59.1.12 problem 12

Internal problem ID [9184]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 12
Date solved : Monday, January 27, 2025 at 05:51:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 82

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

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

✓ Solution by Mathematica

Time used: 0.557 (sec). Leaf size: 54

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

\[ y(t)\to e^{t-\frac {t^2}{2}} (t-1) \left (c_2 \int _1^t\frac {e^{\frac {1}{2} (K[1]-4) K[1]} K[1]}{(K[1]-1)^2}dK[1]+c_1\right ) \]