# Tagged Questions

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

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### Inverse Laplace transform of $\operatorname{arccot}(s)$, $\arctan(s)$

How would one find inverse Laplace transforms of $\operatorname{arccot}(s)$ or of $\arctan(s)$ without knowing in advance that this is related to $\dfrac{\sin x}{x}$?
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### Non-Conservative System

I'm having a bit of trouble understanding the concept of a conservative system mathematically. A problem in a textbook (Arnold's Mathematical Methods for Classical Mechanics) is asking me to give an ...
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### Finding second derivative for $x=\sin t$ and $y= \sin 2t$.

If $x=\sin t$ and $y= \sin 2t$, how to find second derivative of $y$ w.r.t $x$ ? Or rather how to prove $(1-x^{2})\frac{d^{2}y}{dx^{2}}-x\frac {dy}{dx}+4y=0$? Is there any shortcuts to find these ...
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### Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$y''+(\delta(x)-\lambda^2)y=0$$ Then, to find ''bound states'', you solve on the right and find the ...
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### Differential equation with $\sqrt{1-\cos(f)}$

I'm currently trying to solve the differential equation $$\sqrt{2} a \cdot \sqrt{1 - \cos(f)} = f'$$ where $a$ is a constant and I can freely choose $f(0)$ to simplify the solution and calculation. ...
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### Solving recurrence using analogy with continuous $x_{n+1} = \frac{r^2}{2d - x_n}$

What's up lovely friends, I'm facing a physics problem and felt on a recurrence that one does not see everyday. This one: $x_{n+1} = \frac{r^2}{2d - x_n}$ or $f(n+1) = \frac{a}{b-f(n)}$ if you will ...
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### Find a second solution of the given differential equation.

$$xy''+y'=0; y_1=ln(x)$$ I solved this all the way to the end and found my second solution to be $y_2=-1$, but the book says it is $y_2=1$. I am checking my algebra and the method I used was to get ...
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### Regarding continuity and the value of the function at the point of discontinuity.

Suppose while solving a boundary value problem, we have a two piece solution $f_1(x)$ and $f_2(x)$ where $f_1(x)=f(x)$ for $x < x_0$ and $f_2(x) = f(x)$ for $x>x_0$. If there is a matching ...
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### Help me with this differential equation

$$xy'-y=x(1+e^{\frac{y}{x}})$$ Please give me a hint on how to solve this. If I'm not mistaken, this is a Bernoulli equation, but I can't seem to solve it using the substitution $z=y^{\frac{1}{1-a}}$. ...
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### How to Identify a homogeneous first order first degree ODE

The following equation is homogeneous edit: y dx - x dy + 3x^2y^2e^(x^3) dx = 0 (source: Wolfram alpha) but it is not of the form of $f(zx,zy)= z(f(x,y))$. How do I identify such type of special ...
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### How do you read a partial differential equation?

In calculus we can read the "normal derivative", $\frac {df}{dx}$, as the rate of change of our function $f$ with respect to $x$. With partial derivatives of multivariate functions it is very much the ...
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### Homogeneous differential equation $F(x, y, y',y'')$

I'm studying an example of a different equation's solution in my maths textbook. The equation is: $$xy'(yy'' - (y')^2) - y(y')^2 - x^4y^3 = 0$$ The author concludes that it is a homogeneous ...
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### Find the differential equation of all circles of radius 1 and centers on $y=x$

Find the differential equation of all circles of radius 1 and centers on $y=x$, I've answered several problems with circles finding its equation but not like $y=x$ can someone please explain this to ...
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### If $\partial\Omega\in C^{2+\alpha}$ and $-\Delta\Theta=f\text{ in }\Omega$ with $f\in C_0^\infty(\Omega)$, then $\Theta\in C^{2+\alpha}$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with $\partial\Omega\in C^{2+\alpha}$ for some $\alpha>0$ $f\in C_0^\infty(\Omega)$ $\Theta\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be the ...
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### $T: V \to \Bbb{R^2}$ by $T(f)=(f'(0),f(0))$.

Let V be the space of twice differentiable function on $\Bbb{R}$ such that $$f''-2f'+f=0.$$ Define $T: V \to \Bbb{R^2}$ by $$T(f)=(f'(0),f(0)).$$ The I could see that $T$ is one-one, but is $T$ onto?...
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### Differential of a tricky function

I have a function that I'm strugling to take the differential of. $$F(t) = F(t-a)G(t).$$ My attempt is the following: $$dF(t) = F(t-a)dG(t) + G(t) dF(t-a))$$ but I have a feeling something is not ...
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### Finite Difference for Hamilton-Jacobi-Bellman without boundary conditions

Let $t\in\mathbb{R}_+$ denote time, $x \in X$ is the state and $u \in U$ the control. The objective function is $F:X \times U \to\mathbb{R}$ and $f:X \times U \to\mathbb{R}$ is the law of motion for ...
Question: find a curve $x$ so that the area bounded between it's tangent at some point $t$ and the time axis on the interval between the point of contact of $x$ and it's tangent ( $t$ ), and the ...