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 : Monday, January 27, 2025 at 07:40:00 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*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 47

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

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[z^2*D[y[z],{z,2}]-3/2*z*D[y[z],z]+(1+z)*y[z]==0,y[z],{z,0,"6"-1}]
 

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