59.1.203 problem 206

Internal problem ID [9375]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 206
Date solved : Monday, January 27, 2025 at 06:02:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 29

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

\[ y = \sqrt {t}\, \left (c_{1} \sin \left (\sqrt {2}\, \sqrt {t}\right )+c_{2} \cos \left (\sqrt {2}\, \sqrt {t}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.079 (sec). Leaf size: 62

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

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