18.3.6 problem Problem 16.8

Internal problem ID [3506]
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 16.8
Date solved : Monday, January 27, 2025 at 07:40:01 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 32

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

\[ y \left (z \right ) = c_{1} z^{3} \left (1-\frac {1}{10} z^{2}+\frac {1}{280} z^{4}+\operatorname {O}\left (z^{6}\right )\right )+c_{2} \left (12+6 z^{2}-\frac {3}{2} z^{4}+\operatorname {O}\left (z^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 44

AsymptoticDSolveValue[z*D[y[z],{z,2}]-2*D[y[z],z]+z*y[z]==0,y[z],{z,0,"6"-1}]
 

\[ y(z)\to c_1 \left (-\frac {z^4}{8}+\frac {z^2}{2}+1\right )+c_2 \left (\frac {z^7}{280}-\frac {z^5}{10}+z^3\right ) \]