59.1.217 problem 220

Internal problem ID [9389]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 220
Date solved : Wednesday, March 05, 2025 at 07:48:29 AM
CAS classification : [_Lienard]

\begin{align*} z y^{\prime \prime }-2 y^{\prime }+z y&=0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 23

ode:=z*diff(diff(y(z),z),z)-2*diff(y(z),z)+z*y(z) = 0; 
dsolve(ode,y(z), singsol=all);
 

\[ y \left (z \right ) = \left (c_{1} z +c_{2} \right ) \cos \left (z \right )+\sin \left (z \right ) \left (c_{2} z -c_{1} \right ) \]

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 39

ode=z*D[y[z],{z,2}]-2*D[y[z],z]+z*y[z]==0; 
ic={}; 
DSolve[{ode,ic},y[z],z,IncludeSingularSolutions->True]
 

\[ y(z)\to -\sqrt {\frac {2}{\pi }} ((c_1 z+c_2) \cos (z)+(c_2 z-c_1) \sin (z)) \]

✓ Sympy. Time used: 0.181 (sec). Leaf size: 20

from sympy import * 
z = symbols("z") 
y = Function("y") 
ode = Eq(z*y(z) + z*Derivative(y(z), (z, 2)) - 2*Derivative(y(z), z),0) 
ics = {} 
dsolve(ode,func=y(z),ics=ics)
 

\[ y{\left (z \right )} = z^{\frac {3}{2}} \left (C_{1} J_{\frac {3}{2}}\left (z\right ) + C_{2} Y_{\frac {3}{2}}\left (z\right )\right ) \]