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

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1
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4answers
313 views

Finding all functions $f$ satisfying $f'(t)=f(t)+\int_a^bf(t)dt$

I am trying to find all functions f satisfying $f'(t)=f(t)+\int_a^bf(t)dt$. This is a problem from Spivak's Calculus and it is the chapter about Logarithms and Exponential functions. I gave up ...
0
votes
1answer
103 views

Consider Van der Pol’s equation:

Consider Van der Pol’s equation: $$y′′−0.2(1−y^2)y′+y=0,\qquad y(0)=0.1,\ y′(0)=0.1$$ (i) Find the approximate solution for this problem using the Taylor series method. Your expansion should include ...
14
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3answers
4k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
13
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4answers
2k views

Can this gravitational field differential equation be solved, or does it not show what I intended?

This is the equation I'm having trouble with: $G \frac{M m}{r^2} = m \frac{d^2 r}{dt^2}$ That's the non-vector form of the universal law of gravitation on the left and Newton's second law of motion ...
10
votes
2answers
394 views

Fourth Order Nonlinear ODE

I was looking at an ode $w^{(4)} + w^3 = 0$ with initial conditions $[w'''(0),w''(0),w'(0),w(0)]=[1,0,0,0]$. I can see via maple that there is a blowup around 3.7. I was wondering if there was a way ...
7
votes
2answers
1k views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ...
8
votes
1answer
206 views

When do Harmonic polynomials constitute the kernel of a differential operator?

Let $f$ be a real polynomial of two variables. Let $\partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$. Let $H$ denote the space of harmonic polynomials, i.e., ...
15
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5answers
570 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 ...
10
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3answers
11k views

Definition of a Differential Equation?

Here is one definition of a differential equation: "An equation containing the derivatives of one or more dependent variables, with respect to one of more independent variables, is said to be a ...
7
votes
1answer
147 views

solution of $y' = \exp \left(-\frac yx\right) + \frac yx$

Could you help me to solve equation $$y' = \exp \left(-\frac yx\right) + \frac yx;\quad y(e) = 0$$ I know how to solve 1st order linear de like $y' = \exp \bigl(-\frac 1x\bigr) + \frac yx$ but here ...
3
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3answers
2k views

Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
2
votes
3answers
424 views

How does an integrating factor geometrically “uncurl” a vector field?

We know that certain 1-D forms $m(x,y,z)\,dx + n(x,y,z)\,dy + p(x,y,z)\,dz$ admit integrating factors as we teach in basic differential equations. How does the integrating factor geometrically turn ...
11
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8answers
4k views

how do you solve $y''+2y'-3y=0$?

I want to solve this equation: $y''+2y'-3y=0$ I did this: $y' = z$ $y'' = z\dfrac{dz}{dy}$ $z\dfrac{dz}{dy}+2z-3y=0$ $zdz+2zdy-3ydy=0$ $zdz=(3y-2z)dy$ $z=3y-2z$ ...
8
votes
2answers
335 views

Deriving the addition formula of $\sin u$ from a total differential equation

How do we derive the addition formula of $\sin u$ from the following equation? $$\frac{dx}{\sqrt{1 - x^2}} + \frac{dy}{\sqrt{1 - y^2}} = 0$$ Motivation Let $u = \int_{0}^{x}\frac{dt}{\sqrt{1 - ...
8
votes
3answers
881 views

To what extent can you manipulate differentials like dy and dt like actual values?

I have been thinking about the differentials that we use in derivatives and integrals. For example, I have an equation: $${\int{w}{dr}} = \text{other stuff}$$ The context for this strange equation ...
7
votes
1answer
186 views

prove that the following function is: $f(x) = 0$

let $f: [0,1] \to \mathbb R$ , $f$ is differentiable $f(0) = 0$ $|f'(x)|\le|f(x)|$ for $x\in [0,1]$ prove that $f(x) =0$ for $x\in [0,1]$ i believe that i need to somehow use the ...
4
votes
3answers
113 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
4
votes
1answer
86 views

Under which conditions a solution of an ODE is analytic function?

