59.1.624 problem 641

Internal problem ID [9796]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 641
Date solved : Monday, January 27, 2025 at 06:14:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 47

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

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

✓ Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 46

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

\[ y(t)\to e^{-t/2} \left (c_2 \sqrt {t}+\sqrt {2} c_1 \sqrt {-t} \Gamma \left (\frac {1}{2},-\frac {t}{2}\right )\right ) \]