...
7
votes
1answer
370 views

$\frac{dS}{d\rho}$ Factor arising

To get details see: equations 29,30,31,34,44,50,51 We have known some solitary wave solutions, given by(equations 1 to 5) $$ \phi_1=p_1\cos \tau \tag{1}$$ $$\phi_2=\frac16 ...
16
votes
5answers
699 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 ...
4
votes
5answers
588 views

Does this ODE question have closed form solution?

These days, I am struggling with following ODE problem when I build up my research model: $1/2f''(x)+a(b - x) f'(x) -(c+ e^{A+Bx})f(x)=0$ where f(x) is a smooth function, and $a,b,c, A,B$ ...
5
votes
1answer
897 views

Looking for a logically coherent book for the self-study of differential equations

I'm looking for a logically coherent book for the self-study of differential equations. Let me clarify. By logically coherent, I don't mean proofs of the limit laws, uniqueness theorems etc. By ...
3
votes
1answer
226 views

How to solve this recurrence Relation - Varying Coefficient

Sir,I have two questions related to this recurrence relation. It has been messing with me for long. Because of this I couldn't proceed my work for some time .This contains a polynomial term n+2 in ...
1
vote
1answer
1k views

The number of solutions to an $n^{th}$ order differential equation.

For an $n$th order differential equation, why are there always $n$ solutions? Why exactly $n$, not $n - 1, n+1$ or infinite many? Addendum by LePressentiment : This is motivated by P176 on ...
12
votes
2answers
932 views

Sum of derivatives of a polynomial

Let $p(x)$ be a polynomial of degree $n$ satisfying $p(x)\geq 0$ for all $x$. That is, for all $x$, $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \geq 0$, $a_n\neq 0$. Show that ...
12
votes
3answers
320 views

Differential equations that are also functional

I was toying with equations of the type $f(x+\alpha)=f'(x)$ where $f$ is a real function. For example if $\alpha=\frac{\pi}{2}$ then the solutions include the function $f_{\lambda,\mu}(x)=\lambda ...