If I'm not wrong there is a theorem that says that if the conditions for Picard's theorem are satisfied, for an ode $\dot x=f(x,t)$, then the solution of the ode is as smooth as $f$. I think I'm not ...
4
votes
2answers
416 views

Applied ODEs in trajectory problem

I'm having a hard time solving this problem: Let there be a town $A$ in a shore of a river. Let $x=0$ be the shore. Let $(0,0)$ be the location of the town. Let $B$ be another town, in the ...
4
votes
4answers
382 views

Differential equation with a constant in it

Solve $$y'' + s^2y = b \cos sx$$ where $s$ and $b$ are constants. I have tried undetermined coefficients, but it makes such a mess that I keep getting lost, I also tried variation of ...
2
votes
1answer
50 views

does there exist an system which has closed orbit but not constant one?

Can you please give me an example of an ODE system which has no constant orbit or fixes point but closed orbit? Thank u very much
1
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1answer
69 views

number of points of tangency of the zero divergence vector field and the equator of the sphere.

Let $V$ be vector field on the sphere $S^2$ and $\operatorname{div} V=0$. What is the minimum number tangency points of this vector field and the equator of the sphere?
0
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1answer
100 views

Basic Reference material about ODEs such as saparability with calculations and examples?

I am trying to show this kind of non-linear $y''''=y'y''/(1+x)$ in normal form. For example here if $y=e^{x}\rightarrow y^{(n)}=e^{x}\rightarrow x=-1$, where $y^{(n)}$ ...
12
votes
4answers
354 views

Find a continuous function $f$ that satisfies…

Find a continuous function $f$ that satisfies $$ f(x) = 1 + \frac{1}{x}\int_1^x f(t) \ dt $$ Note: I tried differentiating with respect to $x$ to get an ODE but you get one that contains integrals - ...
6
votes
2answers
145 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 $f(x) = ax^b$ ...
5
votes
1answer
714 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 ...
5
votes
2answers
197 views

How to prove $(x^2-1) \frac{d}{dx}(x \frac{dE(x)}{dx})=xE(x)$

$$E(x)=\int_0^{\frac{\pi}{2}} \sqrt{1-x^2 \sin^2 t}\, dt$$ Where $E(x)$ is complete elliptic integral of the second kind. $u=\sin t$ $$E(x)=\int_0^{1} \frac{\sqrt{1-x^2 u^2}}{\sqrt{1-u^2}}\, du$$ ...
4
votes
1answer
113 views

How do you solve this differential equation using variation of parameters?

$\color{green}{question}$: How do you solve this differential equation using variation of parameters? $$y"-\frac{2x}{x^2+1}y'+\frac{2}{x^2+1}y=6(x^2+1)$$ $\color{green}{I~tried}$ . . . ...
4
votes
1answer
265 views

Sturm-Liouville Questions

In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on. The first issue I have is that my book began with the statement that given $$L[y] = a(x)y'' + b(x)y' + ...
4
votes
1answer
340 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
votes
2answers
275 views

Formula for integration bounds of recursively defined polynomial sequence

We can recursively define a sequence of polynomials by $$P_0(x) := 1$$ and then with the definite integral $$P_n(x) := \int_{c_n}^x P_{n-1}(t) ~\mathrm dt$$ where the $c_n$ are to be chosen so ...
3
votes
4answers
263 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
1answer
82 views

Bessel Equations Addition Formula

So, I'm considering yet another tricky proof involving Bessel Functions. Basically, I'm trying to figure out how the following is true: $$J_n(\alpha + \beta) = \sum_{m = -\infty}^\infty ...
3
votes
1answer
337 views

Exercise from Stein with partial differential operator

I have again something from Stein-Shakarchi I would really appreciate some help with. Any references are also welcome! Suppose $L$ is a linear partial differential operator with constant ...
2
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1answer
74 views

:How to find the general solution of $(y+ux)u_x+(x+yu)u_y=u^2-1$?

question : find the general solution of $(y+ux)u_x+(x+yu)u_y=u^2-1$ $\frac{dx}{dt}=y+ux,\frac{dy}{dt}=x+yu,\frac{du}{dt}=u^2-1$ I dont know how to start. is this quasilinear ? edit 1: tried ...
2
votes
2answers
166 views

