# Tagged Questions

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

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### 4 Bugs chasing each other differential equation

This is from a problem seminar and I need help figuring out the solution. Four bugs, $A,B,C,D$ are initially placed at the corners of a unit square. From a given initial moment, all four crawl ...
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A $6$ meter long ladder leans with a vertical wall and top of the ladder is 3 meters above the ground.If it slips at a rate of $2$ m/s then how fast the level is decreasing from the wall? My attempt:...
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### 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 ...
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### Differential equation $\sin \theta \frac{dr}{d \theta}+r\cos \theta =\tan \theta,0<\theta<\pi/2$ [closed]

This problem has been stumping me for over an hour how can I set it up, I think I have done it wrong over and over. Solving for $r$.
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### How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system. I don't know how to use Mathematica/Matlab ... :( If anyone can make this portrait and post a print screen here, I would thank you ...
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### 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 ...
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### solving third-order nonlinear ordinary differential equation

I would like to solve: $(\frac{d^{2}y}{dx^{2}})^2+\frac{d^{3}y}{dx^{3}} \frac{dy}{dx}=0$ Thanks in advance.
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### van der pol equation

Consider the van der Pol equation below: $(x'')+a(x^2-1)(x')+(x)=0$ I need to : Find an equilibrium point and linearize this equation near it Find solutions of the linearized equation depending on ...
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### Riccati differential equation $y'=x^2+y^2$

$$y'=x^2+y^2$$ I know, that this is a kind of Riccati equation, but is it possible to solve it with only simple methods? Thank you
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### Why does the absolute value disappear when taking $e^{\ln|x|}$

I have noticed that if you have an equation (after integrating) such as $$\ln|y| = \ln|x| + c,$$ and you further simplify it using the law of exponents, you get $$e^{\ln|y|} = e^{\ln|x|+с},$$ which is ...
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### Nonlinear 1st order ODE involving a rational function

$$y'=\frac{-6x+y-3}{2x-y-1}$$ Is there a foolproof method for tackling equations of the form $y'=\dfrac{ax+by+c}{dx+ey+f}$ ? I've tried a few substitutions (like $y=vx$ and $v=2x-y-1$, neither of ...
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### “Constrained” numerical solutions of ODEs with conservation laws?

I know little about numerical methods and I was considering the following problem that possibly has standard solution in the literature. Suppose you have an ODE for wich we already know that it must ...
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### An existence of global solution of differential equation of first order

Let $f: (a,b) \times \mathbb{R} \rightarrow \mathbb{R}$ be of class $C^1$ in $D:=(a,b) \times \mathbb{R}$ and satisfies condition $$| f(t,x)| \leq A+B|x| \textrm{ for } (t,x) \in D,$$ where $A,B$ ...
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### 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 $f(x) = g(x) = 0$....
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### 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 ...
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### Second-order nonlinear ODE with Dirac Delta

Can anyone help me with the following differential equation? $$2x(t)x''(t) - x'(t)^2 + kx(t)^2\delta(t - a) =0,$$ where $\delta$ represents the Dirac Delta. I tried Mathematica but with no luck. ...
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### Closed form solutions of $\ddot x(t)-x(t)^n=0$

Given the ODE: $$\ddot x(t)-x(t)=0$$ the solution is: $$x(t)=C_1\exp(-t)+C_2\exp(t)$$ If we square the $x(t)$ we have: $$\ddot x(t)-x(t)^2=0$$ and the solution is given by: $$x(t)=6\wp(t+C_1;0,C_2)$$ ...
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### Series solution to $y''-xy'-y=0$

So I'm learning to solve ODE's with series on my own using Boyce and DiPrima and exercise #3 is irking me...just looking for power series solutions around the ordinary point... $$y''-xy'-y=0$$ So I ...
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### What is an example of a second order differential equation for which it is known that there are no smooth solutions?

I would really appreciate if someone could just write down for me one example of a second order, or higher, differential equation for which it is known that there are no smooth solutions; and it's ...
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### Relation between Heaviside step function to Dirac Delta function

I understand that "delta function" is a distribution, not a function, as in it acts on another integrand, picking out the value of that integrand at a specific point. The discontinuous function is ...
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### Solve $(x^2 + 1)y'' - 6xy' + 10y =0$ using series method

Use series methods to solve: $(x^2 + 1)y'' - 6xy' + 10y =0$ a) Give the recursion formula b) Give the first two non-zero terms of the solution corresponding to $a_0 = 1$ and $a_1 = 0$ c)...
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### Why does acceleration = $v\frac{dv}{dx}$

If we define $x$ = displacement, $v$ = velocity and $a$ = acceleration then I am used to the ideas that $a= \frac{dv}{dt} = \frac{d^2x}{dt^2}$ However I also understand $a=v \frac{dv}{dx}$. Can ...
Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
I have this as homework: $$(xy^2+y)dx+(x^2y-x)dy=0$$ I tried to solve it by substituting $z=xy+1$, but got the answer like $y=Cxe^{xy}$, which, I guess, is wrong. I tried to solve it couple of ...