# Tagged Questions

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

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### General solution of a nonlinear differential equation

Nonlinear differential equation gone beyond my field of expertise but I'd like to know the details of a problem and to do that I should know the general solution of the following nonlinear ...
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### Solving $f'(x) = f(f(x))$ [duplicate]

Is there any solution to the differential equation $f'(x) = f(f(x))$? I couldn't find any information on this kind of DE
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### solution of the ODE $u du =ydx+xdy$

In this case $u=u(x,y)$. When I saw this I just went on to taking iindefinite integral both sides yielding $u^2=4xy+K$. Yet, the book I am using now got $udu=d(xy)$, which yields $u^2=2xy+K$. I'm I ...
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### Solution of $f(x)^2\dfrac{d^2}{dx^2}f(x)=x$

I am stuck in finding the solution of this apparently simple differential equation: $$f(x)^2\dfrac{d^2}{dx^2}f(x)=x$$ with$f(0)=a$ and $f(0)'=b$ Using Maple the solution seems to be a combination of ...
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### How to integrate the following: $\int{\frac{2y'y}{y^2+1}dx}$

I have encountered the following problem: $\int{\frac{2y'y}{y^2+1}dx}$ According to wolfram the solution is: $log(y^2 + 1)$ How was this solution derived and which rules were used?
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### How do you solve the following separable differential equation: y'y = y + 1?

I just started learning about differential equations and encountered following equation: $$y'y = y +1$$ Wolfram alpha provided the following explanation: here But I'm not sure how the integration ...
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### Solving ODE for practice

I'm doing self study and I can't solve this equation: $$ax + \ln y = y + b$$ Where I'm supposed to eliminate the arbitrary constants. The given answer is $(y - y^2)(y'') = (y')^2$ But my workings ...
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### Singularities in a PDE

This is more of a general question rather than anything specific but I was just wondering if someone could point me toward resources which discuss singularities in a PDE rather than in an ODE (by ...
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### Matrix Differentiation of $-a^T X^T y$ on $a.$

In short; what is the correct differentiation of: $$S(a)=-a^TX^Ty$$ when differentiating: $$0=\frac{âˆ‚S}{âˆ‚a}= \;?$$ Long story is; I know that: J(a)=\underbrace{\:\:\:a^TX^TXa\:\:\:}_u\:\...
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### '2nd order' Picard Iteration

I'm self-studying differential equations using MIT's publicly available materials. One of the problem set exercises deals with what I'm calling a second order Picard Iteration. To be explicit, we ...
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### Are there any examples of higher order ireducible linear differential operators?

Given a monic, linear differential operator $L = D^n + f_{n-1}(x)D^{n-1} + \dots + f_1(x) + f_0(x)$, say $f_0, \dots, f_{n-1}$ analytic for simplicity's sake, we say that $L$ is irreducible if there ...
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### is there are specific way to solve coupled first-order differential equations with coefficients varying?

suppose I have "n" coupled differential equation represented by the matrix, Y• = A Y , where Y• is the column matrix containing first derivatives, namely, y1•(t), y2•(t), ... yn&...
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### Polar coordinates for vector field to find sticking flow

I am currently working on an impacting system which is basically just a spring damper and a circular enclosure. Because of the rotational symmetry of the problem I need the vector field in polar ...
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### Differentiation of$f^{-1}(x)$, where $f(x)=e^{x-1}+x^3-4x^{-3}+10$

if $f(x)=e^{x-1}+x^3-4x^{-3}+10$ then find $\frac{d(f^{-1}(x))}{dx}$ at $x=8$..... (here $f^{-1}(x)$ means inverse of $f(x)$) I was trying to solve this problem but was not able to find out the way ....
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### Solve the following IVP with explicit solution

Given: $4 dx + 2 {cos(y)\over sin(y)} dy = 0, \qquad y(0) = {\pi\over 2}$ I've already test the exactness which is $0$ for the result of both derivatives. Then I found the potential function is ...
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### How to find the total derivative of a function $f_a(y(t),x(t))$ subjected to parametric change with the parameter $a$

It is well known to find the total derivative of a function $f(x(t),y(t))$. I consider it as $Td_f$. What, if the function depends upon some parameter, say, $a$. Then, how to find the total derivative ...
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### Finding the polynomial [on hold]

Find a nontrivial polynomial function $p(x)$ such that $p(2x)=p'(x)p''(x)\not=0$
I need to solve the following third order (non-linear) ODE by numerical methods: $$\tag{1} h^{3} \dfrac{d^3 h}{d x^3} = h-1.$$ By assumption, the solution should approach $... 4answers 2k views ### How unique is$e$? Is the property of a function being its own derivative unique to$e^x$, or are there other functions with this property? My working for$e$is that for any$y=a^x$,$ln(y)=x\ln a$, so$\frac{dy}{dx}=\...
$2tx'-x=lnx'$ I differentiated both sides with respect to x: $x'+2tx''=\frac {x''}{x'}$ Substituting $p=x'$, $p+2tp'=\frac{p'}{p}$ But I have no clue what can I do from here on. EDIT: $t$ is the ...