MiniLogo

EqWorld

The World of Mathematical Equations

IPM Logo

Exact Solutions Methods Software For Authors Math Forums

EqArchive: Add Equation/Solution > View Equation

 English only

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):$\displaystyle (at+x)\frac{\partial ^2w}{\partial t^2}=b\frac{\partial^2w}{\partial x^2}$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\hfill\break
$w_0(x,t)=1$,\\
$w_1(x,t)=t$,\\
$\displaystyle w_2(x,t)=t^2+\frac 1b\left(atx^2+\frac {x^3}3\right)$,\\
$\displaystyle w_3(x,t)=t^3+\frac {3t}b\left(atx^2+\frac {x^3}3\right)+\frac {a}{2b^2}\left(atx^4+\frac {3x^5}5\right)$,\\
$\displaystyle w_4(x,t)=t^4+\frac {6t^2}b\left(atx^2+\frac {x^3}3\right)+\frac {3at}{b^2}\left(atx^4+\frac {3x^5}5\right)+\frac {1}{5b^2}\left(atx^5+\frac {2x^6}3\right)+\frac {a^2}{5b^3}\left(atx^6+\frac {5x^7}7\right)$,\\
$\displaystyle w_5(x,t)=t^5+\frac {10t^3}b\left(atx^2+\frac {x^3}3\right)+\frac
{10at^2}{b^2}\left(atx^4+\frac {3x^5}5\right)+\frac {t}{b^2}\left(atx^5+\frac
{2x^6}3\right)+\frac {2a^2t}{b^3}\left(atx^6+\frac {5x^7}7\right)
+\frac {a}{3b^3}\left(atx^7+\frac {3x^8}4\right)+\frac {a^3}{14b^4}\left(atx^8+\frac {7x^9}9\right)$,\\
$\displaystyle w_6(x,t)=t^6+\frac {15t^4}b\left(atx^2+\frac {x^3}3\right)+\frac
{25at^3}{b^2}\left(atx^4+\frac {3x^5}5\right)+\frac {3t^2}{b^2}\left(atx^5+\frac
{2x^6}3\right)+\frac {10a^2t^2}{b^3}\left(atx^6+\frac {5x^7}7\right)+\frac
{18at}{7b^3}\left(atx^7+\frac {3x^8}4\right)+\frac {1}{14b^3}\left(atx^8+\frac
{7x^9}9\right)+\frac {15a^3t}{14b^4}\left(atx^8+\frac {7x^9}9\right)
+\frac {17a^2}{63b^4}\left(atx^9+\frac {4x^{10}}5\right)+\frac {a^4}{42b^5}\left(atx^{10}+\frac {9x^{11}}{11}\right)$.
\medskip

2. Particular solutions:\\
$w_0(x,t)=x$,\\
$w_1(x,t)=tx$,\\
$\displaystyle w_2(x,t)=t^2x+\frac 1{3b}\left(atx^3+\frac {x^4}2\right)$,\\
$\displaystyle w_3(x,t)=t^3x+\frac {t}b\left(atx^3+\frac {x^4}2\right)+\frac {a}{10b^2}\left(atx^5+\frac {2x^6}3\right)$,\\
$\displaystyle w_4(x,t)=t^4x+\frac {2t^2}b\left(atx^3+\frac
{x^4}2\right)+\frac{3at}{5b^2}\left(atx^5+\frac {2x^6}3\right)+\frac
{a}{15b^2}\left(atx^6+\frac{5x^7}7\right)+\frac {a^2}{35b^3}\left(atx^7+\frac{3x^8}4\right)$,\\
$\displaystyle w_5(x,t)=t^5x+\frac {10t^3}{3b}\left(atx^3+\frac
{x^4}2\right)+\frac{2at^2}{b^2}\left(atx^5+\frac {2x^6}3\right)+\frac
{t}{3b^2}\left(atx^6+\frac{5x^7}7\right)+\frac
{2a^2t}{7b^3}\left(atx^7+\frac{3x^8}4\right)
+\frac {5a}{84b^3}\left(atx^8+\frac{7x^9}9\right)+\frac {a^3}{126b^4}\left(atx^9+\frac{4x^{10}}5\right)$,\\
$\displaystyle w_6(x,t)=t^6x+\frac {5t^4}{b}\left(atx^3+\frac
{x^4}2\right)+\frac{5at^3}{b^2}\left(atx^5+\frac {2x^6}3\right)+\frac
{t^2}{b^2}\left(atx^6+\frac{5x^7}7\right)+\frac
{10a^2t^2}{7b^3}\left(atx^7+\frac{3x^8}4\right) +\frac
{13at}{28b^3}\left(atx^8+\frac{7x^9}9\right)+\frac
{5}{252b^3}\left(atx^9+\frac{4x^{10}}5\right)+\frac
{5a^3t}{42b^4}\left(atx^9+\frac{4x^{10}}5\right)+\frac
{43a^2}{1260b^4}\left(atx^{10}+\frac{9x^{11}}{11}\right)
+\frac {a^4}{462b^5}\left(atx^{11}+\frac{5x^{12}}6\right)$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Fri 18 Jul 2008 18:31
Edits by author:0

Edit (Only for author/contributor)


The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.

Copyright © 2006-2011 Andrei D. Polyanin, Alexei I. Zhurov and Alexander L. Levitin