View source for 32.13

{{DISPLAYTITLE:Painlevé Transcendents - 32.13 Reductions of Partial Differential Equations}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/32.13.E1 32.13.E1] || <math qid="Q9474">v_{t}-6v^{2}v_{x}+v_{xxx} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>v_{t}-6v^{2}v_{x}+v_{xxx} = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">v[t]- 6*(v)^(2)* v[x]+ v[x, x, x] = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[v, t]- 6*(v)^(2)* Subscript[v, x]+ Subscript[v, x, x, x] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/32.13#Ex1 32.13#Ex1] || <math qid="Q9475">z = x(3t)^{-1/3}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>z = x(3t)^{-1/3}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(x(+)*y*I) = (x(3*t))^(- 1/3)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(x[+]*y*I) == (x[3*t])^(- 1/3)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/32.13#Ex2 32.13#Ex2] || <math qid="Q9476">v(x,t) = (3t)^{-1/3}w(z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>v(x,t) = (3t)^{-1/3}w(z)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">v(x , t) = (3*t)^(- 1/3)* w*((x + y*I))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">v[x , t] == (3*t)^(- 1/3)* w*((x + y*I))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/32.13.E3 32.13.E3] || <math qid="Q9477">u_{t}+6uu_{x}+u_{xxx} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>u_{t}+6uu_{x}+u_{xxx} = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u[t]+ 6*u*u[x]+ u[x, x, x] = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[u, t]+ 6*u*Subscript[u, x]+ Subscript[u, x, x, x] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/32.13#Ex3 32.13#Ex3] || <math qid="Q9478">z = x(3t)^{-1/3}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>z = x(3t)^{-1/3}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(x(+)*y*I) = (x(3*t))^(- 1/3)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(x[+]*y*I) == (x[3*t])^(- 1/3)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/32.13#Ex5 32.13#Ex5] || <math qid="Q9480">z = x+3\lambda t^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>z = x+3\lambda t^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(x + y*I) = x + 3*lambda*(t)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(x + y*I) == x + 3*\[Lambda]*(t)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/32.13#Ex6 32.13#Ex6] || <math qid="Q9481">u(x,t) = W(z)-\lambda t</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>u(x,t) = W(z)-\lambda t</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u(x , t) = W*((x + y*I))- lambda*t</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u[x , t] == W*((x + y*I))- \[Lambda]*t</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/32.13.E6 32.13.E6] || <math qid="Q9482">u_{xt} = \sin@@{u}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>u_{xt} = \sin@@{u}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>u[x, t] = sin(u)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[u, x, t] == Sin[u]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .70450695e-2+.1624035369*I
Test Values: {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, u[x*t] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.358980334+.5284289409*I
Test Values: {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, u[x*t] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.007045069484300837, 0.16240353677712993]
Test Values: {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[u, Times[t, x]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.3589803343001376, 0.5284289405615687]
Test Values: {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[u, Times[t, x]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/32.13#Ex7 32.13#Ex7] || <math qid="Q9483">z = xt</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>z = xt</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(x + y*I) = x*t</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(x + y*I) == x*t</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/32.13#Ex8 32.13#Ex8] || <math qid="Q9484">u(x,t) = v(z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>u(x,t) = v(z)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u(x , t) = v*((x + y*I))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u[x , t] == v*((x + y*I))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/32.13.E8 32.13.E8] || <math qid="Q9485">u_{tt} = u_{xx}-6(u^{2})_{xx}+u_{xxxx}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>u_{tt} = u_{xx}-6(u^{2})_{xx}+u_{xxxx}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u[t, t] = u[x, x]- 6*(u)^(2)[x, x]+ u[x, x, x, x]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[u, t, t] == Subscript[u, x, x]- 6*Subscript[(u)^(2), x, x]+ Subscript[u, x, x, x, x]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/32.13#Ex9 32.13#Ex9] || <math qid="Q9486">z = x-ct</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>z = x-ct</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(x + y*I) = x - c*t</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(x + y*I) == x - c*t</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/32.13#Ex10 32.13#Ex10] || <math qid="Q9487">u(x,t) = v(z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>u(x,t) = v(z)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u(x , t) = v*((x + y*I))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u[x , t] == v*((x + y*I))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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