18.3.3 problem Problem 16.3

Internal problem ID [3503]
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 16.3
Date solved : Sunday, March 30, 2025 at 01:45:12 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} z y^{\prime \prime }-2 y^{\prime }+9 z^{5} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 28

Order:=7; 
ode:=z*diff(diff(y(z),z),z)-2*diff(y(z),z)+9*z^5*y(z) = 0; 
dsolve(ode,y(z),type='series',z=0);
 

\[ y = c_1 \,z^{3} \left (1-\frac {1}{6} z^{6}+\operatorname {O}\left (z^{7}\right )\right )+c_2 \left (12-6 z^{6}+\operatorname {O}\left (z^{7}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 12

ode=z*D[y[z],{z,2}]-2*D[y[z],z]+9*z^5*y[z]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[z],{z,0,6}]
 

\[ y(z)\to c_2 z^3+c_1 \]

✓ Sympy. Time used: 0.804 (sec). Leaf size: 19

from sympy import * 
z = symbols("z") 
y = Function("y") 
ode = Eq(9*z**5*y(z) + z*Derivative(y(z), (z, 2)) - 2*Derivative(y(z), z),0) 
ics = {} 
dsolve(ode,func=y(z),ics=ics,hint="2nd_power_series_regular",x0=0,n=7)
 

\[ y{\left (z \right )} = C_{2} \left (1 - \frac {z^{6}}{2}\right ) + C_{1} z^{3} + O\left (z^{7}\right ) \]