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View Equation

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Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.5. Higher-Order Equations
Equation(s):$\displaystyle \frac{\partial ^2w}{\partial t^2}+a\frac{\partial^4w}{\partial x^4}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(x,t)=1$,\\
$w_1(x,t)=x$,\\
$w_2(x,t)=x^2$,\\
$w_3(x,t)=x^3$,\\
$\displaystyle w_4(x,t)=x^4-\frac {4!a}{2!}t^2$,\\
$\displaystyle w_5(x,t)=x^5-\frac {5!a}{2!}xt^2$,\\
$\displaystyle w_6(x,t)=x^6-\frac {6!a}{2!2!}x^2t^2$,\\
$\displaystyle w_7(x,t)=x^7-\frac {7!a}{3!2!}x^3t^2$,\\
$\displaystyle w_8(x,t)=x^8-\frac {8!a}{4!2!}x^4t^2+\frac {8!a^2}{4!}t^4$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=x^n+\sum_{k=1}^{n-4k\geq0} \frac {(-1)^kn!a^kt^{k}x^{n-4k}}{(n-4k)!(2k)!}$.
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2. Particular solutions:\\
$w_0(x,t)=t$,\\
$w_1(x,t)=xt$,\\
$w_2(x,t)=x^2t$,\\
$w_3(x,t)=x^3t$,\\
$\displaystyle w_4(x,t)=x^4t-\frac {4!a}{3!}t^3$,\\
$\displaystyle w_5(x,t)=x^5t-\frac {5!a}{3!}xt^3$,\\
$\displaystyle w_6(x,t)=x^6t-\frac {6!a}{2!3!}x^2t^3$,\\
$\displaystyle w_7(x,t)=x^7t-\frac {7!a}{3!3!}x^3t^3$,\\
$\displaystyle w_8(x,t)=x^8t-\frac {8!a}{4!3!}x^4t^3+\frac {8!a^2}{5!}t^5$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=x^nt+\sum_{k=1}^{n-4k\geq0} \frac {(-1)^kn!a^kt^{2k+1}x^{n-4k}}{(n-4k)!(2k+1)!}$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Sun 11 May 2008 12:54
Edits by author:0

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