# Tagged Questions

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

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### Does this ODE question have closed form solution?

These days, I am struggling with following ODE problem when I build up my research model: $1/2f''(x)+a(b - x) f'(x) -(c+ e^{A+Bx})f(x)=0$ where f(x) is a smooth function, and $a,b,c, A,B$ ...
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### How to Solve the Coupled Differential Equations?

I came across the set of following coupled equations while studying cycloid motion in Griffiths' Intro to ED $\ddot{y}=\omega \dot{z}$ $\ddot{z}=\omega (\frac{E}{B}-\dot{y})$ I am at a loss as to ...
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### How can I solve the differential equation $y'+y^{2}=f(x)$?

$$y'+y^{2}=f(x)$$ I know how to find endless series solution via endless integral or endless derivatives and power series solution if we know $f(x)$. I also know how to find general solution if we ...
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### 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 ...
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### Particular solution to a Riccati equation $y' = 1 + 2y + xy^2$

The equation is $y' = 1 + 2y + xy^2$. I've tried $mx+n$, $ax^m$, even $\tan x$ as candidates for particular solution where $a,m,n \in \mathbb Q$, but it did not work. Can anyone find one particular ...
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### Why it is absolutely mistaken to cancel out differentials?

In many of my physics courses, (don't worry, this is a mathematics question!) My teachers cancel out differentials, and every time, they say: "If a mathematician saw me canceling out this ...
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### 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 ...
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### Finite dimensional spaces

What are the finite-dimensional spaces $W$ of differentiable functions with this property: If $f$ is in $W$, then $\frac{df}{dx}$ is in $W$.
When finding the solutions to the simple ODE $$y'- mxy= x^n \text{ ; } y(0) = 0$$ I found the following: Let $P_n$ be the particular solution for each integer exponent $n$. Then if we define $$... 1answer 228 views ### When do Harmonic polynomials constitute the kernel of a differential operator? Let f be a real polynomial of two variables. Let \partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right). Let H denote the space of harmonic polynomials, i.e., ... 1answer 370 views ### \frac{dS}{d\rho} Factor arising To get details see: equations 29,30,31,34,44,50,51 We have known some solitary wave solutions, given by(equations 1 to 5)$$ \phi_1=p_1\cos \tau \tag{1}\phi_2=\frac16 g_2p_1^2\left(\cos(2\tau)-...
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 \| A(t)\|\,...