18.3.7 problem Problem 16.9

Internal problem ID [3507]
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 16.9
Date solved : Monday, January 27, 2025 at 07:40:02 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 37

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

\[ y \left (z \right ) = \left (1+z^{2}+\frac {1}{2} z^{4}\right ) y \left (0\right )+\left (z +\frac {2}{3} z^{3}+\frac {4}{15} z^{5}\right ) D\left (y \right )\left (0\right )+O\left (z^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 38

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

\[ y(z)\to c_2 \left (\frac {4 z^5}{15}+\frac {2 z^3}{3}+z\right )+c_1 \left (\frac {z^4}{2}+z^2+1\right ) \]