59.1.206 problem 209

Internal problem ID [9378]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 209
Date solved : Monday, January 27, 2025 at 06:02:05 PM
CAS classification : [_Lienard]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 33

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

\[ y = \left (c_{2} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, t}{2}\right )+c_{1} \right ) {\mathrm e}^{\frac {t^{2}}{2}}-2 c_{2} t \]

✓ Solution by Mathematica

Time used: 0.150 (sec). Leaf size: 56

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

\[ y(t)\to \frac {1}{2} e \left (\sqrt {2 \pi } c_2 e^{\frac {t^2}{2}} \text {erf}\left (\frac {t}{\sqrt {2}}\right )+2 c_1 e^{\frac {t^2}{2}}-2 c_2 t\right ) \]