The expsols routine returns a basis of the exponential solutions of a linear differential equation having rational function coefficients. If $K\left(x\right)$ denotes the rational field of coefficients, then exponential solutions are those of the form

with $R\left(x\right)$ a rational function from $K\left(x\right)$.

There are two input forms. The first has as the first argument a linear differential equation in diff or D form and as the second argument the variable in the differential equation.

A second input sequence accepts for the first argument the list of coefficients of a linear ode, for the second the right hand side of such an equation, and for the third argument the independent variable of the lode. This input sequence is convenient for programming with the expsols routine.

In the second calling sequence, the list of coefficients is given in order from low differential order to high differential order and does not include the nonhomogeneous term.

In the case of a homogeneous equation a basis is returned (as a list). In the nonhomogeneous case, the returned value is a two-element list, with the first element a basis for the homogeneous case and the second element a particular rational solution (if it exists).

This function is part of the DEtools package, and so it can be used in the form expsols(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[expsols](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{ode}≔\left(t^2+t\right)\mathrm{diff}\left(z\left(t\right)\,\mathrm{`\$`}\left(t\,3\right)\right)-\left(t^2-2\right)\mathrm{diff}\left(z\left(t\right)\,\mathrm{`\$`}\left(t\,2\right)\right)-\left(t+2\right)\mathrm{diff}\left(z\left(t\right)\,t\right)=0\:$

$\mathrm{expsols}\left(\mathrm{ode}\,z\left(t\right)\right)$

$\left[1\,{\ⅇ}^t\right]$

$\mathrm{ode}≔\left(t^2+t\right)\mathrm{diff}\left(z\left(t\right)\,\mathrm{`\$`}\left(t\,3\right)\right)-\left(t^2-2\right)\mathrm{diff}\left(z\left(t\right)\,\mathrm{`\$`}\left(t\,2\right)\right)-\left(t+2\right)\mathrm{diff}\left(z\left(t\right)\,t\right)=3t{\left(t+1\right)}^2\:$

$\mathrm{expsols}\left(\mathrm{ode}\,z\left(t\right)\right)$

$\left[\left[1\,{\ⅇ}^t\right]\,-\frac{1}{3}{t}^3-\frac{3}{2}{t}^2-3t\right]$

$\mathrm{ode}≔\left(t^2+t\right){\mathrm{D}}^{\left(3\right)}\left(z\right)\left(t\right)-\left(t^2-2\right){\mathrm{D}}^{\left(2\right)}\left(z\right)\left(t\right)-\left(t+2\right)\mathrm{D}\left(z\right)\left(t\right)=3t{\left(t+1\right)}^2\:$

$\mathrm{expsols}\left(\mathrm{ode}\,z\left(t\right)\right)$

$\left[\left[1\,{\ⅇ}^t\right]\,-\frac{1}{3}{t}^3-\frac{3}{2}{t}^2-3t\right]$

$\mathrm{ode\_list}≔\left[0\,-\left(t+2\right)\,-\left(t^2-2\right)\,t^2+t\right]\:$

$\mathrm{expsols}\left(\mathrm{ode\_list}\,3t{\left(t+1\right)}^2\,t\right)$

$\left[\left[1\,{\ⅇ}^t\right]\,-\frac{1}{3}{t}^3-\frac{3}{2}{t}^2-3t\right]$