12.3.2 problem Problem 16.2

Internal problem ID [3502]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 16, Series solutions of ODEs. Section 16.6 Exercises, page 550
Problem number : Problem 16.2
Date solved : Tuesday, September 30, 2025 at 06:40:53 AM
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*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 44

Order:=6; 
ode:=4*z*diff(diff(y(z),z),z)+2*(1-z)*diff(y(z),z)-y(z) = 0; 
dsolve(ode,y(z),type='series',z=0);
 

\[ y = c_1 \sqrt {z}\, \left (1+\frac {1}{3} z +\frac {1}{15} z^{2}+\frac {1}{105} z^{3}+\frac {1}{945} z^{4}+\frac {1}{10395} z^{5}+\operatorname {O}\left (z^{6}\right )\right )+c_2 \left (1+\frac {1}{2} z +\frac {1}{8} z^{2}+\frac {1}{48} z^{3}+\frac {1}{384} z^{4}+\frac {1}{3840} z^{5}+\operatorname {O}\left (z^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 85

ode=4*z*D[y[z],{z,2}]+2*(1-z)*D[y[z],z]-y[z]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[z],{z,0,5}]
 

\[ y(z)\to c_1 \sqrt {z} \left (\frac {z^5}{10395}+\frac {z^4}{945}+\frac {z^3}{105}+\frac {z^2}{15}+\frac {z}{3}+1\right )+c_2 \left (\frac {z^5}{3840}+\frac {z^4}{384}+\frac {z^3}{48}+\frac {z^2}{8}+\frac {z}{2}+1\right ) \]

✓ Sympy. Time used: 0.347 (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,hint="2nd_power_series_regular",x0=0,n=6)
 

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