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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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### Special Differential Equation

I ended up with a differential equation that looks like this: $$\frac{d^2y}{dx^2} + \frac 1 x \frac{dy}{dx} - \frac{ay}{x^2} + \left(b -\frac c x - e x \right )y = 0.$$ I tried with Mathematica. But ...
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### differential equation - solving a second-order ODE with variable coefficients

I was analyzing stability for the following system of differential equations: $$z_1'=z_1+(6+e^{-t})z_2$$ $$z_2'=-z_1-4\tanh(t)z_2$$ In an effort to check my answer, I attempted to solve the system, ...
• 323
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### an odd question about solving ODE by MATLAB

Why MATLAB sometimes cannot solve the relatively particular case but can solve the relatively general case? For example: I tried to input (x^2-1)*D2y + 0*x*Dy + 1*x*y = 0 in MATLAB and MATLAB still ...
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### Can one explicitly solve the shifted harmonic oscillator

The quantum harmonic oscillator is described by a Hamiltonian $$H=-\Delta + \left\lvert x \right\rvert^2.$$ By decomposing the eigenvalue problem $H\psi= E\psi$ into its angular and radial part one ...
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### How to solve the Sturm-Liouville problem : $y'' + A(x) y = 0$?

Consider the following Sturm-Liouville problem : $$y'' + A(x) y = 0 \text{ on } [0, 2\pi]$$ where $A$ is a non-constant continuous function on $[0, 2\pi]$. Are there analytical solutions to this ...
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### Is there an analytic solution for this Fokker-Planck equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial t}=-\frac{\partial}{\partial\theta}\Big[[\sin(k\theta)+f]P(\theta,t)-D\frac{\...
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1 vote
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### Bessel-like differential equation

I have the following Bessel-like differential equation: $$r^2T^{''}+K_1rT^{'}+(K_2r^2+K_3r^m)T=0$$ In this equation, $T=f(r)$ and $K_1$, $K_2$, $K_3$, and $m$ are parameters. I need an analytical ...
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1 vote
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### differential equation nondevelopable

I try to solve this differential equation whose solution seems not to be constructable in power series $y''+(x+a/x^2+b)y=0$, where $a$ and $b$ are some positive real numbers. If one can help ...
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### Solve the given initial value problem.I need your help.

Solve the $$x'=tx^2+x-t^3\,,\quad x\left(\, 2\,\right)=1$$ I need its exact solution not a numerical solution.In fact I have to compare the exact solution with the numerical solution.I tried it but I ...
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### Solving differential equation

I want to solve the following differential equation with initial conditions: $$\frac{\mathrm{d}^2 y}{\mathrm{d} x^2}=\frac{x \, y(x)}{\sqrt{1-x}}$$ But do not know how to actually solve it. Any ...
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1 vote
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### Differential equation with nasty coefficients $x^2(1-x)^2 y'' + (Ax + b)y = 0$

I have encountered a differential equation on the form $$x^2(1-x)^2 y'' + (Ax + b)y = 0$$ My math background is too limited to even know where to begin, so any help of solving the equation (if a ...
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We know that the Heun's differential equation is $\dfrac{d^2y}{dx^2}+\left(\dfrac{\gamma}{x}+\dfrac{\delta}{x-1}+\dfrac{\epsilon}{x-a}\right)\dfrac{dy}{dx}+\dfrac{\alpha\beta x-q}{x(x-1)(x-a)}y=0$ , ...