18.3.5 problem Problem 16.6

Internal problem ID [3505]
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 16.6
Date solved : Sunday, March 30, 2025 at 01:45:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} z^{2} y^{\prime \prime }-\frac {3 z y^{\prime }}{2}+\left (1+z \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 47

Order:=6; 
ode:=z^2*diff(diff(y(z),z),z)-3/2*z*diff(y(z),z)+(1+z)*y(z) = 0; 
dsolve(ode,y(z),type='series',z=0);
 

\[ y = c_1 \sqrt {z}\, \left (1+2 z -2 z^{2}+\frac {4}{9} z^{3}-\frac {2}{45} z^{4}+\frac {4}{1575} z^{5}+\operatorname {O}\left (z^{6}\right )\right )+c_2 \,z^{2} \left (1-\frac {2}{5} z +\frac {2}{35} z^{2}-\frac {4}{945} z^{3}+\frac {2}{10395} z^{4}-\frac {4}{675675} z^{5}+\operatorname {O}\left (z^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 84

ode=z^2*D[y[z],{z,2}]-3/2*z*D[y[z],z]+(1+z)*y[z]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[z],{z,0,5}]
 

\[ y(z)\to c_1 \left (-\frac {4 z^5}{675675}+\frac {2 z^4}{10395}-\frac {4 z^3}{945}+\frac {2 z^2}{35}-\frac {2 z}{5}+1\right ) z^2+c_2 \left (\frac {4 z^5}{1575}-\frac {2 z^4}{45}+\frac {4 z^3}{9}-2 z^2+2 z+1\right ) \sqrt {z} \]

✓ Sympy. Time used: 0.964 (sec). Leaf size: 61

from sympy import * 
z = symbols("z") 
y = Function("y") 
ode = Eq(z**2*Derivative(y(z), (z, 2)) - 3*z*Derivative(y(z), z)/2 + (z + 1)*y(z),0) 
ics = {} 
dsolve(ode,func=y(z),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (z \right )} = C_{2} z^{2} \left (- \frac {4 z^{3}}{945} + \frac {2 z^{2}}{35} - \frac {2 z}{5} + 1\right ) + C_{1} \sqrt {z} \left (- \frac {2 z^{4}}{45} + \frac {4 z^{3}}{9} - 2 z^{2} + 2 z + 1\right ) + O\left (z^{6}\right ) \]