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View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.2. Second-Order Hyperbolic Equations
Equation(s):$\displaystyle \frac{\partial ^2w}{\partial t^2}=ax^q\frac{\partial^2w}{\partial
x^2}+bx^{q-1} \frac{\partial w}{\partial x}$, \quad \ $q\neq2$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(x,t)=1$,\\
$w_1(x,t)=t$,\\
$\displaystyle w_2(x,t)=t^2+\frac {2x^{2-q}}{(2-q)(a+b-qa)}$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=t^n+\sum_{k=1}^{n-2k\geq0}\frac {n!t^{n-2k}x^{2k-qk}}{(n-2k)!k!(2-q)^{2k}a^k\left(\frac{a+b-aq}{2a-aq}\right)_k}$.
\medskip

2. Particular solutions:\\
$w_0(x,t)=x^{1-\frac ba}$,\\
$w_1(x,t)=tx^{1-\frac ba}$,\\
$\displaystyle w_2(x,t)=t^2x^{1-\frac ba}+\frac{2x^{3-q-\frac ba}}{(2-q)(3a-b-qa)}$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=t^nx^{1-\frac ba}+x^{1-\frac ba}\sum_{k=1}^{n-2k\geq0}\frac {n!t^{n-2k}x^{2k-qk}}{(n-2k)!k!(2-q)^{2k}a^k\left(\frac{3a-b-aq}{2a-aq}\right)_k}$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Tue 08 Jul 2008 18:42
Edits by author:1
Last edit by author:Wed 07 Oct 2015 18:55

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