59.1.202 problem 205

Internal problem ID [9374]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 205
Date solved : Monday, January 27, 2025 at 06:02:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 56

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

\[ y = c_{1} \sqrt {\pi }\, \left (t^{2}+6 t +3\right ) \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {t}}{2}\right )+5 c_{1} \sqrt {2}\, \left (\sqrt {t}+\frac {t^{{3}/{2}}}{5}\right ) {\mathrm e}^{-\frac {t}{2}}+c_{2} \left (t^{2}+6 t +3\right ) \]

✓ Solution by Mathematica

Time used: 11.026 (sec). Leaf size: 58

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

\[ y(t)\to \left (t^2+6 t+3\right ) \left (c_2 \int _1^t\frac {e^{-\frac {K[1]}{2}-\frac {1}{2}}}{\sqrt {K[1]} \left (K[1]^2+6 K[1]+3\right )^2}dK[1]+c_1\right ) \]