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

Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.5. Higher-Order Equations
Equation(s):$\displaystyle at^2 \frac{\partial ^2w}{\partial t^2}+\frac{\partial^3w}{\partial x^3}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\hfill\break
$w_0(x,t)=1$,\\
$w_1(x,t)=t$,\\
$\displaystyle w_2(x,t)=t^2e^{-x(2a)^{1/3}}$,\\
$\displaystyle w_3(x,t)=t^3e^{-x(6a)^{1/3}}$,\\
$\displaystyle w_4(x,t)=t^4e^{-x(12a)^{1/3}}$,\\
$\displaystyle w_5(x,t)=t^5e^{-x(20a)^{1/3}}$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=t^ne^{-x[n(n-1)a]^{1/3}}$.
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2. Particular solutions:\hfill\break
$w_0(x,t)=x$,\\
$w_1(x,t)=tx$,\\
$\displaystyle w_2(x,t)=t^2\exp\Bigl(\frac {x(2a)^{\frac 13}}2\Bigr)\cos\Bigl(\frac {x(2a)^{\frac 13}\sqrt{3}}2\Bigr)$,\\
$\displaystyle w_3(x,t)=t^3\exp\Bigl(\frac {x(6a)^{\frac 13}}2\Bigr)\cos\Bigl(\frac {x(6a)^{\frac 13}\sqrt{3}}2\Bigr)$,\\
$\displaystyle w_4(x,t)=t^4\exp\Bigl(\frac {x(12a)^{\frac 13}}2\Bigr)\cos\Bigl(\frac {x(12a)^{\frac 13}\sqrt{3}}2\Bigr)$,\\
$\displaystyle w_5(x,t)=t^5\exp\Bigl(\frac {x(20a)^{\frac 13}}2\Bigr)\cos\Bigl(\frac {x(20a)^{\frac 13}\sqrt{3}}2\Bigr)$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=t^n\exp\Bigl(\frac {x[n(n-1)a]^{\frac 13}}2\Bigr)\cos\Bigl(\frac {x[n(n-1)a]^{\frac 13}\sqrt{3}}2\Bigr)$.\\
\medskip

3. Particular solutions:\hfill\break
$w_0(x,t)=x^2$,\\
$w_1(x,t)=tx^2$,\\
$\displaystyle w_2(x,t)=t^2\exp\Bigl(\frac {x(2a)^{\frac 13}}2\Bigr)\sin\Bigl(\frac {x(2a)^{\frac 13}\sqrt{3}}2\Bigr)$,\\
$\displaystyle w_3(x,t)=t^3\exp\Bigl(\frac {x(6a)^{\frac 13}}2\Bigr)\sin\Bigl(\frac {x(6a)^{\frac 13}\sqrt{3}}2\Bigr)$,\\
$\displaystyle w_4(x,t)=t^4\exp\Bigl(\frac {x(12a)^{\frac 13}}2\Bigr)\sin\Bigl(\frac {x(12a)^{\frac 13}\sqrt{3}}2\Bigr)$,\\
$\displaystyle w_5(x,t)=t^5\exp\Bigl(\frac {x(20a)^{\frac 13}}2\Bigr)\sin\Bigl(\frac {x(20a)^{\frac 13}\sqrt{3}}2\Bigr)$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=t^n\exp\Bigl(\frac {x[n(n-1)a]^{\frac 13}}2\Bigr)\sin\Bigl(\frac {x[n(n-1)a]^{\frac 13}\sqrt{3}}2\Bigr)$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Mon 12 May 2008 19:45
Edits by author:0

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