Expressing an oscillator as a series of ODEs

Consider an oscillator satisfying the initial value problem $u''+w^2u=0$, where $u(0)=u_0$, $u'(0)=v_0$. Let $x_1 = u$, $x_2=u'$, and transform the equations given into the form $x' = Ax, x(0)$. Then ...
2
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1answer
114 views

dropping a particle into a vector field

I'm independently studying Colley's Vector Calculus and am on the section on line integrals. I understand that the line integral gives the amount of work done on a vector field for a predetermined ...
2
votes
1answer
143 views

Dimensions analysis in Differential equation

Differential equation of solitary wave oscillons is defined by, $$ \Delta S -S +S^3=0 $$ How can we write this equation as, \begin{equation} \langle(\vec{\nabla}S)^2\rangle+\langle S^2\rangle-\langle ...
2
votes
1answer
188 views

Help with Initial value problem : $y'= x^2+ xy^2, y(0) = 0$; Picard–Lindelöf Approximation.

i need solve this: $$y'=x^2+xy^2 , y(0)= y(t_0)= 0$$ a) Compute, starting from the constant function $u_0=0$ the successive approximations $u_1,u_2,u_3$ (in the sense of the theorem of ...
2
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1answer
183 views

Explain Dot product with Partial derivatives in Polar-coordinates

Related to page 819 prob 4 in this book. I am incorrectly calculating the left-hand-side (def. LHS), some newbie error with commutativity probably. Ideas? Errors? I propose ...
2
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1answer
713 views

Boundary conditions of an elastic bar

I was following some online lecture relating to an elastic bar with length $L$ that obey the differential equation $\displaystyle \frac{d^{2}u}{dx^{2}} = f(x)$, where $f(x)$ is its own weight or some ...
1
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1answer
79 views

Questions concerning the differential operator

Consider the differential equation:- $a \phi + (bD^3 - cD)w =0$, where $a, b$ and $c$ are constants, $D$ denotes the differential operator $\dfrac{d}{dx}$, and $w$ is a function of $x$. I'm ...
1
vote
2answers
530 views

solution of first order differential equation and maximal interval

Find the solution of $x' = x^2t$ with initial value $x(0) = x_{0}$. Determine the maximal interval where it exists, depending on $x_{0}$ Please help me find the maximal interval!
1
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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 ...
1
vote
1answer
285 views

How to show that the geodesics of a metric are the solutions to a second-order differential equation?

On $\mathbb R^n$, let $\rho: \mathbb R^n\to\mathbb R$ be a smooth function, and $g$ be the metric given by scaling the usual flat metric by $e^{2\rho}$. I want to know how to show that the geodesics ...
0
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0answers
74 views

Proof the first-order equation

$$ \begin{align} \tau\dot y + y &= KF(t) \tag{1}\\ y(t) &= C_0 + C_1 e^{-t/\tau} \tag{2} \end{align} $$ How these two equations can form $$y(t)=KA+(y_0-KA)e^{-t/\tau}?$$ Note ...
0
votes
2answers
372 views

Substitution $x=\sinh(\theta)$ and $y=\cosh(\theta)$ to $(1+x^{2})y'-2xy=(1+x^{2})^{2}$?

After this substitution I got to the point $$\cosh^6 (\theta)y'-\sinh(2\theta)-\cosh^4 (\theta)=0$$ and now let $$z=\cosh^2 (\theta)$$ so $$z^3 y'-z^2-\sinh(2\theta)=0$$ but then I ...
10
votes
3answers
355 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 ...
4
votes
2answers
121 views

A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$

Let: $f(x)$ and $g(x)$ be twice differentiable, non-decreasing functions. $f''(x) = g(x)$ and $g''(x) = f(x)$. $f(x) \cdot g(x)$ is a linear function. Then we have to show that $f(x) = g(x) = ...
4
votes
2answers
1k views

How can I solve this Initial Value Problem using the Euler method?

My Problem is this given Initial Value Problem: $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ I am looking for a way to solve this problem using the Euler method. I have a given Interval of $[1,2]$ and a ...