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

learn more… | top users | synonyms (1)

6
votes
2answers
2k views

Question regarding usage of absolute value within natural log in solution of differential equation

The problem from the book. $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6 -y$ I understand the solution till this part. $\ln \vert 6 - y \vert = x + C$ The solution in the book is $6 - Ce^{-x}$ ...
6
votes
2answers
553 views

Show that the equation $y'=f(x,y)$ has a periodic solution

Given information: Let $f$ be a continuous function defined for all $(x,y) \in \mathbb{R}^2$. Let $f$ also satisfy a Lipschitz condition with respect to $y$. Let $f$ be periodic with respect to $x$ ...
6
votes
3answers
276 views

Solving $\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$

I recently came across this equation : $$\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$$where $f \in \mathcal{C}^1(\mathbb{R}, \mathbb{R})$. I've done the following, but I'm stuck at ...
6
votes
1answer
77 views

Nonlinear 1st order ODE

$$y'(x)=\frac{\cos (y(x))+y(x) \cos (x)}{x \sin (y(x))-\sin (x)}$$ Did I enter LaTeX correctly? I am self-learning the differential equation from a textbook and I need some help with above equation. ...
6
votes
1answer
125 views

How to solve the non-linear differential equation $y''=x-y^2$?

$y''(x)=x-y^2(x)$ I'm particularly interested in solutions when $x>0$. I've performed asymptotic analysis and reached the conclusion that solutions must behave as $\pm\sqrt{x}$ when $x\rightarrow ...
6
votes
2answers
119 views

Integrating Factor - Not getting the answer given?

$y' - 4y = t$ My integrating factor is $e^{-4t}$ $\int e^{-4t}y'$ - $\int 4e^{-4t}y$ = $\int te^{-4t}$ $\int (e^{-4t}y)'$ = $\int te^{-4t}$ $e^{-4t}y$ = $-4te^{-4t}$ - $e^{-4t}$ I end up with ...
6
votes
1answer
130 views

Rank of a jet bundle of a vector bundle.

I am trying to understand the jet bundles but currently I am stuck on the following questions: Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) ...
6
votes
2answers
84 views

Geometric series of an operator

In solving a first order linear differential equation $(1-D)y=x^2$ where $D\equiv \frac{d}{dx}$ the way I learnt was that we proceed as ...
6
votes
2answers
342 views

Solving $(y+x^4y^2)dx+xdy=0$

How to solve (in terms of $y$) $(y+x^4y^2)dx+xdy=0$. I know I'm supposed to multiply by an integrating factor to turn this equation into an exact equation. In the previous exercise I proved that in ...
6
votes
1answer
191 views

The Weierstrass substitution in differential equations

\begin{align} y & = \tan\frac\theta2 \\[8pt] \frac{1-y^2}{1+y^2} & = \cos\theta \\[8pt] \frac{2y}{1+y^2} & = \sin\theta \\[8pt] \frac{2\,dy}{1+y^2} & = d\theta \end{align} This, the ...
6
votes
1answer
207 views

numerical integration of equations of motion of large system of particles with lubrication forces

I have a large system of solid particles moving in the liquid. I use traditional Newtonian equations of motion for the particles. There are many different interaction forces between particles and the ...
6
votes
1answer
159 views

How do you solve $f'(x) = f(f(x))$?

A friend told me to solve the following differential equation: $$f'(x)=f(f(x))$$ I have no idea how to solve this! This doesn't seem to be an ordinary differential equation and I can't even solve ...
6
votes
1answer
107 views

Help needed in solving a system of DE

The system of DE is: $$\frac{dI}{db}=-\frac{b}{c}\frac{dJ}{db}-\frac{2ab+1}{2c}J$$ $$\frac{dJ}{db}=\frac{b}{c}\frac{dI}{db}-\frac{2ab-1}{2c}I$$ Assume that $a$ and $c$ are constants and both $I$ and ...
6
votes
1answer
143 views

Confluent Heun equation. Reduction to standard form.

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
6
votes
2answers
255 views

Proving Nonhomogeneous ODE is Bounded

I am trying to prove the following: Let $x(t)$ be a solution of the IVP $$ \dot x=A(t)x+h(t), $$ where $A(t), h(t)$ continuous on $1\le t<\infty$. Further assume that $$ \int_1^\infty \| ...
6
votes
3answers
166 views

What went wrong?

