# Tagged Questions

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### What is the inverse function of $\int{ \frac{1}{{\sqrt{x+1}}{x^n}} dx}$?

I am trying to solve $$\frac{dy}{dt} = \alpha ((y+1)^2 - \gamma)^n \hspace{2cm} y(0)=0$$ Here $y$ is a real-valued, monotonically increasing, positive definite function of $t$ in the interval ...
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### Lyapunov function for non-autonomous non-linear differential equations

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system. Suppose there is a non-autonomous non-linear ...
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### Constant Coefficient Legendre Equation via Change of Variables?

In the introduction to this old book by Craig on ode's it is said that The theory of linear differential equations may almost be said to find its origin in Fuchs's two memoirs published in 1866 ...
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### Sturm Liouville eigenvalues eigenfunctions

The equation/Sturm Liouville problem is: $$u'' + \lambda u = 0, \quad 0≤x≤\frac{\pi}{2}, \quad u'(0) = 0, \quad u(\frac{\pi}{2}) = 0$$ I want to find the eigenvalues and eigenfunctions and the ...
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### Solution to Legendre eq in trig form

Okay I'm having a little trouble in answering this question... so the general solution is $y(x) = AP_n(x) + BQ_n(x)$ umm then what do I do?
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### mathieu function of non integer order asymptotics

I have an asymptotic expression for the integer mathieu functions: $se_\nu(q,z)$ and $ce_\nu(q,z)$, where $\nu$ is an integer. I would like to use these expression for the case $\nu$ real. My question ...
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### eigenfunctions in a Sturm-Liouville problem

I've found that the eigenfunctions in a certain Sturm-Liouville problem satisfy a differential equation whose general solution is $\phi(x)= x^{a}[C_1M(a,2a+2,x)+C_2U(a,2a+2,x)]$, $x\ge0$, where $M$ is ...
1answer
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### What will be the solution of this equation?

What will be the solution of the equation. $(x^2+m^2)\frac{\partial^2y}{\partial(x^2+m^2)}+(x+m)\frac{\partial y}{\partial (x+m)}+(x^2+m^2-n^2)=0$ where $m$ may be a constant
1answer
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### ODE second order and Bessel function

I have a function $xy''-y'+y=0$ that I'm trying to solve. I thought of solving it like this $y''-y'/x+y/x=0$ this you can write as $(y'/x)'+y=0$ and then you can find the solution as a Bessel ...
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### Limits of generalized hypergeometric functions

For a (quite fiddly) asymptotic matching, I would like to be able to write the solution to \frac{\mathrm{d}^5}{\mathrm{d}x^5}f(x) + \frac{10}{15^{1/2}} \frac{\mathrm{d}}{\mathrm{d}x} ...
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### Show that Bessel function $J_n(x)$ satisfies Bessel's differential equation.

here is the question: For each positive integer $n$, the Bessel function $J_n(x)$ may be defined by $$J_n(x) = \frac{x^n}{1\cdot 3\cdot 5\cdots(2n-1)\pi}\int^1_{-1}(1-t^2)^{n-1/2}\cos(xt) \, dt$$ ...
1answer
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### Euler's infinite product for the sine function and differential equation relation

Euler's infinite product for the sine function $$\displaystyle \sin( x) = x \prod_{k=1}^\infty \left( 1 - \frac{x^2}{\pi^2k^2} \right)$$ http://en.wikipedia.org/wiki/Basel_problem We know that ...
2answers
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### Bessel function confirmation

I'm trying to see that $J_0(x)$ is indeed a solution for the Bessel equation $x^2y''+xy'+x^2y=0$, so: $$J_0(x)=\sum_{k=0}^\infty \frac{(-1)^kx^{2k}}{(k!)^22^{2k}}$$ Pluging it in the equation and and ...
0answers
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### Hints/Help studying an Abel Differential Equation

I want to know more than qualitative information about the Abel differential equation $\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$ Since I don´t know how to solve this and as far as could see, this ...
1answer
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### How to solve $(m_{(t)} x')' + kx = 0$ Sturm Liouville equation with bessel functions

I have been working on this problem for a while now and think I need assistance. I am trying to solve with respect to $x_{(t)}$ over the interval $t = [0, \infty]$: $$(m_{(t)} x')' + kx = 0$$ ...
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1answer
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### To solve $U''_{n}(x)-\frac{2n}{x}U'_{n}(x)+(\frac{2n}{x^2}-1)U_{n}(x)=0$

$$e^{x\sqrt{1+t}}=\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!}$$ $$\frac{\partial}{\partial t }(e^{x\sqrt{1+t}})=\frac{\partial}{\partial t }(\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!})$$ ...
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569 views