59.1.218 problem 221

Internal problem ID [9390]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 221
Date solved : Monday, January 27, 2025 at 06:02:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 36

dsolve(z*diff(y(z),z$2)+(2*z-3)*diff(y(z),z)+4/z*y(z)=0,y(z), singsol=all)
 

\[ y \left (z \right ) = 2 \left (c_{2} {\mathrm e}^{-2 z} \left (-\frac {1}{2}+z \right ) \operatorname {Ei}_{1}\left (-2 z \right )+c_{1} \left (-\frac {1}{2}+z \right ) {\mathrm e}^{-2 z}+\frac {c_{2}}{2}\right ) z^{2} \]

✓ Solution by Mathematica

Time used: 0.939 (sec). Leaf size: 55

DSolve[z*D[y[z],{z,2}]+(2*z-3)*D[y[z],z]+4/z*y[z]==0,y[z],z,IncludeSingularSolutions -> True]
 

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