Intrigued by this question, one-dimensional inverse square laws, I started to try to find an answer and came up with what follows. However, I calculated the derivatives to double check myself, and ...
6
votes
1answer
264 views

$f(x)=\int_{0}^{+\infty} e^{-(t+\frac{1}{t})x}dt$ how to find $f(x)$?

$$f(x)=\int_0^{+\infty} e^{-(t+\frac{1}{t})x}\;dt$$ if while $ x>0 $ , $ f(x) $ has values I noticed some interesting relations for $f(x)$ as shown below: $$ \begin{align} t & ...
6
votes
1answer
1k views

Condition of ordinary differential equation to have a periodic solution

Recently I am reading a book about ODE, and I find a question that asks to prove that ODE has a periodic solution under some conditions. Consider the ODE $x'=f(t,x)$, where $x$ is a scalar and ...
6
votes
2answers
2k views

Finding integrating factor when IF will be a function of x and y

I'm not finding any resource or description or systemic methodology to find integrating factors when the integrating factor will be a function of both x and y. I'm on this problem, $$ ( y - xy^2 ) ...
6
votes
1answer
89 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
6
votes
1answer
98 views

Can we reconstruct a DE from its general solution?

If we think of a differential equation as being a 'problem' and the set of all solutions to the DE as being the 'general solution,' then my question is this: Question: Under what circumstances can ...
6
votes
1answer
182 views

Is there an integral form of Newton's method?

Warning : This seems like a silly sort of question, not the kind I'd ask out loud. The contraction mapping theorem is a basic tool for proving existence of, and finding solutions to, equations. Given ...
6
votes
2answers
83 views

I know this DE is solvable…

I need help with a seemingly simple looking diff equ $$ x\frac {d^{2}y} {dx^{2}}+2y=0 $$ $$ \rightarrow \frac {d^{2}y} {dx^{2}}+2\frac {y} {x}=0 $$ $v= (\frac {y} {x})$ substitution isn't working ...
6
votes
2answers
237 views

Differential Equation $y'=x e^y + \cos x$

I am new to differential equations. I tried to find a series solution for this equation, but I don't know how to solve it. \begin{equation} y'=x e^y + \cos x \\y(0)=1 \end{equation} Actually, the ...
6
votes
2answers
163 views

If $f(x)+f'(x)-\frac{1}{x+1}\int_{0}^{x}f(t)dt=0$ and $f(0)=0$, then what is $f'(x)$?

$f\in C^{1}[0,\infty)$, $f(0)=0$ and $$ f(x)+f'(x)-\frac{1}{x+1}\int_{0}^{x}f(t)dt=0 $$ then $f'(x)=$ ? I'v tried in the following ways. First, let $F(x)=\int_{0}^{x}f(t)dt$, then we are left to ...
6
votes
1answer
170 views

Solve $y''+(1-2x \cos x \cos 2x)y=0$

Solve the differential equation $$y''+(1-2x \cos x \cos 2x)y=0 \space $$
6
votes
2answers
226 views

Does $f'(x) = f(\ln(x))$ have an easily expressed solution?

This is motivated by this question, where we can consider a bound of the form $f((n+1)!) - f(n!) \le 2f(n)$. To find a function with a similar growth rate, I wondered if there's a technique for ...
6
votes
1answer
208 views

Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
6
votes
1answer
67 views

Tricky Integral equation - where to start?

How would you go about solving this? $$p(x,t)=C\exp\left[-x+\int_0^t\int_0^\infty y\,p(y,\tau)\,\mathrm{d}y\,\mathrm{d}\tau\right]$$ Here $p(x,t)$ is the time-dependent probability distribution of a ...
6
votes
1answer
82 views

Iterative method for matrix differential equation

Let $A$ and $X(t)$ be $n\times n$ matrices. I want to solve the matrix differential equation $$\dfrac{dX}{dt}(t)=AX(t)$$ with $X(0)=I$ (the $n\times n$ identity matrix) using the Picard iterative ...
6
votes
1answer
77 views

Solution of the IVP $\dot{y}=x^2+y^2,y(0)=0$

The solution of the IVP $$\dot{y}=x^2+y^2,y(0)=0$$ I know this IVP has a unique solution but I can't find the interval in which it has a unique solution can someone help me in finding the interval.
6
votes
2answers
214 views

How to interpret the meaning of “$y$ solves the DE” to have nice properties.

