IsLinearizable - Maple Help

LieAlgebrasOfVectorFields

 IsLinearizable
 Checking if an ODE system can be transformed into a linear one

 Calling Sequence IsLinearizable(DEs, V)

Parameters

 DEs - an equation or a list of differential equations V - a VectorField object

Description

 • The command IsLinearizable(...) checks if an ordinary differential equations (ODEs) system can be transformed to a linear ODE by a point transformation. In other words, let S be a single ODE system with a single dependent variable $u$ and independent variable $x$. Then the method returns true if there exists an invertible transformation $x=\mathrm{\psi }\left(z,w\right),u=\mathrm{\phi }\left(z,w\right)$ to a single linear ODE, for some smooth function $\mathrm{\psi }$ and $\mathrm{\phi }$, and return false otherwise.
 • The second input argument is a VectorField object where the first argument ODEs is associated with. For more detail about how to construct a VectorField object, see LieAlgebrasOfVectorFields[VectorField]
 • This command is part of the LieAlgebrasOfVectorFields package. For more detail, see Overview of the LieAlgebrasOfVectorFields package.
 • This command can be used in the form IsLinearizable(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form :-LieAlgebrasOfVectorFields:-IsLinearizable(...).
 > with(LieAlgebrasOfVectorFields);
 $\left[{\mathrm{Differential}}{,}{\mathrm{DisplayStructure}}{,}{\mathrm{Distribution}}{,}{\mathrm{EliminationLAVF}}{,}{\mathrm{EliminationSystem}}{,}{\mathrm{IDBasis}}{,}{\mathrm{IsLinearizable}}{,}{\mathrm{LAVF}}{,}{\mathrm{LHLibrary}}{,}{\mathrm{LHPDE}}{,}{\mathrm{LHPDO}}{,}{\mathrm{MapDE}}{,}{\mathrm{OneForm}}{,}{\mathrm{SymmetryLAVF}}{,}{\mathrm{VFPDO}}{,}{\mathrm{VectorField}}\right]$ (1)
 > Typesetting:-Settings(userep=true);
 ${\mathrm{false}}$ (2)
 > Typesetting:-Suppress([xi(x,y),eta(x,y)]);
 > V := VectorField(xi(x,u)*D[x] + eta(x,u)*D[u], space = [x,u]);
 ${V}{≔}{\mathrm{\xi }}{}\left({x}{,}{u}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\left({x}{,}{u}\right){}\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)
 > ODE[1] := diff(u(x),x,x,x) + u(x)*diff(u(x),x,x)^2 + 2*u(x) = 0;
 ${{\mathrm{ODE}}}_{{1}}{≔}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right){+}{u}{}\left({x}\right){}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right)}^{{2}}{+}{2}{}{u}{}\left({x}\right){=}{0}$ (4)
 > L := IsLinearizable(ODE[1], V);
 ${L}{≔}{\mathrm{false}}$ (5)
 > ODE[2] := 2*x^2*u(x)*diff(u(x),x,x,x,x) + x^2*u(x)^2 + 8*x^2*diff(u(x),x)*diff(u(x),x,x,x) + 16*x*u(x)*diff(u(x),x,x,x) + 6*x^2*diff(u(x),x,x)^2 + 48*x*diff(u(x),x)*diff(u(x),x,x) + 24*u(x)*diff(u(x),x,x) + 24*diff(u(x),x)^2 = 0;
 ${{\mathrm{ODE}}}_{{2}}{≔}{2}{}{{x}}^{{2}}{}{u}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{{x}}^{{2}}{}{{u}{}\left({x}\right)}^{{2}}{+}{8}{}{{x}}^{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{16}{}{x}{}{u}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{6}{}{{x}}^{{2}}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right)}^{{2}}{+}{48}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{24}{}{u}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{24}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right)}^{{2}}{=}{0}$ (6)
 > IsLinearizable(ODE[2], V);
 ${\mathrm{true}}$ (7)
 > ODE[3] := diff(u(x), x, x, x) + 3*diff(u(x), x)*(diff(u(x), x, x) - diff(u(x), x))/u(x) - 3*diff(u(x), x, x) + 2*diff(u(x), x) - u(x) = 0;
 ${{\mathrm{ODE}}}_{{3}}{≔}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right){+}\frac{{3}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right){-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right)}{{u}{}\left({x}\right)}{-}{3}{}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right){+}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right){-}{u}{}\left({x}\right){=}{0}$ (8)
 > IsLinearizable(ODE[3], V);
 ${\mathrm{true}}$ (9)
 > FalknerEq := diff(u(x), x, x, x) + u(x)*diff(u(x), x, x) + beta*(1 - diff(u(x), x, x)^2) = 0;
 ${\mathrm{FalknerEq}}{≔}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right){+}{u}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{\mathrm{\beta }}{}\left({1}{-}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right)}^{{2}}\right){=}{0}$ (10)
 > IsLinearizable(FalknerEq, V);
 ${\mathrm{false}}$ (11)