Solutions of a linear differential equation of the stretched type via Laguerre functions (Q910539)

For a differential equation of the stretched type in the form \((1)\quad dy/dt=ay(\lambda t)+by(t)\) or the equivalent form \[ (2)\quad y(t)- y(0)=a\int^{t}_{0}y(\lambda t)dt+b\int^{t}_{0}y(t)dt, \] the authors introduce a new approach to the orthogonal series solution. Denoting by \(\Phi_ i(t;a,b)\) the Laguerre function of rank i, \(\Phi (t;a,b)=col(\Phi_ 1,...,\Phi_ n)\) and considering the differential equation \((d/dt)\Phi =D\Phi\) they obtain the form of the matrix D. Then the form of the stretched Laguerre vector is obtained from the Laguerre vector itself. These results are applied to an equation of the form (1). Two examples are analysed using these results.