Assume that $I$ is an open interval $0 \in I$ $x$ varies in $I$ $y$ is a differentiable function of $x$. Now in the context of these assumptions, consider the following problem. ...
6
votes
1answer
100 views

Uniqueness of Ordinary Differential Equations in $D^{'}$, the space of Schwartz distribuitions

Let $m \in \mathbb{N}$. For $k=1,...,m$ let $a_k : \mathbb{R} \rightarrow \mathbb{C}$ be a $C^{\infty}$ function. And suppose that: $a_m(x) \neq 0 \; \forall x \in [x_0, \infty[$ And let P be the ...
6
votes
1answer
245 views

Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
6
votes
1answer
208 views

Completing the differential equation from exercise 10.23 in Tom Apostol's “Mathematical Analysis”

I found this answer, outlining the exercise, to be interesting. However, I have trouble solving the differential equation. The question starts by attempting to solve the following integral without ...
6
votes
2answers
254 views

Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem

A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume". At that time for me to write ...
6
votes
1answer
599 views

Solving differential equation $y^{(5)} + 2y^{(3)} + y' = 2x + \sin(x) + \cos(x)$

I'm trying to solve the differential equation $$L[y] = y^{(5)} + 2y^{(3)} + y' = 2x + \sin(x) + \cos(x)$$ using the method of undetermined coefficients. I'm having a problem in that my solution ...
6
votes
3answers
135 views

Use Taylor Series method to solve $y''-2xy+y=0$

I am doing some practice problems for solving second order ODEs, and I am a bit stuck on this one. Here is what I have: $y''-2xy'+y=0$ Let $y = \sum_{n=0}^{\infty} C_nx^n \implies y' = ...
6
votes
2answers
148 views

Function whose inverse is also its derivative?

What are some good examples of a function $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ ...
6
votes
1answer
69 views

Assistance Solving A Second Order Nonlinear ODE (Converted into a First Order)

I am trying to find the solution to $y''=y+y^2$ I noticed that if I multiplied by $y'$ on both sides and integrated, the result would be $\frac{1}{2}(y')^2=\frac{1}{2}y^2+\frac{1}{3}y^3+c$ I have ...
6
votes
1answer
82 views

Differential equation of a path

Differential equation of a path of a particle is $$\frac{d^2 u}{d \theta^2}-\frac{5}{4}u=\frac{9}{4}\frac{\alpha}{\beta^2}$$ where $u=\frac{1}{r}$ and $r$ is distance from origin to a particle, ...
6
votes
1answer
100 views

Showing the 3D Ricci flow ODE preserves the order of the curvature tensor eigenvalues

The following system of ODEs arises when studying Ricci flow on 3-manifolds: $$ \frac{dm_1}{dt} = m_1^2+m_2m_3 \\ \frac{dm_2}{dt} = m_2^2+m_1m_3 \\ \frac{dm_3}{dt} = m_3^2+m_1m_2 \\ $$ Going back ...
6
votes
1answer
311 views

Nonlinear first-order differential equation with a simple parametric solution.

I have to solve the nonlinear first-order differential equation $$\frac{a-y'}{\sqrt{1+y'^2}}e^{-a \arctan y'}=bx+c,$$ where $a,b,c$ are constants, and $y$ is a function of $x$. Obviously, there is ...
6
votes
1answer
149 views

Solving differential equation from Cauchy problem

I am getting acquainted with the Cauchy equations and I am trying to solve an exercise, taking the examples from my class notes. The exercise is: $$\begin{cases} y'=xy+x\\y(1)=2 \end{cases}$$ I have ...
6
votes
3answers
272 views

Existence of an extremum for the solution of the ODE $\ddot{x}+\frac32x^2=0$

Consider the 2nd order ODE $$ \ddot{x}+\frac32x^2=0. $$ Denote by $u$ the maximal solution of the associated Cauchy problem with initial condition $(x(0),\dot{x}(0))=(0,1)$. The problem is to prove ...
6
votes
1answer
1k views

A Differential Equation with Trigonometric Coefficients

Suppose we have the following second-order differential equation: $\cos^2(x)y'' -\sin(x)y' + y = 0$ How do we determine its general solution? I couldn't even guess a particular solution; all my ...
6
votes
1answer
281 views

Does the Wronskian have anything to do with the product rule in calculus

Does the Wronskian have anything to do with the product rule in calculus. I ask this because i noticed the form looking similar to the product rule. $$W=g(x)f'(x)-g'(x)f(x)$$ where as the ...
6
votes
1answer
100 views

Stability analysis for ODEs with non constant inputs

For a project, I have to deal with systems of ODE's with non constant input such as: $$\begin{cases}\dot x =I(t)x+x^2\\ \dot y=x\end{cases}$$ where I(t) is a random input (for example). In any case, ...
6
votes
1answer
199 views

Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
6
votes
0answers
67 views

Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...