59.1.635 problem 652

Internal problem ID [9807]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 652
Date solved : Sunday, March 30, 2025 at 02:47:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y&=0 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 22

ode:=4*z*diff(diff(y(z),z),z)+2*(1-z)*diff(y(z),z)-y(z) = 0; 
dsolve(ode,y(z), singsol=all);
 

\[ y = {\mathrm e}^{\frac {z}{2}} \left (c_1 \,\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {z}}{2}\right )+c_2 \right ) \]

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 44

ode=4*z*D[y[z],{z,2}]+2*(1-z)*D[y[z],z]-y[z]==0; 
ic={}; 
DSolve[{ode,ic},y[z],z,IncludeSingularSolutions->True]
 

\[ y(z)\to e^{\frac {z}{2}-\frac {1}{4}} \left (\sqrt {e} c_1-\sqrt {2} c_2 \Gamma \left (\frac {1}{2},\frac {z}{2}\right )\right ) \]

✓ Sympy. Time used: 0.899 (sec). Leaf size: 60

from sympy import * 
z = symbols("z") 
y = Function("y") 
ode = Eq(4*z*Derivative(y(z), (z, 2)) + (2 - 2*z)*Derivative(y(z), z) - y(z),0) 
ics = {} 
dsolve(ode,func=y(z),ics=ics)
 

\[ y{\left (z \right )} = C_{2} \left (\frac {z^{5}}{3840} + \frac {z^{4}}{384} + \frac {z^{3}}{48} + \frac {z^{2}}{8} + \frac {z}{2} + 1\right ) + C_{1} \sqrt {z} \left (\frac {z^{4}}{945} + \frac {z^{3}}{105} + \frac {z^{2}}{15} + \frac {z}{3} + 1\right ) + O\left (z^{6}\right ) \]