59.1.409 problem 421

Internal problem ID [9581]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 421
Date solved : Monday, January 27, 2025 at 06:04:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{3} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 34

dsolve(t*diff(y(t),t$2)+ (t^2-1)*diff(y(t),t)+t^3*y(t) = 0,y(t), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {t^{2}}{4}} \left (c_{1} \cos \left (\frac {\sqrt {3}\, t^{2}}{4}\right )+c_{2} \sin \left (\frac {\sqrt {3}\, t^{2}}{4}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 48

DSolve[t*D[y[t],{t,2}]+(t^2-1)*D[y[t],t]+t^3*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-\frac {t^2}{4}} \left (c_2 \cos \left (\frac {\sqrt {3} t^2}{4}\right )+c_1 \sin \left (\frac {\sqrt {3} t^2}{4}\right )\right ) \]