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);

$\left[\mathrm{Differential}\,\mathrm{DisplayStructure}\,\mathrm{Distribution}\,\mathrm{EliminationLAVF}\,\mathrm{EliminationSystem}\,\mathrm{IDBasis}\,\mathrm{IsLinearizable}\,\mathrm{LAVF}\,\mathrm{LHLibrary}\,\mathrm{LHPDE}\,\mathrm{LHPDO}\,\mathrm{MapDE}\,\mathrm{OneForm}\,\mathrm{SymmetryLAVF}\,\mathrm{VFPDO}\,\mathrm{VectorField}\right]$

Typesetting:-Settings(userep=true);

$\mathrm{false}$

Typesetting:-Suppress([xi(x,y),eta(x,y)]);

V := VectorField(xi(x,u)*D[x] + eta(x,u)*D[u], space = [x,u]);

$V≔\mathrm{\xi }\left(x\,u\right)\frac{\partial }{\partial x}+\mathrm{\eta }\left(x\,u\right)\frac{\partial }{\partial u}$

ODE[1] := diff(u(x),x,x,x) + u(x)*diff(u(x),x,x)^2 + 2*u(x) = 0;

${\mathrm{ODE}}_1≔\frac{{\ⅆ}^3}{\ⅆx^3}u\left(x\right)+u\left(x\right){\left(\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right)}^2+2u\left(x\right)=0$

L := IsLinearizable(ODE[1], V);

$L≔\mathrm{false}$

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;

${\mathrm{ODE}}_2≔2x^{2}u\left(x\right)\left(\frac{{\ⅆ}^4}{\ⅆx^4}u\left(x\right)\right)+x^2{u\left(x\right)}^2+8x^2\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}u\left(x\right)\right)+16xu\left(x\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}u\left(x\right)\right)+6x^2{\left(\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right)}^2+48x\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right)+24u\left(x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right)+24{\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)}^2=0$

IsLinearizable(ODE[2], V);

$\mathrm{true}$

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;

${\mathrm{ODE}}_3≔\frac{{\ⅆ}^3}{\ⅆx^3}u\left(x\right)+\frac{3\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)-\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)}{u\left(x\right)}-3\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)+2\frac{\ⅆ}{\ⅆx}u\left(x\right)-u\left(x\right)=0$

IsLinearizable(ODE[3], V);

$\mathrm{true}$

FalknerEq := diff(u(x), x, x, x) + u(x)*diff(u(x), x, x) + beta*(1 - diff(u(x), x, x)^2) = 0;

$\mathrm{FalknerEq}≔\frac{{\ⅆ}^3}{\ⅆx^3}u\left(x\right)+u\left(x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right)+\mathrm{\beta }\left(1-{\left(\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right)}^2\right)=0$

IsLinearizable(FalknerEq, V);

$\mathrm{false}$