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

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1answer
692 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 ...
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2answers
116 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 ...
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3answers
415 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 ...
6
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4answers
476 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 ...
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3answers
226 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 ...
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3answers
698 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} + ...
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3answers
732 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 ...
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3answers
994 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
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1answer
162 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 ...
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2answers
323 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} + ...
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4answers
779 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 ...
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4answers
70k 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 ...
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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
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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 ...
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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 $dy/dx$ not a ratio? What is the practical difference between a ...
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3answers
582 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
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4answers
404 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
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3answers
334 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 ...
4
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2answers
173 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 ...
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1answer
64 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 $8$. ...
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1answer
469 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 ...
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1answer
81 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 = ...
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2answers
97 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 $\lambda\in\mathbb{R}$. ...
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0answers
197 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, ...
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2answers
171 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 ...
0
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1answer
106 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 ...
3
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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? ...
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4answers
876 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 know it can't have any polynomial solutions. If $f$ has degree $n$, then $f(f(x))$ has degree $n^2$, while $f'(x)$ has degree $n-1$. I ...
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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?
12
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1answer
636 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 ...
7
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1answer
368 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 ...
6
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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. ...
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5answers
555 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 $a,b,c, A,B$ ...
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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 $\frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x$ $\frac{\mathrm{d}^2}{\mathrm{d}x^2} e^{-x} = e^{-x}$ ...
3
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1answer
214 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 ...
12
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2answers
798 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 ...
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3answers
305 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 ...
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2answers
284 views

Deriving the addition formula for the lemniscate functions from a total differential equation

The lemniscate of Bernoulli $C$ is a plane curve defined as follows. Let $a > 0$ be a real number. Let $F_1 = (a, 0)$ and $F_2 = (-a, 0)$ be two points of $\mathbb{R}^2$. Let $C = \{P \in ...
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5answers
561 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.
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1answer
932 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
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1answer
493 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 ...
4
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1answer
284 views

Using the Lambert W to express a solution of a differential equation.

I solved a differential equation some time ago and I need to solve for $y$. How can we solve for $y$ using the Lambert W function? $$C_1+x = e^y+Cy$$
3
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3answers
427 views

Integrating factor in linear differential equations

I'm watching various videos on differential equations and they all say that linear differential equations are on the form: $y' + P(x)y = Q(x)$ where $P(x)$ is the integrating factor and equals ...
2
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1answer
95 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
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1answer
6k views

Complementary Solution = Homogenous solution?

I have calculated solutions to homogenous equations but is the complementary solution mentioned here the same as the homogenous solution? Let's take example $y''-3y'+2y=\cos(wx)$ and now ...
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2answers
1k views

`“Variation of Constant”` -method to solve linear DYs?

My school instructs to use some method called "variation of constant" (first page here) to solve linear DY more in my earlier question here. I think I solved the ...
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3answers
290 views

Nicer expression for the following differential operator

I have the following sequence of differential operators: $$D_n = \underbrace{t \partial_t t \partial_t \dots t \partial_t}_{\text{$n$ times}}.$$ Is there any expression involving a sum of "normal" ...
3
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3answers
331 views

Differential equation: autonomous system

This isn't homework. I have no idea what theorems I should be looking at to solve this. Guidance, partial and total solutions are all welcomed. Let $f$ be a locally lipschitz function in an open ...
3
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2answers
138 views

Find a general control and then show that this could have been achieved at x2

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
3
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2answers
327 views

Generating unitary matrices numerically - “close” to the identity element

EDIT: broke this into two parts - for these were two different questions. For numerically obtaining the stabilities of a matricial equation, i need to generate an ensemble of matrices that are ...